Discussion about Mesh independant solution

Seb

I know that John (Mr. C. Chien) often put a lot of emphasis on mesh independant solution when dealing with finite volume methods and doesn't put as much emphasis when dealing with FDM or FEM. I remember that John said it was not as important for FEM because the convergence of these schemes is usually proven.

I want to discuss about this subject because recently convergence results were published by Gallouet, Eymard and Herbin in Hanbook of Numerical Analysis. In this book, these authors demonstrated convergence results and error estimates for "classical" FVM methods (cell centered schemes, one DOF per control volume). These results can be compared to those already published for FEM methods.

I know that getting a "mesh independant solution" is important. But it is often realy hard to do so because the conditioning of the system of equations gets worst when nodes (or elements) are added when the mesh is refined. Thus, the final solution is sometimes (not always) harder to get, specialy when solving poisson type equations. As you well know, in the real world, the solution also depends on the solver (even if it shouldn't.

When building a numerical method, I beleive that we should put more emphasis on the repsect of the rules given by the mathematicians. Because if we follow these rules we can say the scheme that we are using should give a better solution when using mesh refinements.

I suggest that It is important to get a mesh independant solution, but it is more important to use a scheme for which the convergence has been proven and error estimates hold.

Comments?