A Technique for the Quantitative Assessment of the Solution Quality on Particular Finite Elements in COMSOL Multiphysics

To validate the reasonableness of a numerical solution to a partial differential equation on a given mesh, a common approach is to refine the mesh, compute a solution on the finer mesh, and compare the solutions on the two meshes. Comparing graphical representations of the two solutions gives a qualitative assessment of the solution quality. In many cases though, a priori error estimates from the theory of the finite element method are available that provide quantitative predictions of the expected solution quality in terms of the mesh spacing. This note shows how to use tools available in COMSOL Multiphysics to compute numerical estimates that can confirm if the finite element method performs as predicted by the theory. The technique presented does not assume that the true solution of the PDE is known. It is applied to linear Lagrange elements here, and extensions and limitations of the technique are discussed.