FEM Convergence Studies for 2-D and 3-D Elliptic PDEs with Smooth and Non-Smooth Source Terms in COMSOL 5.1

Numerical theory provides the basis for quantification on the accuracy and reliability of a FEM solution by error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents techniques needed in COMSOL 5.1 to perform computational studies for an elliptic test problem in two and three space dimensions that demonstrate this theory by computing the convergence order of the FEM error. For a PDE with smooth right-hand side, linear Lagrange finite elements exhibit second order convergence for all space dimensions. We also show how to perform these techniques for a problem involving a point source modeled by a Dirac delta distribution as forcing term. This demonstrates that PDE problems with a non-smooth source term necessarily have degraded convergence order and thus can be most efficiently solved by low-order FEM such as linear Lagrange elements. Detailed instructions for obtaining the results are included in an appendix.