Effects of basis selection and H-refinement on error estimator reliability and solution efficiency for high-order methods in three space
Peter K. Moore 11 Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
Received by the editors November 23, 2004 and, in revised form, April 12, 2005
Designing efective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, a postscript error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and conditon of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an h-refinement prcedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems. The results demonstrate that popular choices such as the tensor-product basis or the modified Szabo-Babuska basis have significant shortcomings but that promising alternatives exist.
AMS subject classifications: 65N15, 65N30, 65N50
Key words: a posteriori error estimation; adaptivity; high-order finite element basis