Efficiency-based hp-refinement for finite element methods

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is deter- mined based on e±ciency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with a minimal amount of work. Two efficiency measures are discussed, 'work times error' and 'accuracy per computational cost'. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hp-refinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The work times error strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the accuracy per computational cost strategy provides an efficient refinement mechanism for any combination of d and p