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Let's say I want to solve this problem. I know the values on the boundaries and I guess an initial solution on the rectangular grid inside these boundaries, see figure below.Potential=10 on boundaries; potential=0 inside Potential=10 on boundaries; potential=0 inside

I know that eventually the field will converge to the same potential as the boundaries. If I let the program iterate long enough the whole area on the inside will also become yellow. The figure below shows an intermediate step towards equilibrium:

enter image description here

Now, in this project I am working on I am supposed to stop the simulation when the accuracy of the simulation is 1%. Is there a general definition of accuracy in these cases when working with a two dimensional grid? There are several grid nodes, all with different values, are these supposed to be 1% or less from equilibrium (all yellow)?

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Your talking about convergence and simulation so you are really talking about numerical methods or approximations. I'll state the obvious: 1) you need to choose your approximation formula, first order , 2nd order etc., this relates to how non-linear your boundaries are. 2) You also need to decide how fine a mesh of points you will use, this relates to how your simulation data will be used say in experimental verification. Then you can choose an error rule; as you iterate the points will change value, when the delta is <1% you have a solution, this could by 1000 or 10000 or even more iterations. There is another error determination formula that can be worked out, say if you use a 2nd order approximation but you know your system is really 3rd order or higher, there are error formulas for this, need to consult a text book.

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  • $\begingroup$ Thank you for your reply. I am using the finite difference method to discretize the the two-dimensional Laplace equation, this I read on the Wikipedia page for the FDM method. I also read that the accuracy is determined by the difference between the analytical solution and the numerical solution, as you said. When you say delta, do you mean the the ratio between the numerical and analytical solutions? Also, should I stop iterating when all deltas are <1%, is this what you are saying? Finally, the laplace equation I assume is second order PDE, is the FDM illsuited for this problem? Best regards $\endgroup$
    – SimpleProgrammer
    Dec 9, 2018 at 13:55
  • $\begingroup$ Finding an analytic solution is impossible for weird shaped boundaries so make sure your resolution is high. So 1% is the error from one iteration to the next, yes you should do it for every point or at least the areas of difficulty. And you should add a few more iterations to make sure the error is getting smaller. If you have a simple shape the analytic solution is easy and can be compared. $\endgroup$
    – PhysicsDave
    Dec 9, 2018 at 16:16

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