Mesh Shape and Anisotropic Elements: Theory and Practice

The relationship between the shape of nite elements in unstructured meshes and the error that results in the numerical solution is of increasing importance as nite elements are used to solve problems with highly anisotropic and, often, very complex solutions. This issue is explored in terms of a priori and a posteriori error estimates, and through consideration of the practical issues associated with assessing element shape quality and implementing an adaptive nite element solver. 1.1 INTRODUCTION The solutions of many important partial diierential equations (PDEs) possess features whose accurate resolution using nite element (FE) trial functions requires local reene-ment of the underlying computational mesh. Frequently however these solution features are strongly directional, with the principal length scale in one direction being signiicantly smaller than in others. Examples of such features include boundary layers, shocks and edge singularities. The most eecient FE trial spaces for representing these solutions are deened by the use of anisotropic meshes whose elements have an orientation and geometry which reeect the nature of the solution itself. In this paper we present a brief overview of some of our work towards better understanding the practical issues associated with using such anisotropic meshes. This begins by considering how one might deene anisotropic 1