TITLE: When solving a PDE numerically, what's an acceptable level of error?
QUESTION [5 upvotes]: Suppose I solve something like $\vec{\nabla} \cdot \vec{j} = 0$, where $\vec{j}$ is a function of several knowns and one unknown $\Phi$, on a regular mesh using a given finite difference scheme to approximate derivatives and a given matrix-solver to find the solution. Suppose further that the problem is well-posed and that the matrix is full-rank, such that a unique solution for $\Phi$ exists. 
After one solves for $\Phi$ at all points on the mesh, the knowns can be used to construct $\vec{j}$, allowing $\vec{\nabla} \cdot \vec{j}$ to be calculated. 
When solving this numerically, several sources of error cause the solution to be inexact: $\vec{\nabla} \cdot \vec{j} \ne 0$ at all points in the domain. Instead, there's a bit of (usually low-level) noise. 
My Question:
How can one estimate what level of noise is acceptable for a given problem?

REPLY [2 votes]: This is an opinion-based question, so I offer an opinion as an answer: Never solve a PDE more accurately than the error from modeling the phenomenon by the PDE.
When you solve a PDE, there exists a more complicated PDE which models the phenomenon better, and often you can cheaply get an estimate of the difference between the two. For example, the Schrodinger equation cannot describe spin-orbit coupling, but the Dirac equation can. However, you can estimate the magnitude of the spin-orbit coupling by a numerical integration. So when solving the Schrodinger equation, get a rough estimate of the solution, then use it to calculate the spin-orbit coupling. You now know your solution can never get more accurate than this, so don't try for more.