text: PREPRINT SUBMITTED TO IEEE TRANSACTIONS ON MOBILE COMPUTING, 2018 8 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 (a) captioncolorSAPHE (b) captioncolorWalkie-Talkie 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 (c) captioncolorBANDANA (d) captioncolorIPI Fig. 8: captioncolorDistribution of p-values achieved for keys after 20 runs of the DieHarder set of statistical tests. Tests are: (1) birthdays (2) operm5 (3) rank32x32 (4) rank6x8 (5) bitstream (6) opso (7) oqso (8) dna (9) count-1s-str (10) count-1s-byt (11) parking (12) 2D circle (13) 3D sphere (14) squeeze (15) runs (16) craps (17) marsaglia (18) sts monobit (19) sts runs (20) sts serial [1-16] (21) rgb bitdistr. [1-12] (22) rgb min dist. [2-5] (23) rgb perm. [2-5] (24) rgb lagged sum [0-32] (25) rgb kstest (26) dab bytedistr. (27) dab dct (28) dab ﬁlltree (29) dab ﬁlltree 2 (30) dab monobit 2 TABLE 2: Results for keys generated by the evaluated protocols after error correction, and ﬁnally agree on a key. An obvious running the ENT Pseudorandom Number Sequence Test Program. attack surface is exposed by the sensors themself (A). A device or its owner could be forced to behave in a certain SAPHE Walkie-Talkie BANDANA IPI way, e.g., by an adversary controlling stride speed with a Sequence size (bit) 1444864 7758808 113792 456104 Entropy (bits per bit) 0.9999 0.9958 0.9999 0.8929 treadmill. Depending on the pairing protocol, it could be Optimum compression rate 0% 0 % 0% 10% also possible to bypass the sensor data acquisition (B) and Chi square distribution 6.91 44993.29 0.3586 65969.75 Arithmetic mean (random=.5) 0.501094 0.461924 0.5 0.690156 reuse data from the past. With a biased quantization, a Monte Carlo value for Pi (er- 3.122155 3.084985 3.642194 2.056830 naïve brute force attack would become feasible (C). Some ror) protocols employ a special communication phase before the Serial correlation coefﬁcient 0.008204 0.080251 -0.644796 -0.002701 (uncorrelated=0.0) actual key agreement (SAPHE: random seed and distance ordering, Walkie-Talkie: reconciliation phase, BANDANA: exchange of reliability indices). The exchanged indices could A B C potentially be exploited as a side channel (D). After error correction (e.g. in BANDANA and IPI), the key agreement is Pre-Processing Error Sensors +Quantization Correction executed between both participants. Here, the risk of a Man- in-the-Middle (MitM) attack (E) or impersonation attack on E one participant (G) must be considered. Finally, the key Quantization Key D Protocol Agreement F agreement itself could be weak or based on false security assumptions, especially if it has been designed for this G protocol only and is not based on established standards (F). Sensors Pre-Processing Error In the following, we will discuss attacks we found during +Quantization Correction our analysis. No discussion on attack vector B is included as it assumes that the device is already compromised by Fig. 9: captioncolorConceptual view of gait-based pairing protocols malware, which falls outside the focus of this work. with attack vectors (blue line depicts device boundary, dashed parts are optional) sectioncolor 5.1 One-Shot Success Probability (E, G) Without requiring additional knowledge about the vic- 5 SECURITY ANALYSIS C tim’s gait, an attacker may want to exhaust the keyspace of As in most security applications, although much effort all keys k to execute a MitM (E) or impersonation attack (G). against it may have been taken, unwanted vulnerabilities However, in all discussed protocols, after each single try, a might anyway sustain within an approach’s conceptual de- completely new authentication process (new k independent sign. In [55], [68], a general concept to analyse security appli- from theCprevious one) is started. Thus, it is impossible to cations and to identify possible pitfalls has been introduced. exhaust making this a one-shot attack. For comparison Furthermore, adversaries of different strength necessitate between protocols, we target the same length of 16 bit for k. the weighting of the impact of certain weaknesses. Figure 9 The length of sequences sampled for a target key k of 16 bit shows such a conceptual view for sensor-based device-to- may vary depending on the quantization scheme. device pairing including possible attack vectors A-G. A pair sectioncolor 5.1.1 Candidate Key Protocol Variants of devices measure data from sensors, quantize the data to bit strings after some preprocessing, apply potentially The candidate key protocol is, for instance, realized ##LTFigure## ##LTFigure## ##LTFigure## ##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## 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text: PREPRINT

text: SUBMITTED

text: TO

text: IEEE

text: TRANSACTIONS

text: ON

text: MOBILE

text: COMPUTING,

text: (a)

text: captioncolorSAPHE

text: (b)

text: captioncolorWalkie-Talkie

text: (c)

text: captioncolorBANDANA

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text: captioncolorIPI

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text: (cid:88) The reminder term R (T) satisﬁes the following estimate 1 | | ≤ (cid:107) (cid:107) (cid:107)∇ (cid:107) R (T) Ch2 u v . 1 3 d b 0 ∈T T h Proof. From (6.6) and the structure of χ , we have i (cid:104)∇ · − Q ∇ · S − (cid:105) u(cid:90) n u (cid:90)n, (v ) v (cid:90) (cid:90) h b b ∂T (cid:104)∇ · S − (cid:105) = u n, (v ) v b b ∂T (cid:90)− (cid:90) − (6.11) = u χ dy + u χ dy u χ dx + u χ dx x 1 x 1 y 2 y 2 e1 e2 e3 e4 = u χ dT + u χ dT. xx 1 yy 2 T T Since χ is linear in the y-direction and constant in the x-direction, then 1 − χ (y) = χ (M ) + (y y )∂ χ . 1 1 1 c y 1 Thus, we have (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) − u χ dT = u χ (M )dT + u (y y )∂ χ dT xx 1 xx 1 1 xx c y 1 T T T = u χ (M )dT + u E (y)∂ χ dT, xx 1 1 xxy 3 y 1 T T | | − − where E (y) = 1 e 2 1(y y )2. 3 8 1 2 c Similarly, one may derive the following (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) − u χ dT = u χ (M )dT + u (x x )∂ χ dT yy 2 yy 2 3 yy c x 2 T T T = u χ (M )dT + u E (x)∂ χ dT, yy 2 3 yyx 4 x 2 T T | | − − where E (x) = 1 e 2 1(x x )2. 4 8 3 2 c Substituting the last two formulas into (6.11) yields (cid:90) (cid:90) (cid:104)∇ · − Q ∇ · S − (cid:105) u n u n, (v ) v h b b ∂T (cid:90) (cid:90) = u dTχ (M ) + u dTχ (M ) (6.12) xx 1 1 yy 2 3 T T + u E (y)dT∂ χ + u E (x)dT∂ χ . xxy 3 y 1 yyx 4 x 2 T T 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: The

text: reminder

text: term

text: R

text: (T)

text: satisﬁes

text: the

text: following

text: estimate

text: R

text: (T)

text: u

text: v

text: d

text: b

text: ∈T

text: T

text: h

text: Proof.

text: From

text: and

text: the

text: structure

text: of

text: χ

text: we

text: have

text: i

label: equation

text: Q

text: S

text: n

text: u

text: (v

text: v

text: h

text: b

text: b

text: ∂T

text: S

text: u

text: n,

text: (v

text: v

text: b

text: b

text: ∂T

text: u

text: χ

text: dy

text: u

text: χ

text: dy

text: u

text: χ

text: dx

text: u

text: χ

text: dx

text: x

text: x

text: y

text: y

text: u

text: χ

text: dT

text: u

text: χ

text: dT.

text: xx

text: yy

text: T

text: T

text: Since

text: χ

text: is

text: linear

text: in

text: the

text: y-direction

text: and

text: constant

text: in

text: the

text: x-direction,

text: then

text: χ

text: (y)

text: χ

text: (M

text: (y

text: y

text: χ

text: c

text: y

text: Thus,

text: we

text: have

text: u

text: χ

text: dT

text: u

text: χ

text: (M

text: )dT

text: u

text: (y

text: y

text: χ

text: dT

text: xx

text: xx

text: xx

text: c

text: y

text: T

text: T

text: T

text: u

text: χ

text: (M

text: )dT

text: u

text: E

text: (y)∂

text: χ

text: dT,

text: xx

text: xxy

text: y

text: T

text: T

text: where

text: E

text: (y)

text: e

text: y

text: c

text: Similarly,

text: one

text: may

text: derive

text: the

text: following

text: u

text: χ

text: dT

text: u

text: χ

text: (M

text: )dT

text: u

text: (x

text: x

text: χ

text: dT

text: yy

text: yy

text: yy

text: c

text: x

text: T

text: T

text: T

text: u

text: χ

text: (M

text: )dT

text: u

text: E

text: (x)∂

text: χ

text: dT,

text: yy

text: yyx

text: x

text: T

text: T

text: where

text: E

text: (x)

text: e

text: x

text: c

text: Substituting

text: the

text: last

text: two

text: formulas

text: into

text: yields

text: Q

text: S

text: u

text: n

text: u

text: n,

text: (v

text: v

text: h

text: b

text: b

text: ∂T

text: u

text: dTχ

text: (M

text: u

text: dTχ

text: (M

text: xx

text: yy

text: T

text: T

text: u

text: E

text: (y)dT∂

text: χ

text: u

text: E

text: (x)dT∂

text: χ

text: xxy

text: y

text: yyx

text: x

text: T

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Now using the Euler-MacLaurin formula we arrive at (cid:90) (cid:90) | | | | e e (Q u(M ) + Q u(M )) 1 3 b 1 b 2 | (cid:90)| | (cid:90)| (cid:90) = e u(x − , y)dy + e u(x , y)dy (6.17) 3 i 1 3 i e1 e2 | | | | = 2 u(x, y)dT + A e 2 u dT + A e 3 u E (x)dT, 1 3 xx 2 3 xxx 1 T T T where A = 1/6 and A is another constant, and E is a cubic polynomial in the 1 2 1 x-direction. Analogously, we have (cid:90) (cid:90) | | | | e e (Q u(M ) + Q u(M )) 1 3 b 3 b 4 | (cid:90)| | (cid:90)| (cid:90) = e u(x, y − )dx + e u(x, y )dx (6.18) 1 j 1 1 j e3 e4 | | | | = 2 u(x, y)dT + A e 2 u dT + A e 3 u E (y)dT, 1 1 yy 2 1 yyy 2 T T T where E is a cubic polynomial in the y-direction. 2 Substituting (6.17) and (6.18) into (6.16) yields (cid:16) (cid:90) (cid:90) − (cid:104) S − S − (cid:105) ρh 1 Q (Q u) Q u, Q (v ) v b b b b b b ∂T − − | | (cid:90) | | | |− − |(cid:90) || |− (cid:17) = ρh 1 e χ (M ) A e e 1 u dT A e e 1 u dT 1 1 1 1 3 1 xx 1 1 3 yy T T | | |(cid:18)|− (cid:90) − | | | |−(cid:90) (cid:19) + A e 2 e 1 u E (x)dT A e 2 e 1 u E (y)dT 2 3 1 xxx 1 2 1 3 yyy 2 T T − − | | | | = ρh 1A e χ (M ) u dT + e χ (M ) u dT + R (T), 1 3 1 1 xx 1 2 3 yy 2 T T | | −| | where we have used the relation e χ (M ) = e χ (M ), and R is given by 1 (cid:18)1 1 3(cid:90) 2 3 2 − − | (cid:90)| (cid:19) R (T) = A ρh 1 e 2χ (M ) u E (x)dT 2 2 3 1 1 xxx 1 (6.19) T | | + e 2χ (M ) u E (y)dT , 1 2 3 yyy 2 T which can be seen to satisfy (6.14). We are now in a position to state the main result of this paper regarding super- convergence. For simplicity, we consider the homogeneous Dirichlet boundary value 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Now

text: using

text: the

text: Euler-MacLaurin

text: formula

text: we

text: arrive

text: at

text: e

text: e

text: (Q

text: u(M

text: Q

text: u(M

text: b

text: b

text: e

text: u(x

text: y)dy

text: e

text: u(x

text: y)dy

text: i

text: i

text: u(x,

text: y)dT

text: A

text: e

text: u

text: dT

text: A

text: e

text: u

text: E

text: (x)dT,

text: xx

text: xxx

text: T

text: T

text: T

text: where

text: A

text: and

text: A

text: is

text: another

text: constant,

text: and

text: E

text: is

text: a

text: cubic

text: polynomial

text: in

text: the

text: x-direction.

text: Analogously,

text: we

text: have

text: e

text: e

text: (Q

text: u(M

text: Q

text: u(M

text: b

text: b

text: e

text: u(x,

text: y

text: )dx

text: e

text: u(x,

text: y

text: )dx

text: j

text: j

text: u(x,

text: y)dT

text: A

text: e

text: u

text: dT

text: A

text: e

text: u

text: E

text: (y)dT,

text: yy

text: yyy

text: T

text: T

text: T

text: where

text: E

text: is

text: a

text: cubic

text: polynomial

text: in

text: the

text: y-direction.

text: Substituting

text: and

text: into

text: yields

text: S

text: S

text: ρh

text: Q

text: (Q

text: u)

text: Q

text: u,

text: Q

text: (v

text: v

text: b

text: b

text: b

text: b

text: b

text: b

text: ∂T

text: ρh

text: e

text: χ

text: (M

text: A

text: e

text: e

text: u

text: dT

text: A

text: e

text: e

text: u

text: dT

text: xx

text: yy

text: T

text: T

text: A

text: e

text: e

text: u

text: E

text: (x)dT

text: A

text: e

text: e

text: u

text: E

text: (y)dT

text: xxx

text: yyy

text: T

text: T

text: ρh

text: e

text: χ

text: (M

text: u

text: dT

text: e

text: χ

text: (M

text: u

text: dT

text: R

text: (T),

text: xx

text: yy

text: T

text: T

text: where

text: we

text: have

text: used

text: the

text: relation

text: e

text: χ

text: (M

text: e

text: χ

text: (M

text: and

text: R

text: is

text: given

text: by

text: R

text: (T)

text: A

text: ρh

text: e

text: 2χ

text: (M

text: u

text: E

text: (x)dT

text: xxx

text: T

text: e

text: 2χ

text: (M

text: u

text: E

text: (y)dT

text: yyy

text: T

text: which

text: can

text: be

text: seen

text: to

text: satisfy

text: We

text: are

text: now

text: in

text: a

text: position

text: to

text: state

text: the

text: main

text: result

text: of

text: this

text: paper

text: regarding

text: super-

text: convergence.

text: For

text: simplicity,

text: we

text: consider

text: the

text: homogeneous

text: Dirichlet

text: boundary

text: value

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 7.6 Test Case 2: Convergence of the lowest order WG-FEM on the unit square domain with exact solution u = sin(x)cos(y), uniform rectangular partitions, stabilization parameter ρ = 1, mesh parameter h = max(hx,hy), and perturbed L2 projection of the Dirichlet boundary data by (6.34)- (6.35). (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 1.1667e-02 8.3456e-03 1.4734e-02 1.1884e-02 1.4008e-02 1/8 3.1447e-03 2.1001e-03 3.6994e-03 2.9877e-03 3.5184e-03 1/16 8.1077e-04 5.2619e-04 9.2630e-04 7.4845e-04 8.8109e-04 1/32 2.0539e-04 1.3163e-04 2.3168e-04 1.8722e-04 2.2038e-04 1/64 5.1652e-05 3.2912e-05 5.7926e-05 4.6813e-05 5.5101e-05 1/128 1.2949e-05 8.2283e-06 1.4482e-05 1.1704e-05 1.3776e-05 1/256 3.2415e-06 2.0571e-06 3.6205e-06 2.9259e-06 3.4440e-06 Rate 2.00 2.00 2.00 2.00 2.00 Tables 7.7 and 7.8 show the performance of the WG ﬁnite element scheme (3.1) when the exact L2 projection of the boundary data is employed in (3.1). On uniform square partitions, the numerical solutions are seen to be convergent at the rate of r = 2 in the discrete H1 norm as shown in Table 7.7. It should be pointed out that on uniform square partitions, one may carry out the analysis further to derive a superconvergence with the full rate of r = 2 if the boundary data satisﬁes u = xx u , which is the case for Test Case 2. On the other hand, Table 7.8 illustrates a yy convergence at a rate lower than r = 2 on uniform rectangular partitions. The result with rectangular partitions reveals a sub-optimal order of superconvergence for the scheme (3.1) when the boundary value is approximated by the L2 projection or the usual nodal point interpolation. This sub-optimal order can be theoretically justiﬁed to be r = 1.5 with proper regularity assumptions on the exact solution, though the computation indicates a superconvergence with an order around r = 1.9. Table 7.7 Test Case 2: Convergence of the lowest order WG-FEM on the unit square domain with exact solution u = sin(x)cos(y), uniform square partitions, stabilization parameter ρ = 1, mesh parameter h = max(hx,hy), and L2 projection of the Dirichlet boundary data. (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 7.8025e-03 3.4806e-03 8.0885e-04 3.9540e-03 1.0155e-03 1/8 2.0827e-03 8.7414e-04 2.0366e-04 9.8946e-04 2.5557e-04 1/16 5.3530e-04 2.1884e-04 5.1091e-05 2.4754e-04 6.4140e-05 1/32 1.3547e-04 5.4730e-05 1.2788e-05 6.1899e-05 1.6056e-05 1/64 3.4060e-05 1.3684e-05 3.1980e-06 1.5476e-05 4.0154e-06 1/128 8.5379e-06 3.4210e-06 7.9956e-07 3.8690e-06 1.0039e-06 1/256 2.1372e-06 8.5526e-07 1.9989e-07 9.6726e-07 2.5111e-07 Rate 2.00 2.00 2.00 2.00 2.00 Tables 7.9-7.11 illustrate the numerical performance of the lowest order WG-FEM on uniform square or rectangular partitions with the stabilization parameter ρ = 6. The results are similar to the case of ρ = 1. The following conclusions seem to be appropriate from Test Case 2: • If the Dirichlet boundary value is approximated by the modiﬁed L2 projec- tion, the theory-predicted superconvergence of order 2 by Corollary 6.6 is computationally valid for the WG scheme (3.1). • If the Dirichlet boundary value is approximated by the exact L2 projection, 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Table

text: Test

label: caption

text: Case

text: Convergence

text: of

text: the

text: lowest

text: order

text: WG-FEM

text: on

text: the

text: unit

text: square

text: domain

text: with

text: exact

text: solution

text: u

text: sin(x)cos(y),

text: uniform

text: rectangular

text: partitions,

text: stabilization

text: parameter

text: ρ

text: mesh

text: parameter

text: h

text: max(hx,hy),

text: and

text: perturbed

text: projection

text: of

text: the

text: Dirichlet

text: boundary

text: data

text: by

label: table

text: S

text: S

text: S

text: h

text: u

text: (ub)

text: u

text: (ub)

text: d(eb)

text: dub

text: u

text: (ub))

text: Rate

text: Tables

text: and

text: show

text: the

text: performance

text: of

text: the

text: WG

text: ﬁnite

text: element

text: scheme

text: when

text: the

text: exact

text: projection

text: of

text: the

text: boundary

text: data

text: is

text: employed

text: in

text: On

text: uniform

text: square

text: partitions,

text: the

text: numerical

text: solutions

text: are

text: seen

text: to

text: be

text: convergent

text: at

text: the

text: rate

text: of

text: r

text: in

text: the

text: discrete

text: norm

text: as

text: shown

text: in

text: Table

text: It

text: should

text: be

text: pointed

text: out

text: that

text: on

text: uniform

text: square

text: partitions,

text: one

text: may

text: carry

text: out

text: the

text: analysis

text: further

text: to

text: derive

text: a

text: superconvergence

text: with

text: the

text: full

text: rate

text: of

text: r

text: if

text: the

text: boundary

text: data

text: satisﬁes

text: u

text: xx

text: u

text: which

text: is

text: the

text: case

text: for

text: Test

text: Case

text: On

text: the

text: other

text: hand,

text: Table

text: illustrates

text: a

text: yy

text: convergence

text: at

text: a

text: rate

text: lower

text: than

text: r

text: on

text: uniform

text: rectangular

text: partitions.

text: The

text: result

text: with

text: rectangular

text: partitions

text: reveals

text: a

text: sub-optimal

text: order

text: of

text: superconvergence

text: for

text: the

text: scheme

text: when

text: the

text: boundary

text: value

text: is

text: approximated

text: by

text: the

text: projection

text: or

text: the

text: usual

text: nodal

text: point

text: interpolation.

text: This

text: sub-optimal

text: order

text: can

text: be

text: theoretically

text: justiﬁed

text: to

text: be

text: r

text: with

text: proper

text: regularity

text: assumptions

text: on

text: the

text: exact

text: solution,

text: though

text: the

text: computation

text: indicates

text: a

text: superconvergence

text: with

text: an

text: order

text: around

text: r

text: Table

text: Test

text: Case

text: Convergence

text: of

text: the

text: lowest

text: order

text: WG-FEM

text: on

text: the

text: unit

text: square

text: domain

text: with

text: exact

text: solution

text: u

text: sin(x)cos(y),

text: uniform

text: square

text: partitions,

text: stabilization

text: parameter

text: ρ

text: mesh

text: parameter

text: h

text: max(hx,hy),

text: and

text: projection

text: of

text: the

text: Dirichlet

text: boundary

text: data.

text: S

text: S

text: S

text: h

text: u

text: (ub)

text: u

text: (ub)

text: d(eb)

text: dub

text: u

text: (ub))

text: Rate

text: Tables

text: illustrate

text: the

text: numerical

text: performance

text: of

text: the

text: lowest

text: order

text: WG-FEM

text: on

text: uniform

text: square

text: or

text: rectangular

text: partitions

text: with

text: the

text: stabilization

text: parameter

text: ρ

text: The

text: results

text: are

text: similar

text: to

text: the

text: case

text: of

text: ρ

text: The

text: following

text: conclusions

text: seem

text: to

text: be

text: appropriate

text: from

text: Test

text: Case

text: If

text: the

text: Dirichlet

text: boundary

text: value

text: is

text: approximated

text: by

text: the

text: modiﬁed

text: projec-

text: tion,

text: the

text: theory-predicted

text: superconvergence

text: of

text: order

text: by

text: Corollary

text: is

text: computationally

text: valid

text: for

text: the

text: WG

text: scheme

text: If

text: the

text: Dirichlet

text: boundary

text: value

text: is

text: approximated

text: by

text: the

text: exact

text: projection,

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##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Table 7.8 Test Case 2: Convergence of the lowest order WG-FEM on the unit square domain with exact solution u = sin(x)cos(y), uniform rectangular partitions, stabilization parameter ρ = 1, mesh parameter h = max(hx,hy), and L2 projection of the Dirichlet boundary data. (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 5.2855e-03 2.2786e-03 6.3200e-03 6.8392e-03 6.3871e-03 1/8 1.5497e-03 5.7273e-04 2.0952e-03 2.1923e-03 2.1099e-03 1/16 4.3514e-04 1.4374e-04 6.1241e-04 6.3312e-04 6.1571e-04 1/32 1.1742e-04 3.5992e-05 1.7103e-04 1.7568e-04 1.7179e-04 1/64 3.0790e-05 9.0027e-06 4.6707e-05 4.7771e-05 4.6880e-05 1/128 7.9255e-06 2.2511e-06 1.2579e-05 1.2826e-05 1.2619e-05 1/256 2.0190e-06 5.6279e-07 3.3544e-06 3.4124e-06 3.3640e-06 Rate 1.97 2.00 1.91 1.91 1.91 the numerical solutions of (3.1) do not have a full rate of convergence at r = 2, ≈ but a convergence at a lower rate of r 1.9 is observed numerically. Table 7.9 Test Case 2: Convergence of the lowest order WG-FEM on the unit square domain with exact solution u = sin(x)cos(y), uniform square partitions, stabilization parameter ρ = 6, mesh parameter h = (hx + hy)/2, perturbed L2 projection of the Dirichlet boundary data by (6.34)-(6.35). (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 7.8248e-03 3.4864e-03 7.9599e-04 3.9619e-03 1.0135e-03 1/8 2.0842e-03 8.7474e-04 2.0322e-04 9.9044e-04 2.5604e-04 1/16 5.3544e-04 2.1888e-04 5.1080e-05 2.4762e-04 6.4192e-05 1/32 1.3548e-04 5.4733e-05 1.2788e-05 6.1904e-05 1.6060e-05 1/64 3.4060e-05 1.3684e-05 3.1980e-06 1.5476e-05 4.0156e-06 1/128 8.5379e-06 3.4210e-06 7.9956e-07 3.8690e-06 1.0040e-06 1/256 2.1373e-06 8.5526e-07 1.9989e-07 9.6726e-07 2.5111e-07 Rate 2.00 2.00 2.00 2.00 2.00 Table 7.10 Test Case 2: Convergence of the lowest order WG-FEM on the unit square domain with exact solution u = sin(x)cos(y), uniform rectangular partitions, stabilization parameter ρ = 6, h = (hx + hy)/2, and perturbed L2 projection of the Dirichlet boundary data by (6.34)-(6.35). (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 2.08e-01 5.6939e-03 2.4689e-03 7.5368e-04 2.5643e-03 6.6175e-04 1.04e-01 1.5092e-03 6.1891e-04 1.9018e-04 6.4106e-04 1.6697e-04 5.21e-02 3.8698e-04 1.5483e-04 4.7657e-05 1.6027e-04 4.1846e-05 2.60e-02 9.7868e-05 3.8715e-05 1.1921e-05 4.0067e-05 1.0468e-05 1.30e-02 2.4601e-05 9.6791e-06 2.9808e-06 1.0017e-05 2.6174e-06 6.51e-03 6.1664e-06 2.4198e-06 7.4522e-07 2.5042e-06 6.5438e-07 3.26e-03 1.5436e-06 6.0495e-07 1.8630e-07 6.2605e-07 1.6451e-07 Rate 2.00 2.00 2.00 2.00 2.00 Test Case 3 (Nonhomogeneous BVP): The model problem in Test Case 3 has exact solution u = exp(x) sin(y) on the unit square domain. Unlike Test Case 2, the boundary data for Test Case 3 does not satisfy u = u so that no superconvergence xx yy of full order of r = 2 is predicted by the theory when the usual L2 projection is applied for the boundary data. 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text: numerical scheme (3.1). The results are similar to those on square partitions. Most importantly, Table 7.15 provides a numerical validation of the superconvergence the- ory developed in Corollary 6.6. Table 7.14 Test Case 3: convergence of the lowest order WG-FEM on the unit square domain with exact solution u = exp(x)sin(y), uniform rectangular partitions, stabilization parameter ρ = 1, meshsize { } h = max hx,hy , and L2 projection of the boundary data. (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 1.6184e-02 1.0739e-02 7.9121e-02 7.8678e-02 7.9028e-02 1/8 6.8896e-03 3.1242e-03 2.6904e-02 2.6796e-02 2.6881e-02 1/16 2.3176e-03 8.2994e-04 8.1057e-03 8.0813e-03 8.1005e-03 1/32 6.7670e-04 2.1190e-04 2.3194e-03 2.3139e-03 2.3182e-03 1/64 1.8508e-04 5.3335e-05 6.4474e-04 6.4349e-04 6.4447e-04 1/128 4.8913e-05 1.3362e-05 1.7591e-04 1.7563e-04 1.7585e-04 1/256 1.2678e-05 3.3426e-06 4.7374e-05 4.7308e-05 4.7361e-05 Rate 1.95 2.00 1.89 1.89 1.89 Table 7.15 Test Case 3: convergence of the lowest order WG-FEM on the unit square domain with exact solution u = exp(x)sin(y), uniform rectangular partitions, stabilization parameter ρ = 1, meshsize { } h = max hx,hy , and perturbed L2 projection of the boundary data by (6.34)-(6.35). (cid:107) − S (cid:107) (cid:107) − S (cid:107) (cid:107)∇ (cid:107) (cid:107)∇ − ∇ (cid:107) (cid:107)∇ − S (cid:107) h u (ub) ∞,(cid:63) u (ub) 0 d(eb) 0 dub u 0,(cid:63) (Q0u (ub)) 0 1/4 1.0775e-02 8.5720e-03 4.2529e-02 4.1088e-02 4.3546e-02 1/8 3.0986e-03 2.1106e-03 1.0669e-02 1.0307e-02 1.0923e-02 1/16 8.2422e-04 5.2608e-04 2.6702e-03 2.5797e-03 2.7340e-03 1/32 2.1187e-04 1.3144e-04 6.6775e-04 6.4512e-04 6.8370e-04 1/64 5.3652e-05 3.2854e-05 1.6695e-04 1.6129e-04 1.7094e-04 1/128 1.3495e-05 8.2132e-06 4.1739e-05 4.0324e-05 4.2736e-05 1/256 3.3836e-06 2.0532e-06 1.0435e-05 1.0081e-05 1.0683e-05 Rate 2.00 2.00 2.00 2.00 2.00 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: numerical

text: scheme

text: The

text: results

text: are

text: similar

text: to

text: those

text: on

text: square

text: partitions.

text: Most

text: importantly,

text: Table

text: provides

text: a

text: numerical

text: validation

text: of

text: the

text: superconvergence

text: the-

text: ory

text: developed

text: in

text: Corollary

text: Table

text: Test

text: Case

text: convergence

text: of

text: the

text: lowest

text: order

text: WG-FEM

text: on

text: the

text: unit

text: square

text: domain

text: with

text: exact

text: solution

text: u

text: exp(x)sin(y),

text: uniform

text: rectangular

text: partitions,

text: stabilization

text: parameter

text: ρ

text: meshsize

text: h

text: max

text: hx,hy

text: and

text: projection

text: of

text: the

text: boundary

text: data.

text: S

text: S

text: S

text: h

text: u

text: (ub)

text: u

text: (ub)

text: d(eb)

text: dub

text: u

text: (ub))

text: Rate

text: Table

text: Test

text: Case

text: convergence

text: of

text: the

text: lowest

text: order

text: WG-FEM

text: on

text: the

text: unit

text: square

text: domain

text: with

text: exact

text: solution

text: u

text: exp(x)sin(y),

text: uniform

text: rectangular

text: partitions,

text: stabilization

text: parameter

text: ρ

text: meshsize

text: h

text: max

text: hx,hy

text: and

text: perturbed

text: projection

text: of

text: the

text: boundary

text: data

text: by

text: S

text: S

text: S

text: h

text: u

text: (ub)

text: u

text: (ub)

text: d(eb)

text: dub

text: u

text: (ub))

text: Rate

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text: The exact dCS metric for a spinning BH possesses a fourth constant of motion: ...13 With the background spacetime, the equations of motion and an integration scheme deﬁned, we now proceed to study the dynamics of test particles in detail and compute Poincar´e surfaces of section and rotation curves. We will split our analysis into a study of bounded orbits and one of unbounded orbits. 4.1. Bound Orbits For bounded motion, the test particle is conﬁned to a particular region of the eﬀective potential. Within this region, there is a local minimum, depicted in detail in the right-most embedded diagrams of the left panel of Fig. 4, implying the existence of stable periodic orbits. In the same sense as in Newtonian gravity, the angular momentum of the test particle is responsible for a centrifugal barrier that prevents the particle from falling into the central object. Here, since the SMBH background is spinning, there is an extra contribution to the centrifugal barrier due to the BH spin. Bearing in mind the discussion in Sec. about perturbed systems, we expect geodesics of a slowly-rotating Kerr metric to be chaotic. Figure 5 shows Poincar´e surfaces of section for two diﬀerent sets of parameters that produce bounded orbits, O using the slowly-rotating metric to (χ2). The surfaces of section look regular, seemingly without any signatures of chaos, but this can be deceiving because features of chaos may be small and hard to resolve on this scale. Figure 5. Poincar´e surfaces of section for a SMBH with χ = 0.1, E = 0.98 and angular momentum Lz = 3.74265M (left), and χ = 0.2, E = 0.995 and angular momentumOLz = 3.75365M (right), using the slow-rotation expansion of the Kerr metric to (χ2). The rotation curve, however, can signal the presence of chaotic behavior, even when this cannot be resolved with the naked eye from the surfaces of section. The left panel of Fig. 6 shows the rotation curve for the same sets of orbits as those in the right-panel of Fig. 5. Observe that there is a plateau in the rotation number around ≈ r 4.016M. In practice, such behavior is found by studying the derivative of the rotation number with respect to radius. This plateau is a tell-tale sign of chaos, which we can use to reﬁne our search to a subregion of the surfaces of section. Doing so, we ﬁnd Birkhoﬀ chains of islands in the surfaces of section, as shown in the right panel of Fig. 6. The emergence of Birkhoﬀ islands is generic and not dependent on the particular parameters of the geodesic or of the SMBH. The left-panel of Fig. 7 is a zoom of the left-panel of Fig. 5 to a regime where the rotation number suggests the presence ##LTFigure## ##LTFigure##

text: The

text: exact

text: dCS

text: metric

text: for

text: a

text: spinning

text: BH

text: possesses

text: a

text: fourth

text: constant

text: of

text: motion:

text: With

text: the

text: background

text: spacetime,

text: the

text: equations

text: of

text: motion

text: and

text: an

text: integration

text: scheme

text: deﬁned,

text: we

text: now

text: proceed

text: to

text: study

text: the

text: dynamics

text: of

text: test

text: particles

text: in

text: detail

text: and

text: compute

text: Poincar´e

text: surfaces

text: of

text: section

text: and

text: rotation

text: curves.

text: We

text: will

text: split

text: our

text: analysis

text: into

text: a

text: study

text: of

text: bounded

text: orbits

text: and

text: one

text: of

text: unbounded

text: orbits.

label: section

text: Bound

text: Orbits

text: For

text: bounded

text: motion,

text: the

text: test

text: particle

text: is

text: conﬁned

text: to

text: a

text: particular

text: region

text: of

text: the

text: eﬀective

text: potential.

text: Within

text: this

text: region,

text: there

text: is

text: a

text: local

text: minimum,

text: depicted

text: in

text: detail

text: in

text: the

text: right-most

text: embedded

text: diagrams

text: of

text: the

text: left

text: panel

text: of

text: Fig.

text: implying

text: the

text: existence

text: of

text: stable

text: periodic

text: orbits.

text: In

text: the

text: same

text: sense

text: as

text: in

text: Newtonian

text: gravity,

text: the

text: angular

text: momentum

text: of

text: the

text: test

text: particle

text: is

text: responsible

text: for

text: a

text: centrifugal

text: barrier

text: that

text: prevents

text: the

text: particle

text: from

text: falling

text: into

text: the

text: central

text: object.

text: Here,

text: since

text: the

text: SMBH

text: background

text: is

text: spinning,

text: there

text: is

text: an

text: extra

text: contribution

text: to

text: the

text: centrifugal

text: barrier

text: due

text: to

text: the

text: BH

text: spin.

text: Bearing

text: in

text: mind

text: the

text: discussion

text: in

text: Sec.

text: about

text: perturbed

text: systems,

text: we

text: expect

text: geodesics

text: of

text: a

text: slowly-rotating

text: Kerr

text: metric

text: to

text: be

text: chaotic.

text: Figure

text: shows

text: Poincar´e

text: surfaces

text: of

text: section

text: for

text: two

text: diﬀerent

text: sets

text: of

text: parameters

text: that

text: produce

text: bounded

text: orbits,

text: O

text: using

text: the

text: slowly-rotating

text: metric

text: to

text: The

text: surfaces

text: of

text: section

text: look

text: regular,

text: seemingly

text: without

text: any

text: signatures

text: of

text: chaos,

text: but

text: this

text: can

text: be

text: deceiving

text: because

text: features

text: of

text: chaos

text: may

text: be

text: small

text: and

text: hard

text: to

text: resolve

text: on

text: this

text: scale.

text: Figure

text: Poincar´e

text: surfaces

text: of

text: section

text: for

text: a

text: SMBH

text: with

text: χ

text: E

text: and

text: angular

text: momentum

text: Lz

text: (left),

text: and

text: χ

text: E

text: and

text: angular

text: momentumOLz

text: (right),

text: using

text: the

text: slow-rotation

text: expansion

text: of

text: the

text: Kerr

text: metric

text: to

text: The

text: rotation

text: curve,

text: however,

text: can

text: signal

text: the

text: presence

text: of

text: chaotic

text: behavior,

text: even

text: when

text: this

text: cannot

text: be

text: resolved

text: with

text: the

text: naked

text: eye

text: from

text: the

text: surfaces

text: of

text: section.

text: The

text: left

text: panel

text: of

text: Fig.

text: shows

text: the

text: rotation

text: curve

text: for

text: the

text: same

text: sets

text: of

text: orbits

text: as

text: those

text: in

text: the

text: right-panel

text: of

text: Fig.

text: Observe

text: that

text: there

text: is

text: a

text: plateau

text: in

text: the

text: rotation

text: number

text: around

text: r

text: In

text: practice,

text: such

text: behavior

text: is

text: found

text: by

text: studying

text: the

text: derivative

text: of

text: the

text: rotation

text: number

text: with

text: respect

text: to

text: radius.

text: This

text: plateau

text: is

text: a

text: tell-tale

text: sign

text: of

text: chaos,

text: which

text: we

text: can

text: use

text: to

text: reﬁne

text: our

text: search

text: to

text: a

text: subregion

text: of

text: the

text: surfaces

text: of

text: section.

text: Doing

text: so,

text: we

text: ﬁnd

text: Birkhoﬀ

text: chains

text: of

text: islands

text: in

text: the

text: surfaces

text: of

text: section,

text: as

text: shown

text: in

text: the

text: right

text: panel

text: of

text: Fig.

text: The

text: emergence

text: of

text: Birkhoﬀ

text: islands

text: is

text: generic

text: and

text: not

text: dependent

text: on

text: the

text: particular

text: parameters

text: of

text: the

text: geodesic

text: or

text: of

text: the

text: SMBH.

text: The

text: left-panel

text: of

text: Fig.

text: is

text: a

text: zoom

text: of

text: the

text: left-panel

text: of

text: Fig.

text: to

text: a

text: regime

text: where

text: the

text: rotation

text: number

text: suggests

text: the

text: presence

text: ##LTFigure##

text: ##LTFigure##