When solving a PDE numerically, what's an acceptable level of error? 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? 
 A: 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.
A: Well it's not exactly a mathematics question as it is a physics / experimental sciences question. There is no golden number for such questions in general.
Let's say you have made measurements and found some feature that you believe (through means of calculations) to be $99.9\%$ real. For example you may have just discovered a new particle in a CERN experiment. Are you satisfied with the above results? Is a medical test that has a $1\%$ chance for false positives and $98\%$ of detection a satisfying test or not?
All these types of questions can only be answered by preferences of the decision making humans involved. You have endless statistical tests you can put your data through, but in the end there is just an inherent "leap of faith" that you must make, and it's only a question of how good do the test results need to be to satisfy your needs.
In physics, and particularly experimental physics, you usually allow errors of an order small enough to "blend" in the background noise, i.e. to be indistinguishable from the rest of the error and/or the sensitivity of your devices (detectors etc.), although sometimes you may just need to do with what you have...
As you can see, math has little to do here, so if you want to ask what makes physicists satisfied with experimental results, you should ask in Physics SE.
In terms of math, the more relevant question would be what techniques are available to estimate the errors of your results. But I don't think this is what you were asking, since it seems you have taken some course in computational physics. If not, please correct me and I'll try to adapt my answer to your needs.
