Finite Element Convergence Studies Using COMSOL 4.0a and LiveLink for MATLAB

Abstract

In order to gauge how reasonable a finite element solution to a partial differential equation is on a given mesh, a common strategy is to refine the mesh, compute the solution on the finer mesh, and use the solutions on the two meshes for a qualitative comparison. The theory of the finite element method (FEM) makes these comparisons quantitative by estimating the convergence order of the FEM error on a sequence of progressively finer meshes obtained by uniform mesh refinement. We show in detail how to carry out convergence studies of this type using the graphical user interface (GUI) of COMSOL 4.0a as well as using COMSOL’s LiveLink for MATLAB on the example of Lagrange elements of varying polynomial degrees. Conducting the convergence study in this manner shows how to quantify the convergence of FEM solutions and brings out the potential benefit of using higher order elements. The interconnection of COMSOL with MATLAB allows for a convenient automization of the study that is not possible through the use of the GUI alone, but is vital for reproducible studies and useful for running studies in batch mode on computing clusters.