Performance of hp-Adaptive Strategies for 3D Elliptic Problems

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations (PDEs) because it can achieve an exponential convergence rate in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or p. Several strategies for making this determination have been proposed over the years. In a recent study [Mitchell, W.F. and McClain, M.A., ACM Trans. Math. Software 41(1), 2:1-2:39 (Oct 2014)], the effectiveness of 13 strategies for 2D elliptic PDEs were compared in terms of the number of degrees of freedom and computation time needed to reach an error tolerance. This paper presents the results of a similar study for 3D elliptic PDEs. It was found that the results are very similar to those of the 2D study. For minimizing the computation time, the method based on a priori knowledge is very effective when there are known point singularities, and the method based on the decay rate of the expansion coefficients is very effective across all categories of problems. For minimizing the number of degrees of freedom, the method based on a priori knowledge works well for easy problems and problems with singularities, and the method that uses a “type parameter” works well for hard problems.