Finite Element Convergence for Time-Dependent PDEs with a Point Source in COMSOL 4.2

The FEM theory provides the basis for quantification on the accuracy and reliability of a numerical solution by the a priori error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents information on the techniques needed in COMSOL 4.2 to enable computational studies that demonstrate this theory for time-dependent problems, in extension of previous work on stationary problems. These techniques can be used in many general settings, including when the analytic PDE solution is not known. We consider the time-dependent linear heat equation with homogeneous Dirichlet boundary conditions in both two and three spatial dimensions with both a smooth source term and a nonsmooth point source term modeled by one Dirac delta function located at the center of the domain. The presence of the point source for all positive times results in a problem for which no analytic solution is known. The observed constant slope for errors at several representative times in conventional log-log plots of the FEM error versus the reciprocal of the mesh size confirms the exponent of the mesh size in the error estimate to agree with recent theory for the problem. This paper presents information on the techniques needed in COMSOL 4.2 to enable the studies, including how to correctly implement the Dirac delta function for a time-dependent problem, how to set up a study with repeated uniform mesh refinement at several points in time as needed for a time-dependent PDE problem, how to use a reference solution since no analytic PDE solution is known, how to collect data from each refinement level and compute the convergence order from them, and more.