Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions

Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires
the selection of a finite element basis, a posteriori error estimator and refinement strategy.  Estimator accuracy may depend on the basis chosen, which in turn, may lead to unrelability or inefficiency via under- or over-refinement, respectively.  The basis may also have an impact on the size and condition 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 procedure.  The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems.