text: 8Draft version May 28, 2020 Li et al. Preprint typeset using LATEX style emulateapj v. 01/23/15 ASSESSMENT OF SYSTEMATIC CHROMATIC ERRORS THAT IMPACT SUB–1% PHOTOMETRIC PRECISION IN LARGE-AREA SKY SURVEYS T. S. Li1, D. L. DePoy1, J. L. Marshall1, D. Tucker2, R. Kessler3, J. Annis2, G. M. Bernstein4, S. Boada1, D. L. Burke5,6, D. A. Finley2, D. J. James7, S. Kent2, H. Lin2, J. Marriner2, N. Mondrik1, D. Nagasawa1, E. S. Rykoff6,5, D. Scolnic3, A. R. Walker7, W. Wester2, T. M. C. Abbott7, S. Allam2, A. Benoit-Le´vy8,9,10, E. Bertin8,10, D. Brooks9, D. Capozzi11, A. Carnero Rosell12,13, M. Carrasco Kind14,15, J. Carretero16,17, M. Crocce16, C. E. Cunha6, C. B. D’Andrea11,18, L. N. da Costa12,13, S. Desai19,20, H. T. Diehl2, P. Doel9, B. Flaugher2, P. Fosalba16, J. Frieman2,3, E. Gaztanaga16, D. A. Goldstein21,22, D. Gruen6,23,5,24, R. A. Gruendl14,15, G. Gutierrez2, K. Honscheid25,26, K. Kuehn27, N. Kuropatkin2, M. A. G. Maia12,13, P. Melchior28, C. J. Miller29,30, R. Miquel31,17, J. J. Mohr19,20,23, E. Neilsen2, R. C. Nichol11, B. Nord2, R. Ogando12,13, A. A. Plazas32, A. K. Romer33, A. Roodman6,5, M. Sako4, E. Sanchez34, V. Scarpine2, M. Schubnell30, I. Sevilla-Noarbe34,14, R. C. Smith7, M. Soares-Santos2, F. Sobreira2,12, E. Suchyta4, G. Tarle30, D. Thomas11, V. Vikram35 (The DES Collaboration) Keywords: 1 George P. and Cynthia Woods Mitchell Institute for Funda- mental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA 2 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 3 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 4 Department of Physics and Astronomy, University of Penn- sylvania, Philadelphia, PA 19104, USA 5 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 6 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA 7 Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile 8 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F- 75014, Paris, France 9 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 10 Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 11 Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 12 Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 13 Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de Maximilians Universit¨at Mu¨nchen, Scheinerstr. 1, 81679 Janeiro, RJ - 20921-400, Brazil Mu¨nchen, Germany 14 Department of Astronomy, University of Illinois, 1002 W. 25 Center for Cosmology and Astro-Particle Physics, The Ohio Green Street, Urbana, IL 61801, USA State University, Columbus, OH 43210, USA 15 National Center for Supercomputing Applications, 1205 26 Department of Physics, The Ohio State University, Colum- West Clark St., Urbana, IL 61801, USA bus, OH 43210, USA 16 Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, 27 Australian Astronomical Observatory, North Ryde, NSW Carrer de Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain 2113, Australia 17 Institut de F´ısica d’Altes Energies (IFAE), The Barcelona 28 Department of Astrophysical Sciences, Princeton Univer- Institute of Science and Technology, Campus UAB, 08193 Bel- sity, Peyton Hall, Princeton, NJ 08544, USA laterra (Barcelona) Spain 29 Department of Astronomy, University of Michigan, Ann Ar- 18 School of Physics and Astronomy, University of Southamp- bor, MI 48109, USA ton, Southampton, SO17 1BJ, UK 30 Department of Physics, University of Michigan, Ann Arbor, 19 Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garch- MI 48109, USA ing, Germany 31 Instituci´o Catalana de Recerca i Estudis Avan¸cats, E-08010 20 Faculty of Physics, Ludwig-Maximilians University, Schein- Barcelona, Spain erstr. 1, 81679 Munich, Germany 32 Jet Propulsion Laboratory, California Institute of Technol- 21 Department of Astronomy, University of California, Berke- ogy, 4800 Oak Grove Dr., Pasadena, CA 91109, USA ley, 501 Campbell Hall, Berkeley, CA 94720, USA 33 Department of Physics and Astronomy, Pevensey Building, 22 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, University of Sussex, Brighton, BN1 9QH, UK Berkeley, CA 94720, USA 34 Centro de Investigaciones Energ´eticas, Medioambientales y 23 Max Planck Institute for Extraterrestrial Physics, Giessen- Tecnol´ogicas (CIEMAT), Madrid, Spain bachstrasse, 85748 Garching, Germany 35 Argonne National Laboratory, 9700 South Cass Avenue, 24 Universita¨ts-Sternwarte, Fakult¨at fu¨r Physik, Ludwig- Lemont, IL 60439, USA

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text: 10 Li et al. 1.6 1.2 0.8 0.4 0.0 400 500 600 700 800 900 1000 1100 wavelength[nm] 10 10 0 0 −10 −10 10 10 0 0 −10 −10 10 10 0 0 −10 −10 −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 g−i g−i Figure 2. Top panel: The ratio of the atmospheric transmission for two airmass values, X = 1.2 and X = 1.8. The ﬁducial instrumental throughput φinst(λ) for ugrizY -bands is also shown in the plot as reference. Bottom panels: Synthetic SCE in ugrizY -bands for O5V-M6V ref − stars introduced by this airmass change, as a function of g i color. − ∼Table 1 − ∼ Synthetic SCE on a M6V star (g i 4) with respect to a G2V star (g i 0.6). Synthetic SCE (mmag) Component Fiducial Changeda u g r i z Y − − − − − Pressure P = 779 hpa P = 789 hpa 0.2 0.4 0.08 0.06 0.04 0.02 0 0 − − − − − − AOD = 0.02, α = 1 AOD = 0.20, α = 1 3 11 6 4 1 0.2 Aerosol 550 550 − − − − − − AOD = 0.02, α = 1 AOD = 0.20, α = 0.5 41 5 3 2 0.8 0.1 550 550 − − PWV PWV = 3 mm PWV = 10 mm 0 0 3 0.6 +10 3 − Ozone Ozone = 270 DU Ozone = 230 DU 0 0.7 0.9 0.1 0 0 − − − − − Airmass X = 1.2 X = 1.8 9 13 6 2 +2 0.8 − − − − − − Instrument DECal scan shift 2 nm 24 15 16 19 10 5 aThe “changed” conditions here are just examples. For pressure and ozone, we used the extreme examples since the SCE are small; for aerosol, PWV and airmass, we give the examples where the change could introduce about 1% or 10 mmag SCE. For the instrument, we choose 2nm shift as it’s about the average value from the DECal scans. In this section, we study the synthetic SCE, ∆m, due throughput over its survey period (Doi et al. 2010). Such to the variation in the instrumental throughput. For variations could introduce SCE similar to those caused reference, SDSS discovered variations of the instrumental by the variation in the atmospheric transmission. DES so mta φ/ mta φ )gamm(u∆ )gamm(r∆ )gamm(z∆ 8.1= X 2.1= X )gamm(g∆ )gamm(i∆ )gamm(Y∆ ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Li

text: et

text: al.

font: default

text: wavelength[nm]

text: −1

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text: g−i

text: g−i

text: Figure

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text: ratio

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text: the

text: atmospheric

text: transmission

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text: two

text: airmass

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text: X

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text: mm

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text: survey

text: period

text: (Doi

text: et

text: al.

text: Such

text: to

text: the

text: variation

text: in

text: the

text: instrumental

text: throughput.

text: For

text: variations

text: could

text: introduce

text: SCE

text: similar

text: to

text: those

text: caused

text: reference,

text: SDSS

text: discovered

text: variations

text: of

text: the

text: instrumental

text: by

text: the

text: variation

text: in

text: the

text: atmospheric

text: transmission.

text: DES

text: so

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text: φ/

text: mta

text: φ

text: )gamm(u∆

text: )gamm(r∆

text: )gamm(z∆

text: X

text: X

text: )gamm(g∆

text: )gamm(i∆

text: )gamm(Y∆

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text: cite the Ising model on the square lattice with competing NN and NNN interactions, where diﬀerent ordered phases, transitions and universality classes are observed (for a recent survey see [14]). In polymers, competing (on-site) interactions in the multiple monomer per site (MMS) model by Krawczyk et al. [15] is known to change the coil-globule transition in a certain region of its phase diagram, but the order of transition still remain unclear [16]. Another interesting feature of the ISAW model when the force between NNN monomers is repulsive is its semiﬂexibility, because (cid:15) < 0 acts as a bending energy, though it also 2 repels NNN monomers that are not part of a bend. Anyway, for large enough (cid:15) > 0 and 1 (cid:15) < 0, a crystalline phase could be expected in this model. However, at least in the range 2 of energies analyzed in Ref. [13], this crystalline phase was not observed. In order to analyze in more detail whether competing forces between monomers can change or not the coil-globule transition, as well as whether it yields or not an ordered (crystalline) phase in the ISAW with NNN interactions, here, we solve this model on a Husimi lattice built with squares. Diﬀerent deﬁnitions of next nearest-neighbors (and interactions between them) on this lattice are analyzed, but in all cases the same qualitative results are obtained: no crystalline phase is found and the θ-line extends over the whole phase −∞ ∞ diagram, for (cid:15) in the range [ , ), which includes the regions of competing interactions. 1 Only in the extremal case of an inﬁnite repulsive force between NNN monomers there is a breakdown of the coil-globule transition, which is quite expected since in this case the chains are straight. The rest of this work is organized as follows. In Sec. the model is deﬁned on a Husimi lattice built with squares and solved in terms of recursion relations. The thermodynamic properties of the model are presented in Sec. . In Sec. our ﬁnal discussions and conclusions are summarized. The calculations of the free energy and of the θ-lines are demonstrated in appendices and , respectively. II. DEFINITION OF THE MODEL AND ITS SOLUTION IN TERMS OF RE- CURSION RELATIONS We investigate interacting self- and mutually avoiding walks on a Husimi lattice - the core of a Cayley tree [17] - built with squares (see Fig.1). The endpoints of the walks are placed on the surface of the tree. In our grand-canonical solution, the thermodynamic variables of interest are the monomer fugacity z and the Boltzmann weights ω = exp((cid:15) /k T ) and 1 1 B 3

text: cite

text: the

text: Ising

text: model

text: on

text: the

text: square

text: lattice

text: with

text: competing

text: NN

text: and

text: NNN

text: interactions,

text: where

text: diﬀerent

text: ordered

text: phases,

text: transitions

text: and

text: universality

text: classes

text: are

text: observed

text: (for

text: a

text: recent

text: survey

text: see

text: In

text: polymers,

text: competing

text: (on-site)

text: interactions

text: in

text: the

text: multiple

text: monomer

text: per

text: site

text: (MMS)

text: model

text: by

text: Krawczyk

text: et

text: al.

text: is

text: known

text: to

text: change

text: the

text: coil-globule

text: transition

text: in

text: a

text: certain

text: region

text: of

text: its

text: phase

text: diagram,

text: but

text: the

text: order

text: of

text: transition

text: still

text: remain

text: unclear

text: Another

text: interesting

text: feature

text: of

text: the

text: ISAW

text: model

text: when

text: the

text: force

text: between

text: NNN

text: monomers

text: is

text: repulsive

text: is

text: its

text: semiﬂexibility,

text: because

text: acts

text: as

text: a

text: bending

text: energy,

text: though

text: it

text: also

text: repels

text: NNN

text: monomers

text: that

text: are

text: not

text: part

text: of

text: a

text: bend.

text: Anyway,

text: for

text: large

text: enough

text: and

text: a

text: crystalline

text: phase

text: could

text: be

text: expected

text: in

text: this

text: model.

text: However,

text: at

text: least

text: in

text: the

text: range

text: of

text: energies

text: analyzed

text: in

text: Ref.

text: this

text: crystalline

text: phase

text: was

text: not

text: observed.

text: In

text: order

text: to

text: analyze

text: in

text: more

text: detail

text: whether

text: competing

text: forces

text: between

text: monomers

text: can

text: change

text: or

text: not

text: the

text: coil-globule

text: transition,

text: as

text: well

text: as

text: whether

text: it

text: yields

text: or

text: not

text: an

text: ordered

text: (crystalline)

text: phase

text: in

text: the

text: ISAW

text: with

text: NNN

text: interactions,

text: here,

text: we

text: solve

text: this

text: model

text: on

text: a

text: Husimi

text: lattice

text: built

text: with

text: squares.

text: Diﬀerent

text: deﬁnitions

text: of

text: next

text: nearest-neighbors

text: (and

text: interactions

text: between

text: them)

text: on

text: this

text: lattice

text: are

text: analyzed,

text: but

text: in

text: all

text: cases

text: the

text: same

text: qualitative

text: results

text: are

text: obtained:

text: no

text: crystalline

text: phase

text: is

text: found

text: and

text: the

text: θ-line

text: extends

text: over

text: the

text: whole

text: phase

text: diagram,

text: for

text: in

text: the

text: range

text: which

text: includes

text: the

text: regions

text: of

text: competing

text: interactions.

text: Only

text: in

text: the

text: extremal

text: case

text: of

text: an

text: inﬁnite

text: repulsive

text: force

text: between

text: NNN

text: monomers

text: there

text: is

text: a

text: breakdown

text: of

text: the

text: coil-globule

text: transition,

text: which

text: is

text: quite

text: expected

text: since

text: in

text: this

text: case

text: the

text: chains

text: are

text: straight.

text: The

text: rest

text: of

text: this

text: work

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: the

text: model

text: is

text: deﬁned

text: on

text: a

text: Husimi

text: lattice

text: built

text: with

text: squares

text: and

text: solved

text: in

text: terms

text: of

text: recursion

text: relations.

text: The

text: thermodynamic

text: properties

text: of

text: the

text: model

text: are

text: presented

text: in

text: Sec.

text: In

text: Sec.

text: our

text: ﬁnal

text: discussions

text: and

text: conclusions

text: are

text: summarized.

text: The

text: calculations

text: of

text: the

text: free

text: energy

text: and

text: of

text: the

text: θ-lines

text: are

text: demonstrated

text: in

text: appendices

text: and

text: respectively.

text: II.

text: DEFINITION

text: OF

text: THE

text: MODEL

text: AND

text: ITS

text: SOLUTION

text: IN

text: TERMS

text: OF

text: RE-

text: CURSION

text: RELATIONS

text: We

text: investigate

text: interacting

text: self-

text: and

text: mutually

text: avoiding

text: walks

text: on

text: a

text: Husimi

text: lattice

text: the

text: core

text: of

text: a

text: Cayley

text: tree

text: built

text: with

text: squares

text: (see

text: The

text: endpoints

text: of

text: the

text: walks

text: are

text: placed

text: on

text: the

text: surface

text: of

text: the

text: tree.

text: In

text: our

text: grand-canonical

text: solution,

text: the

text: thermodynamic

text: variables

text: of

text: interest

text: are

text: the

text: monomer

text: fugacity

text: z

text: and

text: the

text: Boltzmann

text: weights

text: ω

text: /k

text: T

text: and

text: B


text: FIG. 1. (Color online) a) Example of contribution to the partition function of the model on a Husimi tree with three generations. The polymer chains are represented by full thick (blue) lines and the dashed lines gives examples of each type of monomer interaction. The weight of (cid:48) this conﬁguration is z18ω2ω8ω 4. b) Deﬁnition of the second neighbor (circles) - by the chemical 1 2 2 distance - of the site i (square). ω = exp((cid:15) /k T ) associated to each pair of nonconsecutive monomers on nearest-neighbor 2 2 B (NN) sites and pairs of monomers on next-NN (NNN) sites on the lattice, respectively. Hereafter, we will refer to them as NN and NNN monomers. Then, the grand-canonical partition function of the model is given by (cid:88) Y = zMωMNN ωMNNN , (1) 1 2 where the sum runs over all conﬁgurations of the walks on the tree, and M, M and NN M are respectively the total number of monomers and the number of NN and NNN NNN monomers. At this point, one notices that on the Husimi lattice there exists an ambiguity in deﬁnition of NNN sites: a) the neighborhood of a given site, let us say i, can be deﬁned according to the chemical distance (associated to the number of steps along the lattice edges to reach a site j starting at i). So, at ﬁrst, any site would have ten second neighbors, as shown in Fig. 1b. However, four of these sites (7-10 in Fig. 1b) would correspond to third neighbors on the square lattice. Since our aim is to compare the Husimi solution with results for the ordinary lattice, it is 4 ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: a)

text: Example

text: of

text: contribution

text: to

text: the

text: partition

text: function

text: of

text: the

text: model

text: on

text: a

text: Husimi

text: tree

text: with

text: three

text: generations.

text: The

text: polymer

text: chains

text: are

text: represented

text: by

text: full

text: thick

text: (blue)

text: lines

text: and

text: the

text: dashed

text: lines

text: gives

text: examples

text: of

text: each

text: type

text: of

text: monomer

text: interaction.

text: The

text: weight

text: of

text: this

text: conﬁguration

text: is

text: b)

text: Deﬁnition

text: of

text: the

text: second

text: neighbor

text: (circles)

text: by

text: the

text: chemical

text: distance

text: of

text: the

text: site

text: i

text: (square).

text: ω

text: /k

text: T

text: associated

text: to

text: each

text: pair

text: of

text: nonconsecutive

text: monomers

text: on

text: nearest-neighbor

text: B

text: (NN)

text: sites

text: and

text: pairs

text: of

text: monomers

text: on

text: next-NN

text: (NNN)

text: sites

text: on

text: the

text: lattice,

text: respectively.

text: Hereafter,

text: we

text: will

text: refer

text: to

text: them

text: as

text: NN

text: and

text: NNN

text: monomers.

text: Then,

text: the

text: grand-canonical

text: partition

text: function

text: of

text: the

text: model

text: is

text: given

text: by

text: Y

text: zMωMNN

text: ωMNNN

text: where

text: the

text: sum

text: runs

text: over

text: all

text: conﬁgurations

text: of

text: the

text: walks

text: on

text: the

text: tree,

text: and

text: M,

text: M

text: and

text: NN

text: M

text: are

text: respectively

text: the

text: total

text: number

text: of

text: monomers

text: and

text: the

text: number

text: of

text: NN

text: and

text: NNN

text: NNN

text: monomers.

text: At

text: this

text: point,

text: one

text: notices

text: that

text: on

text: the

text: Husimi

text: lattice

text: there

text: exists

text: an

text: ambiguity

text: in

text: deﬁnition

text: of

text: NNN

text: sites:

text: a)

text: the

text: neighborhood

text: of

text: a

text: given

text: site,

text: let

text: us

text: say

text: i,

text: can

text: be

text: deﬁned

text: according

text: to

text: the

text: chemical

text: distance

text: (associated

text: to

text: the

text: number

text: of

text: steps

text: along

text: the

text: lattice

text: edges

text: to

text: reach

text: a

text: site

text: j

text: starting

text: at

text: i).

text: So,

text: at

text: ﬁrst,

text: any

text: site

text: would

text: have

text: ten

text: second

text: neighbors,

text: as

text: shown

text: in

text: Fig.

text: However,

text: four

text: of

text: these

text: sites

text: in

text: Fig.

text: would

text: correspond

text: to

text: third

text: neighbors

text: on

text: the

text: square

text: lattice.

text: Since

text: our

text: aim

text: is

text: to

text: compare

text: the

text: Husimi

text: solution

text: with

text: results

text: for

text: the

text: ordinary

text: lattice,

text: it

text: is

text: ##LTFigure##

text: ##LTFigure##


text: 16 STÉPHANE BALLET, JULIA PIELTANT, MATTHIEU RAMBAUD, AND JEROEN SIJSLING • (cid:70) to obtain a type (d) coo∈rdinate, w∈e need 4 multiplications in since we have (cid:70) (cid:70) q to determine either U or V such that: q q (cid:48) (cid:48) · · (cid:48) (f g) (R ) = f (R ) g(R ) + f (R ) g (R ) = U + αV i i i i i so we need to compute · (cid:48) − · (cid:48) · (cid:48) − · (cid:48) U = f (R ) g (R ) f (R ) g (R ) + g(R ) f (R ) g(R ) f (R ) i 1 i 1 i 2 i 2 i 1 i 1 i 2 i 2 or · (cid:48) · (cid:48) · (cid:48) · (cid:48) V = f (R ) g (R ) + f (R ) g (R ) + g(R ) f (R ) + g(R ) f (R ) . i 2 i 1 i 1 i 2 i 2 i 1 i 1 i 2 So far, it seems that we ne−ed L + 2(cid:96) + 2L + 4(cid:96) multiplications in (cid:70) to obtain 1 1 2 2 q the L + (cid:96) + L + (cid:96) = 2n + g 1 coordinates of a product, which would be bounded 1 1 2 2 by N + 2a + 4N + 8a according to (10). We have to be a bit more precise to obtain 1 1 2 2 (cid:70) a better bound. Indeed, when we use more than half the coordinates in coming q from places of degree 2, we know that we can be more efﬁcient since we will have to compute some coordinates which come from the same evaluation. Namely, if we know that we will hav·e to compute both (f g)(R ) and (f g)(R ) for some i, then we would not need 2 2 = 4 multiplications in (cid:70)i ,1 but only 3, tih2anks to Karatsuba q algorithm. The same reasonning(cid:48)holds for deriv(cid:48)ated evalutations at places of deg·ree 2: if we need to compute both (f g) (R ) and (f g) (R ) , then we would not need 2 4 = 8 (cid:70) i 1 i 2 multiplications in but only 6. q We therefore have to distinguish cases were we know how many “paired” coordi- nates we have. Here is how we proceed: ≤ ≤ ≤ L2 N2 N2 < L2 2N2 (cid:96)2 ≤a2 Case 1 Case 2 a < (cid:96) 2a Case 3 Case 4 2 2 2 Thus, for −the L type (c) coordinates, we kno≤w that in cases 1 and 3, there are at 2 least 2(L N ) “paired” coordinates (since−L N ), and that each couple requires 3 mult−iplic2ation2s in (cid:70) , so we perform 3(L N2 ) su2ch multiplications. The remaining q 2 2 2N L coordinates have to be computed independently: it costs 2 multiplications in (cid:70) 2 2 for each. q The≤same reasoning applie−s to the type (d) coordinates in cases 2 and 4: since N < L 2N , there are 2(L N ) coordinates which can be c−omputed “pairwise”, ea2ch co2uple n2eeding 3 mu−ltipl2icatio2ns in (cid:70) , so we perform 3(L N ) multiplications (cid:70) q 2 2 in . The remaining 2N L coordinates have to be computed independently; it costs q (cid:70) 2 2 4 multiplications in for each. q From this reasoning and the−inequalities (9) and (10), we get the following bounds for the obtention of the 2n + g 1 coordinates of a product: Case 1: L1 + 2(cid:96)1 + 2L2 + 4(cid:96)2 =≤ (L1 + (cid:96)1−+ L2 + (cid:96)2) + (cid:96)1 + L2 + 3(cid:96)2 2n + g 1 + a + N + 3a 1 2 2 Case 2: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: AXXTMN+CharterBT-Roman

text: STÉPHANE

text: BALLET,

text: JULIA

text: PIELTANT,

text: MATTHIEU

text: RAMBAUD,

text: AND

text: JEROEN

text: SIJSLING

label: list

text: to

text: obtain

text: a

text: type

text: (d)

text: coo∈rdinate,

text: w∈e

text: need

text: multiplications

text: in

text: since

text: we

text: have

text: q

font: JWUBOF+CharterBT-Italic

text: to

text: determine

text: either

text: U

text: or

text: V

text: such

text: that:

text: q

text: q

text: (f

text: g)

text: (R

text: f

text: (R

text: g(R

text: f

text: (R

text: g

text: (R

text: U

text: αV

text: i

text: i

text: i

text: i

text: i

text: so

text: we

text: need

text: to

text: compute

text: U

text: f

text: (R

text: g

text: (R

text: f

text: (R

text: g

text: (R

text: g(R

text: f

text: (R

text: g(R

text: f

text: (R

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: or

text: V

text: f

text: (R

text: g

text: (R

text: f

text: (R

text: g

text: (R

text: g(R

text: f

text: (R

text: g(R

text: f

text: (R

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: So

text: far,

text: it

text: seems

text: that

text: we

text: ne−ed

text: L

text: multiplications

text: in

text: to

text: obtain

text: q

text: the

text: L

text: L

text: g

text: coordinates

text: of

text: a

text: product,

text: which

text: would

text: be

text: bounded

text: by

text: N

text: according

text: to

text: We

text: have

text: to

text: be

text: a

text: bit

text: more

text: precise

text: to

text: obtain

text: a

text: better

text: bound.

text: Indeed,

text: when

text: we

text: use

text: more

text: than

text: half

text: the

text: coordinates

text: in

text: coming

text: q

text: from

text: places

text: of

text: degree

text: we

text: know

text: that

text: we

text: can

text: be

text: more

text: efﬁcient

text: since

text: we

text: will

text: have

text: to

text: compute

text: some

text: coordinates

text: which

text: come

text: from

text: the

text: same

text: evaluation.

text: Namely,

text: if

text: we

text: know

text: that

text: we

text: will

text: hav·e

text: to

text: compute

text: both

text: (f

text: g)(R

text: and

text: (f

text: g)(R

text: for

text: some

text: i,

text: then

text: we

text: would

text: not

text: need

text: multiplications

text: in

text: but

text: only

text: to

text: Karatsuba

text: q

text: algorithm.

text: The

text: same

text: for

text: evalutations

text: at

text: places

text: of

text: deg·ree

text: if

text: we

text: need

text: to

text: compute

text: both

text: (f

text: g)

text: (R

text: and

text: (f

text: g)

text: (R

text: then

text: we

text: would

text: not

text: need

text: i

text: i

text: multiplications

text: in

text: but

text: only

text: q

text: We

text: therefore

text: have

text: to

text: distinguish

text: cases

text: were

text: we

text: know

text: how

text: many

text: “paired”

text: coordi-

text: nates

text: we

text: have.

text: Here

text: is

text: how

text: we

text: proceed:

text: Case

text: Case

text: a

text: Case

text: Case

text: Thus,

text: for

text: −the

text: L

text: type

text: (c)

text: coordinates,

text: we

text: kno≤w

text: that

text: in

text: cases

text: and

text: there

text: are

text: at

text: least

text: N

text: “paired”

text: coordinates

text: (since−L

text: N

text: and

text: that

text: each

text: couple

text: requires

text: in

text: so

text: we

text: perform

text: multiplications.

text: The

text: remaining

text: q

text: L

text: coordinates

text: have

text: to

text: be

text: computed

text: independently:

text: it

text: costs

text: multiplications

text: in

text: for

text: each.

text: q

text: The≤same

text: reasoning

text: applie−s

text: to

text: the

text: type

text: (d)

text: coordinates

text: in

text: cases

text: and

text: since

text: N

text: L

text: there

text: are

text: N

text: coordinates

text: which

text: can

text: be

text: c−omputed

text: “pairwise”,

text: in

text: so

text: we

text: perform

text: N

text: multiplications

text: q

text: in

text: The

text: remaining

text: L

text: coordinates

text: have

text: to

text: be

text: computed

text: independently;

text: it

text: costs

text: q

text: multiplications

text: in

text: for

text: each.

text: q

text: From

text: this

text: reasoning

text: and

text: the−inequalities

text: and

text: we

text: get

text: the

text: following

text: bounds

text: for

text: the

text: obtention

text: of

text: the

text: g

text: coordinates

text: of

text: a

text: product:

text: Case

text: g

text: a

text: N

text: Case

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON SOME BOUNDS FOR SYMMETRIC TENSOR RANK OF MULTIPLICATION IN FINITE FIELDS 19 (cid:129) (cid:139) ≤ Φ(X) 3X + 2g k,s+1 2g = 3 1 + k,s+1 X. 3X We want to−give a bound for Φ(X) that is independent of k and s. Recall that D := (p 1)psqk, and k,s ≥ 1 − − nk,s qk 1ps(q + 1)(q 3) by Lemma 2.4 0 2 and ≤ − g qk 1(q + 1)ps+1 by Lemma 2.2 (iii). k,s+1 So we have 2g 2g k,s+1 = k,s+1 3X 3(nk,s + D ) 0 k,s − ≤ 2qk 1(q + 1)ps+1 − − − 3(1qk 1ps(q +(cid:128)1)(q 3) + (p 1)psqk(cid:138)) 2 − 2qk 1(q + 1)psp = − − − qk 1(q + 1)ps 3(q 3) + 3(p 1) q 2 q+1 = − 43 p − . (cid:16) (cid:17) (q 3) + 2(p 1) q q+1 Thus the graph of t(cid:130)he function Φ lies below the(cid:140)line y = 3 1 + − 43 p− x. (q 3)+2(p 1)q+q1 In particular, we obtain ≤ 4 p Φ(n) 3 1 + − 3 − n. (q 3) + 2(p 1) q ≥ q+1 (ii) Let n 1(p + 1 + ε(p≤)); in the complementary case, we already know from 2 Section that µsym(n) 2n. According to Lemma 2.12, there exists a step of the tower T/(cid:70) p on which we can apply Theorem 2.14 with a = a = 0. We p (cid:70) 1 2 denote by H / the ﬁrst step of the tower that satisﬁes the hypotheses of k+1 p Theor≥em 2.14 wit−h a = a = 0, i.e. k is an−integer such that 1 2 N 2n + 2g 1 and N < 2n + 2g 1, where Nk+:1= N ((cid:8)H /(cid:70)k)+1+ 2N (H /(cid:70) ) and(cid:9)g := gk(H ). W≥e deknote by−nk the biggest k 1 k p 2 k p k k 0 integer s∈uch| ≤that − we have N 2nk + 2g 1, i.e. nk = sup n (cid:78) 2n N 2g + 1 . To performk multi0plicatikon in (cid:70) , we 0 k k pn have the following alternative approaches: (a) use the algorithm at the step H . In this case, a bound for the bilinear k+1 complexity is given by Theorem 2.14 applied with a = a = 0: ≤ − 1 2 µsym(n) 3n + 2g = 3nk + 2g + 3(n nk) + 2∆g . q k+1 0 k 0 k (b) use the algorithm at the step H with an −appropriate nu≥mber of deri−va- k tive evaluations. If we let a≥+ 2a := 2−(n nk), then N 2≤nk + 2g 1 1 2 0 k 0 k implies that N + a + 2a 2n + 2g 1. Thus if a + 2a N , we can k 1 2 k 1 2 k ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: SOME

text: BOUNDS

text: FOR

text: SYMMETRIC

text: TENSOR

text: RANK

text: OF

text: MULTIPLICATION

text: IN

text: FINITE

text: FIELDS

label: equation

text: Φ(X)

text: X.

text: We

text: want

text: to−give

text: a

text: bound

text: for

text: Φ(X)

text: that

text: is

text: independent

text: of

text: k

text: and

text: s.

text: Recall

text: that

text: D

text: (p

text: and

text: k,s

text: nk,s

text: qk

text: by

text: Lemma

text: and

text: g

text: qk

text: by

text: Lemma

text: (iii).

text: So

text: we

text: have

text: D

text: k,s

text: (p

text: qk

text: q

text: p

text: (q

text: q

text: Thus

text: the

text: graph

text: of

text: function

text: Φ

text: lies

text: below

text: y

text: p−

text: x.

text: (q

text: In

text: particular,

text: we

text: obtain

text: p

text: Φ(n)

text: n.

text: (q

text: q

text: (ii)

text: Let

text: n

text: ε(p≤));

text: in

text: the

text: complementary

text: case,

text: we

text: already

text: know

text: from

text: Section

text: that

text: µsym(n)

text: According

text: to

text: Lemma

text: there

text: exists

text: a

text: step

text: of

text: the

text: tower

text: p

text: on

text: which

text: we

text: can

text: apply

text: Theorem

text: with

text: a

text: a

text: We

text: p

text: denote

text: by

text: H

text: the

text: ﬁrst

text: step

text: of

text: the

text: tower

text: that

text: satisﬁes

text: the

text: hypotheses

text: of

text: p

text: Theor≥em

text: wit−h

text: a

text: a

text: i.e.

text: k

text: is

text: an−integer

text: such

text: that

text: N

text: and

text: N

text: where

text: N

text: (H

text: gk(H

text: W≥e

text: deknote

text: by−nk

text: the

text: biggest

text: k

text: k

text: p

text: k

text: p

text: k

text: k

text: integer

text: s∈uch|

text: ≤that

text: we

text: have

text: N

text: i.e.

text: nk

text: sup

text: n

text: N

text: To

text: performk

text: in

text: we

text: k

text: k

text: pn

text: have

text: the

text: following

text: alternative

text: approaches:

text: (a)

text: use

text: the

text: algorithm

text: at

text: the

text: step

text: H

text: In

text: this

text: case,

text: a

text: bound

text: for

text: the

text: bilinear

text: complexity

text: is

text: given

text: by

text: Theorem

text: applied

text: with

text: a

text: a

text: µsym(n)

text: nk)

text: q

text: k

text: k

text: (b)

text: use

text: the

text: algorithm

text: at

text: the

text: step

text: H

text: with

text: an

text: −appropriate

text: nu≥mber

text: of

text: deri−va-

text: k

text: tive

text: evaluations.

text: If

text: we

text: let

text: a≥+

text: nk),

text: then

text: N

text: k

text: k

text: implies

text: that

text: N

text: a

text: Thus

text: if

text: a

text: N

text: we

text: can

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON SOME BOUNDS FOR SYMMETRIC TENSOR RANK OF MULTIPLICATION IN FINITE FIELDS 3 Let us recall that the original algorithm of D.V. and G.V. Chudnovsky introduced in [15] is symmetric by deﬁnition and leads to the following result from [2]: THEOREM 1.3. Let q be a power of a prime p. The symmetric tensor rank µsym(n) of (cid:70) q multiplication in any ﬁnite ﬁeld is linear with respect to the extension degree; more qn precisely, there exists a constant C such that q ≤ µsym(n) C n. q q General expressions for C have been obtained, such as the following best current q  published estimates:   (cid:39)   if q = 2, then 4824 19,6 [7] and [13]  247     (cid:128) (cid:138) else if q = 3, then 27 [3]   ≥ Cq =  else if q = p 5, then 3(cid:128)1 + q−43 (cid:138) [4]    ≥  else if q = p2 25, then 2(cid:128)1 + −2 (cid:138) [4] p 3 ≥ else if q 4, then 6 1 + −p [3] q 3 Now we introduce the generalized Chudnovsky-Chudnovskytype algorithm described in [13]; the original algorithm given in [15] by D.V. and G.V. Chudnovsky being the case where deg P = 1 and u = 1 for i = 1,..., N. Here a wider notion of complexity i i is involved: the quantity µsym(m,(cid:96)), which corresponds to the symmetric bilinear com- q plexity of the multiplication over (cid:70) in (cid:70) [X]/(X(cid:96)), the (cid:70) -algebra of polynomials in q (cid:70) qm q one indeterminate with coefﬁcients in truncated at order (cid:96). qm THE•OREM 1.4. Let • q be a prime power, (cid:70) • F/ be an algebraic function ﬁeld, q (cid:70) • Q(cid:68) be a degree n place of F/ , (cid:70) q • be a{ divisor of}F/ , q • P = P ,..., P be a set of N places of arbitrary degree, 1 N u ,...,u be positive integers. 1 N (cid:68) We suppose that Q and all the pla(cid:12)ces in P are not in the support of and that: (cid:12) [a)]the map (cid:12) (cid:12) (cid:12) (cid:68) → (cid:39) Ev : L ( ) (cid:55)−→ (cid:70)qn FQ Q f f (Q) (cid:12) (cid:12) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) is onto,(cid:12) the map (cid:12) (cid:0) (cid:1) (cid:12) (cid:68) −→ × × ··· × Ev : L (2 ) (cid:55)−→ (cid:70)qdegP1 u1 (cid:70)qdegP2 u2 (cid:70)qdegPN uN P f ϕ (f ),ϕ (f ),...,ϕ (f ) 1 2 N ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: SOME

text: BOUNDS

text: FOR

text: SYMMETRIC

text: TENSOR

text: RANK

text: OF

text: MULTIPLICATION

text: IN

text: FINITE

text: FIELDS

text: Let

text: us

text: recall

text: that

text: the

text: original

text: algorithm

text: of

text: D.V.

text: and

text: G.V.

text: Chudnovsky

text: introduced

text: in

text: is

text: symmetric

text: by

text: deﬁnition

text: and

text: leads

text: to

text: the

text: following

text: result

text: from

text: THEOREM

text: Let

text: q

text: be

text: a

text: power

text: of

text: a

text: prime

text: p.

text: The

text: symmetric

text: tensor

text: rank

text: µsym(n)

text: of

text: q

text: multiplication

text: in

text: any

text: ﬁnite

text: ﬁeld

text: is

text: linear

text: with

text: respect

text: to

text: the

text: extension

text: degree;

text: more

text: qn

text: precisely,

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: q

text: µsym(n)

text: C

text: n.

text: q

text: q

text: General

text: expressions

text: for

text: C

text: have

text: been

text: obtained,

text: such

text: as

text: the

text: following

text: best

text: current

text: q

text: published

text: estimates:

text: if

text: q

text: then

text: and

text: else

text: if

text: q

text: then

text: Cq

text: else

text: if

text: q

text: p

text: then

text: else

text: if

text: q

text: then

text: −2

text: p

text: else

text: if

text: q

text: then

text: −p

text: q

text: Now

text: we

text: introduce

text: the

text: generalized

text: Chudnovsky-Chudnovskytype

text: algorithm

text: described

text: in

text: the

text: original

text: algorithm

text: given

text: in

text: by

text: D.V.

text: and

text: G.V.

text: Chudnovsky

text: being

text: the

text: case

text: where

text: deg

text: P

text: and

text: u

text: for

text: i

text: N.

text: Here

text: a

text: wider

text: notion

text: of

text: complexity

text: i

text: i

text: is

text: involved:

text: the

text: quantity

text: which

text: corresponds

text: to

text: the

text: symmetric

text: bilinear

text: com-

text: q

text: plexity

text: of

text: the

text: multiplication

text: over

text: in

text: the

text: -algebra

text: of

text: polynomials

text: in

text: q

text: qm

text: q

text: one

text: indeterminate

text: with

text: coefﬁcients

text: in

text: truncated

text: at

text: order

text: qm

text: THE•OREM

text: Let

text: q

text: be

text: a

text: prime

text: power,

text: F/

text: be

text: an

text: algebraic

text: function

text: ﬁeld,

text: q

text: be

text: a

text: degree

text: n

text: place

text: of

text: F/

text: q

text: be

text: a{

text: divisor

text: of}F/

text: q

text: P

text: P

text: P

text: be

text: a

text: set

text: of

text: N

text: places

text: of

text: arbitrary

text: degree,

text: N

text: u

text: ,...,u

text: be

text: positive

text: integers.

text: N

text: We

text: suppose

text: that

text: Q

text: and

text: all

text: the

text: in

text: P

text: are

text: not

text: in

text: the

text: support

text: of

text: and

text: that:

text: [a)]the

text: map

text: Ev

text: L

text: FQ

text: Q

text: f

text: f

text: (Q)

text: is

text: the

text: map

text: Ev

text: L

text: uN

text: P

text: f

text: ϕ

text: (f

text: ),ϕ

text: (f

text: ),...,ϕ

text: (f

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON SOME BOUNDS FOR SYMMETRIC TENSOR RANK OF MULTIPLICATION IN FINITE FIELDS 27 (cid:48) p stand for the two primes over the split prime 3). The two columns in the middle 3 describe whether the primes 2 and 3 are inert in F.3 TABLE 1. Counterexamples curve (cid:112)F 2 inert 3 inert Df e2d13D4 (cid:81)( (cid:112)13) yes no p2 (cid:48) e2d13D36 p p p e3d8D9 (cid:81)( 2) no yes 2p3 3 3 For these three curves of genus 1, the canonical models, deﬁned over F, do not (cid:81) descend to . Let X be the curve e2d13D36. X is deﬁned over F, of genus 1, but doesn’t neces- sarily have a rational point. However, we were able to derive properties of its jacobian J, which is an elliptic curve over F: • • Its conductor equals 6, by [38, Proposition 2.1.6]. (cid:48) The valuation of its j-invariant at p is equal to -10 (resp. -2 at p and p ). 2 3 3 Let us detail this result for the valuation at p2. First, deﬁne the q(cid:48)uaterni(cid:79)on algebra H ramiﬁed exactly at both inﬁnite places of F and at p p . C(cid:79)all 3 3 H the maximal order of(cid:79)H. As in [38, Proposition 3.1.9 (ii)] , consider (cid:79) (p ), H 2 a level p su(cid:79)border of . Consider the set of classes of right ideals of (p ), 2 H H 2 n|(cid:79)oted Pic ( (|p )). To each ideal class [I(p )] in this set, associate the weight (I(p ))r./(cid:90)H. (2equal to the cardinality of2 the projectivized group of units l 2 F of the left-order of I(p )). These weights can be computed by running the 2 Magma ([9]) ﬁle PadInit in [37]. The sum of these weights is then equal to the opposite of the valuation of j at p , by [38, Proposition 3.1.14 (iii)]. 2 (cid:81) Now if the curve X were deﬁned over , then the jacobian J would descend to an elliptic curve J(cid:81) over (cid:81), by the argument of [27, Proposition 1.9]. So, let us suppose that suc•h a rational model J(cid:81) does exist, then · the conductor of J(cid:81) is either equal to 6, or to 6 132. Indeed: (cid:81) – at every place p but 13, the extension F does not ramify, so the P p conductor of J(cid:81) has the same valuation than that of J, by Proposition 5.4 (a) of [39]. (As regards the particular cases of 2 and 3, note that J has multiplicative reduction at these places, so the valuation of the conductor of J(cid:81) is necessarily equal to 1 at these places.) (cid:81) – at the place 13 where the extension F / ramiﬁes, J(cid:81) cannot have P 13 multiplicative reduction. For that if it were the case, then J would also have multiplicative reduction at 13 (by [39, Proposition 5.4 (b)]). This • contradicts the above result on the conductor of J. the j-invariant of J(cid:81) should be equal to the one of J. So, in particular, it should have the valuations at 2 and 3 predicted above. Then, by a lookup in the tables of Cremona (proved to be exhaustive, see the in- troduction of [16]), only two elliptic curves E and E over (cid:81) fulﬁll the conditions 1 2 3Thus, one can see that we are unlucky because these counterexamples would not have been, anyway, good candidates for reduction modulo or 3 (these primes either meet the discriminant, or they are not inert). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: SOME

text: BOUNDS

text: FOR

text: SYMMETRIC

text: TENSOR

text: RANK

text: OF

text: MULTIPLICATION

text: IN

text: FINITE

text: FIELDS

text: p

text: stand

text: for

text: the

text: two

text: primes

text: over

text: the

text: split

text: prime

text: The

text: two

text: columns

text: in

text: the

text: middle

text: describe

text: whether

text: the

text: primes

text: and

text: are

text: inert

text: in

text: TABLE

text: Counterexamples

label: caption

text: curve

label: table

text: inert

text: inert

text: Df

text: yes

text: no

text: p

text: p

text: p

text: no

text: yes

text: For

text: these

text: three

text: curves

text: of

text: genus

text: the

text: canonical

text: models,

text: deﬁned

text: over

text: F,

text: do

text: not

text: descend

text: to

text: Let

text: X

text: be

text: the

text: curve

text: X

text: is

text: deﬁned

text: over

text: F,

text: of

text: genus

text: but

text: doesn’t

text: neces-

text: sarily

text: have

text: a

text: rational

text: point.

text: However,

text: we

text: were

text: able

text: to

text: derive

text: properties

text: of

text: its

text: jacobian

text: J,

text: which

text: is

text: an

text: elliptic

text: curve

text: over

text: F:

text: Its

text: conductor

text: equals

text: by

text: Proposition

text: The

text: valuation

text: of

text: its

text: j-invariant

text: at

text: p

text: is

text: equal

text: to

text: (resp.

text: at

text: p

text: and

text: p

text: Let

text: us

text: detail

text: this

text: result

text: for

text: the

text: valuation

text: at

text: First,

text: deﬁne

text: the

text: algebra

text: H

text: ramiﬁed

text: exactly

text: at

text: both

text: inﬁnite

text: places

text: of

text: F

text: and

text: at

text: p

text: p

text: H

text: the

text: maximal

text: order

text: As

text: in

text: Proposition

text: (ii)]

text: consider

text: (p

text: H

text: a

text: level

text: p

text: of

text: Consider

text: the

text: set

text: of

text: classes

text: of

text: right

text: ideals

text: of

text: (p

text: H

text: H

text: Pic

text: (|p

text: To

text: each

text: ideal

text: class

text: [I(p

text: in

text: this

text: set,

text: associate

text: the

text: weight

text: (I(p

text: to

text: the

text: cardinality

text: the

text: projectivized

text: group

text: of

text: units

text: l

text: F

text: of

text: the

text: left-order

text: of

text: I(p

text: These

text: weights

text: can

text: be

text: computed

text: by

text: running

text: the

text: Magma

text: ﬁle

text: PadInit

text: in

text: The

text: sum

text: of

text: these

text: weights

text: is

text: then

text: equal

text: to

text: the

text: opposite

text: of

text: the

text: valuation

text: of

text: j

text: at

text: p

text: by

text: Proposition

text: (iii)].

text: Now

text: if

text: the

text: curve

text: X

text: were

text: deﬁned

text: over

text: then

text: the

text: jacobian

text: J

text: would

text: descend

text: to

text: an

text: elliptic

text: curve

text: over

text: by

text: the

text: argument

text: of

text: Proposition

text: So,

text: let

text: us

text: suppose

text: that

text: suc•h

text: a

text: rational

text: model

text: does

text: exist,

text: then

text: the

text: conductor

text: of

text: is

text: either

text: equal

text: to

text: or

text: to

text: Indeed:

text: at

text: every

text: place

text: p

text: but

text: the

text: extension

text: F

text: does

text: not

text: ramify,

text: so

text: the

text: P

text: p

text: conductor

text: of

text: has

text: the

text: same

text: valuation

text: than

text: that

text: of

text: J,

text: by

text: Proposition

text: (a)

text: of

text: (As

text: regards

text: the

text: particular

text: cases

text: of

text: and

text: note

text: that

text: J

text: has

text: multiplicative

text: reduction

text: at

text: these

text: places,

text: so

text: the

text: valuation

text: of

text: the

text: conductor

text: of

text: is

text: necessarily

text: equal

text: to

text: at

text: these

text: places.)

text: at

text: the

text: place

text: where

text: the

text: extension

text: F

text: ramiﬁes,

text: cannot

text: have

text: P

text: multiplicative

text: reduction.

text: For

text: that

text: if

text: it

text: were

text: the

text: case,

text: then

text: J

text: would

text: also

text: have

text: multiplicative

text: reduction

text: at

text: (by

text: Proposition

text: (b)]).

text: This

text: contradicts

text: the

text: above

text: result

text: on

text: the

text: conductor

text: of

text: J.

text: the

text: j-invariant

text: of

text: should

text: be

text: equal

text: to

text: the

text: one

text: of

text: J.

text: So,

text: in

text: particular,

text: it

text: should

text: have

text: the

text: valuations

text: at

text: and

text: predicted

text: above.

text: Then,

text: by

text: a

text: lookup

text: in

text: the

text: tables

text: of

text: Cremona

text: (proved

text: to

text: be

text: exhaustive,

text: see

text: the

text: in-

text: troduction

text: of

text: only

text: two

text: elliptic

text: curves

text: E

text: and

text: E

text: over

text: fulﬁll

text: the

text: conditions

text: one

text: can

text: see

text: that

text: we

text: are

text: unlucky

text: because

text: these

text: counterexamples

text: would

text: not

text: have

text: been,

text: anyway,

text: good

text: candidates

text: for

text: reduction

text: modulo

text: or

text: (these

text: primes

text: either

text: meet

text: the

text: discriminant,

text: or

text: they

text: are

text: not

text: inert).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: A UNIFIED ANALYSIS OF QUASI-OPTIMAL CONVERGENCE FOR ADAPTIVE MIXED FINITE ELEMENT METHODS ∗ JUN HU1 AND GUOZHU YU2, 1LMAM and School of Mathematical Sciences, Peking University, Beijing 100871,China 2School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China Abstract. In this paper, we present a uniﬁed analysis of both convergence and optimality of adaptive mixed ﬁnite element methods for a class of problems when the ﬁnite element spaces and corresponding a posteriori error estimates under consideration satisfy ﬁve hypotheses. We prove that these ﬁve conditions are suﬃcient for convergence and optimality of the adaptive algorithms under consideration. The main ingredient for the analysis is a new method to analyze both discrete reliability and quasi-orthogonality. This new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces, namely, a mesh-dependent discrete H1 norm for the former and a L2 norm for the latter, and a newly deﬁned projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the ﬁner mesh. As applications, we prove these ﬁve hypotheses for the Raviart–Thomas and Brezzi–Douglas–Marini elements of the Poisson and Stokes problems in both 2D and 3D. 1. Introduction This paper is devoted to convergence and optimality of adaptive mixed ﬁnite element methods ∈ ∈ × (AMFEMs) for the problem of the following form: Given f L2(Ω), ﬁnd (σ, u) Σ U such that A − ∈ ( σ, τ) (div τ, u) = 0, for any τ Σ, L2(Ω) L2(Ω) (1.1) − ∈ (div σ, v) (f, v) = 0, for any v U. L2(Ω) L2(Ω) In the paper, we refer to Σ as the stress space, and U as the displacement space. Here, Ω is a R simply connected bounded domain in d(d = 2, 3) with the boundary ∂Ω, and Σ, U are Sobolev spaces deﬁned as R × R Σ := H(div, Ω; d n), U := L2(Ω; n) A with n some positive integer. Furthermore, we assume is a linear, bounded and semi-deﬁnite operator, satisfying *: Corresponding author. The research of the ﬁrst author was supported by NSFC projection 11271035, 91430213 and 11421101. The research of the second author was supported by NSFC projection 11401492. Email addresses: hujun@math.pku.edu.cn (J. Hu), yuguozhumail@163.com (G. Yu). 1 ##LTLine##

text: A

text: UNIFIED

text: ANALYSIS

text: OF

text: QUASI-OPTIMAL

text: CONVERGENCE

text: FOR

text: ADAPTIVE

text: MIXED

text: FINITE

text: ELEMENT

text: METHODS

text: JUN

text: AND

text: GUOZHU

text: and

text: School

text: of

text: Mathematical

text: Sciences,

text: Peking

text: University,

text: Beijing

text: of

text: Mathematics,

text: Southwest

text: Jiaotong

text: University,

text: Chengdu

text: China

text: Abstract.

text: In

label: abstract

text: this

text: paper,

text: we

text: present

text: a

text: uniﬁed

text: analysis

text: of

text: both

text: convergence

text: and

text: optimality

text: of

text: adaptive

text: mixed

text: ﬁnite

text: element

text: methods

text: for

text: a

text: class

text: of

text: problems

text: when

text: the

text: ﬁnite

text: element

text: spaces

text: and

text: corresponding

text: a

text: posteriori

text: error

text: estimates

text: under

text: consideration

text: satisfy

text: ﬁve

text: hypotheses.

text: We

text: prove

text: that

text: these

text: ﬁve

text: conditions

text: are

text: suﬃcient

text: for

text: convergence

text: and

text: optimality

text: of

text: the

text: adaptive

text: algorithms

text: under

text: consideration.

text: The

text: main

text: ingredient

text: for

text: the

text: analysis

text: is

text: a

text: new

text: method

text: to

text: analyze

text: both

text: discrete

text: reliability

text: and

text: quasi-orthogonality.

text: This

text: new

text: method

text: arises

text: from

text: an

text: appropriate

text: and

text: natural

text: choice

text: of

text: the

text: norms

text: for

text: both

text: the

text: discrete

text: displacement

text: and

text: stress

text: spaces,

text: namely,

text: a

text: mesh-dependent

text: discrete

text: norm

text: for

text: the

text: former

text: and

text: a

text: norm

text: for

text: the

text: latter,

text: and

text: a

text: newly

text: deﬁned

text: projection

text: operator

text: from

text: the

text: discrete

text: stress

text: space

text: on

text: the

text: coarser

text: mesh

text: onto

text: the

text: discrete

text: divergence

text: free

text: space

text: on

text: the

text: ﬁner

text: mesh.

text: As

text: applications,

text: we

text: prove

text: these

text: ﬁve

text: hypotheses

text: for

text: the

text: Raviart–Thomas

text: and

text: Brezzi–Douglas–Marini

text: elements

text: of

text: the

text: Poisson

text: and

text: Stokes

text: problems

text: in

text: both

text: and

text: Introduction

text: This

text: paper

text: is

text: devoted

text: to

text: convergence

text: and

text: optimality

text: of

text: adaptive

text: mixed

text: ﬁnite

text: element

text: methods

text: (AMFEMs)

text: for

text: the

text: problem

text: of

text: the

text: following

text: form:

text: Given

text: f

text: ﬁnd

text: (σ,

text: u)

text: Σ

text: U

text: such

text: that

text: A

text: σ,

text: τ)

text: (div

text: τ,

text: u)

text: for

text: any

text: τ

text: Σ,

text: (div

text: σ,

text: v)

text: (f,

text: v)

text: for

text: any

text: v

text: U.

text: In

text: the

text: paper,

text: we

text: refer

text: to

text: Σ

text: as

text: the

text: stress

text: space,

text: and

text: U

text: as

text: the

text: displacement

text: space.

text: Here,

text: Ω

text: is

text: a

text: R

text: simply

text: connected

text: bounded

text: domain

text: in

text: d(d

text: with

text: the

text: boundary

text: ∂Ω,

text: and

text: Σ,

text: U

text: are

text: Sobolev

text: spaces

text: deﬁned

text: as

text: R

text: R

text: Σ

text: H(div,

text: Ω;

text: d

text: n),

text: U

text: n)

text: A

text: with

text: n

text: some

text: positive

text: integer.

text: Furthermore,

text: we

text: assume

text: is

text: a

text: linear,

text: bounded

text: and

text: semi-deﬁnite

text: operator,

text: satisfying

text: Corresponding

text: author.

text: The

text: research

text: of

text: the

text: ﬁrst

text: author

text: was

text: supported

text: by

text: NSFC

text: projection

text: and

text: The

text: research

text: of

text: the

text: second

text: author

text: was

text: supported

text: by

text: NSFC

text: projection

text: Email

text: addresses:

text: hujun@math.pku.edu.cn

text: (J.

text: Hu),

text: (G.

text: Yu).

text: ##LTLine##


text: QUASI-OPTIMAL CONVERGENCE FOR AMFEMS 17 M Theorem 4.6. Let be a set of marked elements with minimal cardinality, (σ, u) the solution of T k (1.1), and ( , Σ , σ , u ) the sequence of triangulations, ﬁnite element spaces and discrete solutions k k k k produced by the adaptive ﬁnite element methods with the marking parameter θ in Lemma 4.3. Then it holds that (cid:107) − (cid:107) T (cid:46) | | T − T − ∈ A σ σ 2A + osc2(f, ) σ, f (# # ) s, for (σ, f) . N N s N 0 s (cid:80) Proof. Let µ = (α(cid:48)M)−1s |u, f|s1s C11s C(cid:88)2. We use the(cid:88)result #Tk −#T0 (cid:46) kj=−01 #Mj from [39], and the upper bound of # in Theorem 4.5, to obtain that j − − T − T (cid:46) N 1 M ≤ N 1 (cid:107) − (cid:107) T − # N # 0 # j µ( σ σj 2A + osc2(f, j)) 1s . j=0 j=0 ≤ − ≤ ≤ − From the convergence result in Theorem 4.2 we have (cid:15) αN j(cid:15) for any 0 j N 1, which, ≈ (cid:107) − (cid:107) T N j (cid:88) along with the fact (cid:15) σ σ 2A + osc2(f, ), implies j j j − T − T (cid:46) (cid:107) − (cid:107) T − N 1 #(cid:80) N # 0 µ( σ σN 2A + osc2(f, N)) 1s αsj . j=0 − Since α < 1, the term Nj=01 αsj is bounded. The deﬁnition of µ leads to (cid:107) − (cid:107) T (cid:46) | | T − T − ∈ A σ σ 2A + osc2(f, ) u, f (# # ) s, for (u, f) . N N s N 0 s (cid:3) 5. Applications In this section, we present two examples which satisfy these ﬁve hypotheses. The ﬁrst example is the mixed ﬁnite element of the Poisson equation; the second one is the mixed ﬁnite element of the Stokes problem within the pseudostress-velocity formulation. For both the 2D and 3D, we prove that the Raviart–Thomas and Brezzi–Douglas–Marini elements satisfy Hypotheses 1–5 in Section 2. Hence the corresponding adaptive algorithms converge at the optimal rate in the nonlinear approximation sense. In the sequel, we use superscript P to denote the subspace or operator for the Poisson problem, and S to denote the subspace or operator for the Stokes problem. ≥ The Raviart–Thomas element spaces [10] are deﬁned for k 0 by × RTP = ΣP UP , h h,k h,k where { ∈ R | ∈ ∀ ∈ T } ΣP := τ H(div, Ω; d) : τ P (K)d + xP (K), K , h,k K k k h and { ∈ R | ∈ ∀ ∈ T } UP := v L2(Ω; ) : v P (K), K . h,k K k h Here P (K) denotes the space of polynomials of degree at most k over K. k ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUASI-OPTIMAL

text: CONVERGENCE

text: FOR

text: AMFEMS

text: M

text: Theorem

text: Let

text: be

text: a

text: set

text: of

text: marked

text: elements

text: with

text: minimal

text: cardinality,

text: (σ,

text: u)

text: the

text: solution

text: of

text: T

text: k

text: and

text: Σ

text: σ

text: u

text: the

text: sequence

text: of

text: triangulations,

text: ﬁnite

text: element

text: spaces

text: and

text: discrete

text: solutions

text: k

text: k

text: k

text: k

text: produced

text: by

text: the

text: adaptive

text: ﬁnite

text: element

text: methods

text: with

text: the

text: marking

text: parameter

text: θ

text: in

text: Lemma

text: Then

text: it

text: holds

text: that

text: T

text: T

text: T

text: A

text: σ

text: σ

text: σ,

text: f

text: s,

text: for

text: (σ,

text: f)

text: N

text: N

text: s

text: N

text: s

text: Proof.

text: Let

text: µ

text: |u,

text: We

text: use

text: #Tk

text: #Mj

text: from

text: and

text: the

text: upper

text: bound

text: of

text: in

text: Theorem

text: to

text: obtain

text: that

text: j

text: T

text: T

text: N

text: M

text: N

text: T

text: N

text: j

text: µ(

text: σ

text: σj

text: j))

text: From

text: the

text: convergence

text: result

text: in

text: Theorem

text: we

text: have

text: αN

text: for

text: any

text: j

text: N

text: which,

text: T

text: N

text: j

text: along

text: with

text: the

text: fact

text: σ

text: σ

text: implies

text: j

text: j

text: j

text: T

text: T

text: T

text: N

text: N

text: µ(

text: σ

text: σN

text: N))

text: αsj

text: Since

text: α

text: the

text: term

text: αsj

text: is

text: bounded.

text: The

text: deﬁnition

text: of

text: µ

text: leads

text: to

text: T

text: T

text: T

text: A

text: σ

text: σ

text: u,

text: f

text: s,

text: for

text: (u,

text: f)

text: N

text: N

text: s

text: N

text: s

text: Applications

text: In

text: this

text: section,

text: we

text: present

text: two

text: examples

text: which

text: satisfy

text: these

text: ﬁve

text: hypotheses.

text: The

text: ﬁrst

text: example

text: is

text: the

text: mixed

text: ﬁnite

text: element

text: of

text: the

text: Poisson

text: equation;

text: the

text: second

text: one

text: is

text: the

text: mixed

text: ﬁnite

text: element

text: of

text: the

text: Stokes

text: problem

text: within

text: the

text: pseudostress-velocity

text: formulation.

text: For

text: both

text: the

text: and

text: we

text: prove

text: that

text: the

text: Raviart–Thomas

text: and

text: Brezzi–Douglas–Marini

text: elements

text: satisfy

text: Hypotheses

text: in

text: Section

text: Hence

text: the

text: corresponding

text: adaptive

text: algorithms

text: converge

text: at

text: the

text: optimal

text: rate

text: in

text: the

text: nonlinear

text: approximation

text: sense.

text: In

text: the

text: sequel,

text: we

text: use

text: superscript

text: P

text: to

text: denote

text: the

text: subspace

text: or

text: operator

text: for

text: the

text: Poisson

text: problem,

text: and

text: S

text: to

text: denote

text: the

text: subspace

text: or

text: operator

text: for

text: the

text: Stokes

text: problem.

text: The

text: Raviart–Thomas

text: element

text: spaces

text: are

text: deﬁned

text: for

text: k

text: by

text: RTP

text: ΣP

text: UP

text: h

text: h,k

text: h,k

text: where

text: R

text: T

text: ΣP

text: τ

text: H(div,

text: Ω;

text: d)

text: τ

text: P

text: (K)d

text: xP

text: (K),

text: K

text: h,k

text: K

text: k

text: k

text: h

text: and

text: R

text: T

text: UP

text: v

text: v

text: P

text: (K),

text: K

text: h,k

text: K

text: k

text: h

text: Here

text: P

text: (K)

text: denotes

text: the

text: space

text: of

text: polynomials

text: of

text: degree

text: at

text: most

text: k

text: over

text: K.

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: QUASI-OPTIMAL CONVERGENCE FOR AMFEMS 3 of art of this ﬁeld and also a wonderful simultaneous axiomatic analysis of both convergence and optimality of adaptive ﬁnite element methods of several classes of linear and nonlinear problems. This paper aims at a uniﬁed convergence and optimality analysis of AMFEMs of the problem (1.1) in both two and three dimensional cases. The main result states that if the mixed ﬁnite element methods and associated a posteriori error estimates satisfy ﬁve hypotheses, see more de- tails in next section, the corresponding adaptive algorithms converge with optimal rates in the nonlinear approximate sense. The uniﬁed analysis is based on a new method to establish both discrete reliability and quasi-orthogonality, which are two main and indispensable ingredients for the convergence and optimality analysis of adaptive ﬁnite element methods. In fact, in contrary to [18, 6, 22, 20, 24, 29, 17], the discrete displacement space is endowed with a mesh-dependent discrete H1 norm, which deﬁnes one component of the new method. Hence the L2 norm becomes a natural norm for the discrete stress space. The other component of the new method is to introduce a pro- jection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the ﬁner mesh. As applications, the Raviart–Thomas and Brezzi–Douglas–Marini elements of any order of both the Poisson problem and the Stokes problem within the pseudostress-velocity formulation in 2D and 3D are proved to satisfy these ﬁve hypotheses. Therefore the corresponding adaptive schemes admit optimal convergence. As a result, it extends the optimal convergence re- sult for the ﬁrst order Raviart–Thomas element of the Stokes problem in 2D from [17] to the more general case. (cid:46) Throughout this paper, the notation a b represents that there exists a generic positive constant C, which is independent of the mesh parameter h and may not be the same at diﬀerent occurrences, ≤ ≈ (cid:46) (cid:46) such that a Cb. The symbol a b means a b a. · · · Let v, β = (β , , β )T and τ = (τ ) × be scalar, vector and tensor functions of two or three 1 d ij·d· · d · · · variables respectively, and(cid:18)let τi = (τi1, (cid:19), τid)T denote the ith row for τ with i = 1, , d. We deﬁne the grad, div, curl and rot operators by ∂v · · · ∂v T · · · grad v := , , , grad β := (grad β , , grad β )T , ∂x ∂x 1 d 1 d (cid:18) ∂β · · ·(cid:19) ∂β · · · div β := 1 + + d , div τ := (div τ , , div τ )T , ∂x ∂x 1 d 1 d ∂v − ∂v T curl v := , , curl β := (curl β , curl β )T , d = 2, ∂x ∂x 1 2 2 1 × curl β := grad β, curl τ := (curl τ , curl τ , curl τ )T , d = 3, 1 2 3 ∂β − ∂β rot β := 1 2 , rot τ := (rot τ , rot τ )T , d = 2. ∂x ∂x 1 2 2 1 Moreover, for the tensor function τ, we deﬁne its tangential component by · · · × × × × τ t := (τ t, τ t)T for d = 2, τ ν := (τ ν, τ ν, τ ν)T for d = 3. 1 2 1 2 3 ⊂ R R R × For a given Lebesgue measurable set G d, we use L2(G; ) or L2(G; d n) to denote the Hilbert space of square integrable functions or matrix-value ﬁelds, respectively, with inner product · · R R × ( , ) . Here and thereafter we will omit or d n for simplicity when there is no risk of L2(G) confusion. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUASI-OPTIMAL

text: CONVERGENCE

text: FOR

text: AMFEMS

text: of

text: art

text: of

text: this

text: ﬁeld

text: and

text: also

text: a

text: wonderful

text: simultaneous

text: axiomatic

text: analysis

text: of

text: both

text: convergence

text: and

text: optimality

text: of

text: adaptive

text: ﬁnite

text: element

text: methods

text: of

text: several

text: classes

text: of

text: linear

text: and

text: nonlinear

text: problems.

text: This

text: paper

text: aims

text: at

text: a

text: uniﬁed

text: convergence

text: and

text: optimality

text: analysis

text: of

text: AMFEMs

text: of

text: the

text: problem

text: in

text: both

text: two

text: and

text: three

text: dimensional

text: cases.

text: The

text: main

text: result

text: states

text: that

text: if

text: the

text: mixed

text: ﬁnite

text: element

text: methods

text: and

text: associated

text: a

text: posteriori

text: error

text: estimates

text: satisfy

text: ﬁve

text: hypotheses,

text: see

text: more

text: de-

text: tails

text: in

text: next

text: section,

text: the

text: corresponding

text: adaptive

text: algorithms

text: converge

text: with

text: optimal

text: rates

text: in

text: the

text: nonlinear

text: approximate

text: sense.

text: The

text: uniﬁed

text: analysis

text: is

text: based

text: on

text: a

text: new

text: method

text: to

text: establish

text: both

text: discrete

text: reliability

text: and

text: quasi-orthogonality,

text: which

text: are

text: two

text: main

text: and

text: indispensable

text: ingredients

text: for

text: the

text: convergence

text: and

text: optimality

text: analysis

text: of

text: adaptive

text: ﬁnite

text: element

text: methods.

text: In

text: fact,

text: in

text: contrary

text: to

text: the

text: discrete

text: displacement

text: space

text: is

text: endowed

text: with

text: a

text: mesh-dependent

text: discrete

text: norm,

text: which

text: deﬁnes

text: one

text: component

text: of

text: the

text: new

text: method.

text: Hence

text: the

text: norm

text: becomes

text: a

text: natural

text: norm

text: for

text: the

text: discrete

text: stress

text: space.

text: The

text: other

text: component

text: of

text: the

text: new

text: method

text: is

text: to

text: introduce

text: a

text: pro-

text: jection

text: operator

text: from

text: the

text: discrete

text: stress

text: space

text: on

text: the

text: coarser

text: mesh

text: onto

text: the

text: discrete

text: divergence

text: free

text: space

text: on

text: the

text: ﬁner

text: mesh.

text: As

text: applications,

text: the

text: Raviart–Thomas

text: and

text: Brezzi–Douglas–Marini

text: elements

text: of

text: any

text: order

text: of

text: both

text: the

text: Poisson

text: problem

text: and

text: the

text: Stokes

text: problem

text: within

text: the

text: pseudostress-velocity

text: formulation

text: in

text: and

text: are

text: proved

text: to

text: satisfy

text: these

text: ﬁve

text: hypotheses.

text: Therefore

text: the

text: corresponding

text: adaptive

text: schemes

text: admit

text: optimal

text: convergence.

text: As

text: a

text: result,

text: it

text: extends

text: the

text: optimal

text: convergence

text: re-

text: sult

text: for

text: the

text: ﬁrst

text: order

text: Raviart–Thomas

text: element

text: of

text: the

text: Stokes

text: problem

text: in

text: from

text: to

text: the

text: more

text: general

text: case.

text: Throughout

text: this

text: paper,

text: the

text: notation

text: a

text: b

text: represents

text: that

text: there

text: exists

text: a

text: generic

text: positive

text: constant

text: C,

text: which

text: is

text: independent

text: of

text: the

text: mesh

text: parameter

text: h

text: and

text: may

text: not

text: be

text: the

text: same

text: at

text: diﬀerent

text: occurrences,

text: such

text: that

text: a

text: Cb.

text: The

text: symbol

text: a

text: b

text: means

text: a

text: b

text: a.

text: Let

text: v,

text: β

text: (β

text: β

text: )T

text: and

text: τ

text: (τ

text: be

text: scalar,

text: vector

text: and

text: tensor

text: functions

text: of

text: two

text: or

text: three

text: d

text: ij·d·

text: d

text: variables

text: respectively,

text: τi

text: τid)T

text: denote

text: the

text: ith

text: row

text: for

text: τ

text: with

text: i

text: d.

text: We

text: deﬁne

text: the

text: grad,

text: div,

text: curl

text: and

text: rot

text: operators

text: by

text: ∂v

text: ∂v

text: T

text: grad

text: v

text: grad

text: β

text: (grad

text: β

text: grad

text: β

text: )T

text: ∂x

text: ∂x

text: d

text: d

text: ∂β

text: ∂β

text: div

text: β

text: d

text: div

text: τ

text: (div

text: τ

text: div

text: τ

text: )T

text: ∂x

text: ∂x

text: d

text: d

text: ∂v

text: ∂v

text: T

text: curl

text: v

text: curl

text: β

text: (curl

text: β

text: curl

text: β

text: )T

text: d

text: ∂x

text: ∂x

text: curl

text: β

text: grad

text: β,

text: curl

text: τ

text: (curl

text: τ

text: curl

text: τ

text: curl

text: τ

text: )T

text: d

text: ∂β

text: ∂β

text: rot

text: β

text: rot

text: τ

text: (rot

text: τ

text: rot

text: τ

text: )T

text: d

text: ∂x

text: ∂x

text: Moreover,

text: for

text: the

text: tensor

text: function

text: τ,

text: we

text: deﬁne

text: its

text: tangential

text: component

text: by

text: τ

text: t

text: (τ

text: t,

text: τ

text: t)T

text: for

text: d

text: τ

text: ν

text: (τ

text: ν,

text: τ

text: ν,

text: τ

text: ν)T

text: for

text: d

text: R

text: R

text: R

text: For

text: a

text: given

text: Lebesgue

text: measurable

text: set

text: G

text: d,

text: we

text: use

text: or

text: d

text: n)

text: to

text: denote

text: the

text: Hilbert

text: space

text: of

text: square

text: integrable

text: functions

text: or

text: matrix-value

text: ﬁelds,

text: respectively,

text: with

text: inner

text: product

text: R

text: R

text: Here

text: and

text: thereafter

text: we

text: will

text: omit

text: or

text: d

text: n

text: for

text: simplicity

text: when

text: there

text: is

text: no

text: risk

text: of

text: confusion.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: QUASI-OPTIMAL CONVERGENCE FOR AMFEMS 9 We need to estima(cid:18)te the term on the right hand s(cid:90)ide of (3.8). For(cid:19)the 2D case, an integration (cid:88) (cid:88) by parts plus (2.10) yield (cid:107) (cid:107) − A A · ξ 2A =  (rot( σ ), ψ) + σ t ψds  h (cid:18) H L2(K) H E (cid:19)  K(cid:88)∈R˜ E∈K(cid:88)∩EH E  1/2 (cid:46) (cid:107) A (cid:107) (cid:107) A · (cid:107) (cid:107) (cid:107) h rot( σ ) 2 + h1/2[ σ t ] 2 ξ . K H L2(K) E H E L2(E) h L2(Ω) ∈R ∈ ∩E K ˜ E K H Since div ξ = 0, it follows from (1.2) that h (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ξ ξ A, h L2(Ω) h (cid:18) (cid:19) which immediately imp(cid:88)lies (cid:88) (cid:107) (cid:107) (cid:46) (cid:107) A (cid:107) (cid:107) A · (cid:107) (3.9) ξ 2A h rot( σ ) 2 + h1/2[ σ t ] 2 . h K H L2(K) E H E L2(E) ∈R ∈ ∩E K ˜ E K H (cid:18) (cid:90) (cid:19) (cid:88) (cid:88) For the 3D case, a similar argument shows (cid:107) (cid:107) − A A × ξ 2A =  (curl( σ ), ψ) + σ ν ψds  h (cid:18) H L2(K) H E (cid:19)  K(cid:88)∈R˜ E∈K(cid:88)∩EH E  1/2 (cid:46) (cid:107) A (cid:107) (cid:107) A × (cid:107) (cid:107) (cid:107) h curl( σ ) 2 + h1/2[ σ ν ] 2 ξ A. K H L2(K) E H E L2(E) h ∈R ∈ ∩E K ˜ E K H (cid:18) (cid:19) Therefore, (cid:88) (cid:88) (cid:107) (cid:107) (cid:46) (cid:107) A (cid:107) (cid:107) A × (cid:107) (3.10) ξ 2A h curl( σ ) 2 + h1/2[ σ ν ] 2 . h K H L2(K) E H E L2(E) ∈R ∈ ∩E K ˜ E K H (cid:3) Finally, the desired result follows from (3.2), (3.5), (3.9) and (3.10). To establish quasi-orthogonality, we follow the idea of [22] to introduce the following problem: ∈ × Find (σ˜ , u˜ ) Σ U such that h h h h A − ∈ ( σ˜ , τ ) (div τ , u˜ ) = 0, for any τ Σ , h h L2(Ω) h h L2(Ω) h h (3.11) − ∈ (div σ˜ , v ) (f , v ) = 0, for any v U . h h L2(Ω) H h L2(Ω) h h ∈ Lemma 3.2. Given f L2(Ω), let (σ, u) be the solution of (1.1), (σ , u ) and (σ , u ) be the Th h T H H solutions to the discrete problem (2.1) over the nested triangulations and respectively, and h H let (σ˜ , u˜ ) be the solution of (3.11). Then h h A − − (3.12) ( (σ σ ), σ˜ σ ) = 0, h h H L2(Ω)

text: QUASI-OPTIMAL

text: CONVERGENCE

text: FOR

text: AMFEMS

text: We

text: need

text: to

text: the

text: term

text: on

text: the

text: right

text: hand

text: of

text: case,

text: an

text: integration

text: by

text: parts

text: plus

text: yield

text: A

text: A

text: ξ

text: (rot(

text: σ

text: ψ)

text: σ

text: t

text: ψds

text: h

text: H

text: H

text: E

text: E

text: A

text: A

text: h

text: rot(

text: σ

text: σ

text: t

text: ξ

text: K

text: H

text: E

text: H

text: E

text: h

text: ∈R

text: ∩E

text: K

text: E

text: K

text: H

text: Since

text: div

text: ξ

text: it

text: follows

text: from

text: that

text: h

text: ξ

text: ξ

text: A,

text: h

text: h

text: which

text: immediately

text: A

text: A

text: ξ

text: h

text: rot(

text: σ

text: σ

text: t

text: h

text: K

text: H

text: E

text: H

text: E

text: ∈R

text: ∩E

text: K

text: E

text: K

text: H

text: For

text: the

text: case,

text: a

text: similar

text: argument

text: shows

text: A

text: A

text: ξ

text: (curl(

text: σ

text: ψ)

text: σ

text: ν

text: ψds

text: h

text: H

text: H

text: E

text: E

text: A

text: A

text: h

text: curl(

text: σ

text: σ

text: ν

text: ξ

text: A.

text: K

text: H

text: E

text: H

text: E

text: h

text: ∈R

text: ∩E

text: K

text: E

text: K

text: H

text: Therefore,

text: A

text: A

text: ξ

text: h

text: curl(

text: σ

text: σ

text: ν

text: h

text: K

text: H

text: E

text: H

text: E

text: ∈R

text: ∩E

text: K

text: E

text: K

text: H

text: Finally,

text: the

text: desired

text: result

text: follows

text: from

text: and

text: To

text: establish

text: quasi-orthogonality,

text: we

text: follow

text: the

text: idea

text: of

text: to

text: introduce

text: the

text: following

text: problem:

text: Find

text: (σ˜

text: u˜

text: Σ

text: U

text: such

text: that

text: h

text: h

text: h

text: h

text: A

text: σ˜

text: τ

text: (div

text: τ

text: u˜

text: for

text: any

text: τ

text: Σ

text: h

text: h

text: h

text: h

text: h

text: h

text: (div

text: σ˜

text: v

text: (f

text: v

text: for

text: any

text: v

text: U

text: h

text: h

text: H

text: h

text: h

text: h

text: Lemma

text: Given

text: f

text: let

text: (σ,

text: u)

text: be

text: the

text: solution

text: of

text: (σ

text: u

text: and

text: (σ

text: u

text: be

text: the

text: Th

text: h

text: T

text: H

text: H

text: solutions

text: to

text: the

text: discrete

text: problem

text: over

text: the

text: nested

text: triangulations

text: and

text: respectively,

text: and

text: h

text: H

text: let

text: (σ˜

text: u˜

text: be

text: the

text: solution

text: of

text: Then

text: h

text: h

text: A

text: (σ

text: σ

text: σ˜

text: σ

text: h

text: h

text: H