Abstracts for the Conference on
Grid Adaptivity in Computational PDEs,
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Adaptive solution techniques using h-r-refinement

P K Jimack & P J Capon

School of Computer Studies,
University of Leeds, Leeds LS2 9JT, UK

Abstract

In this paper we consider the application of the finite element method to a variety of two-dimensional PDEs using unstructured meshes of triangles. Conventional mesh refinement strategies involve estimating the local error on each triangle and then subdividing those triangles for which the error exceeds some predefined tolerance. This form of adaptivity (i.e. h-refinement) is generally quite robust provided a reliable error estimator is available. However, we will demonstrate in this paper that the efficiency of this approach may be significantly enhanced by the additional use of mesh movement via nodal relocation (r-refinement).

The judicious use of nodal movement can allow the quality of a mesh of a given topology to be improved, or even optimized, for the representation of a particular solution. As a strategy for mesh adaptivity on its own however this approach has a number of weaknesses. In particular, for a mesh with a fixed topology, one can never guarantee to satisfy any predetermined accuracy criteria by relocation of the node points alone: the best possible mesh of that topology may still be inadequate. Nevertheless, when this approach is combined with h-refinement we will demonstrate that it is possible to utilize the strengths of both strategies: the robustness and reliability of h-refinement and the mesh quality (and therefore improved efficiency) of r-refinement.

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