Discussion Forum

The unknown functions in Partial Differential Equations (PDEs) describing physics problems are in most cases continuous, e.g. stress or displacement in an elastic solid.

With numerical methods, you obtain approximate solutions of Partial Differential Equations (PDEs), because you solve the PDEs on a finite number of points. In other words, you obtain a discrete approximation - at a finite number of points - of a continous function.

The more points you have, the closer your discrete approximation gets from the continuous solution.

Refining your mesh should lead to a stable solution, very close to the continuous solution. If it is not case, it means you are not close to the real solution of the problem: your problem may be ill-posed or you have to refine more.

There are error estimates, which computes the difference between your approximate solution and the theoretical solution. If your problem is well-posed, the finer the mesh, the smaller the error, e.g. your error converge to 0 with finer meshes, e.g. your approximate solution converges to the real solution.

Najib gives a good explanation. Another application of the term mesh convergence is a little more practical, and that is how the solution with respect to a relevant variable of interest changes with increasing mesh size. Since there is a trade-off between mesh size and solution time, one aims for a mesh which is 'good enough'. By solving your model using increasing mesh sizes, one can find where the variable of interest does not change much with increasing mesh size. Once then selects this mesh size for the model.

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Steven Conrad, MD PhD
LSU Health