Measuring Mesh Qualities and Application to Variational Mesh Adaptation

Abstract

The mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior. Three elementwise mesh quality measures characterizing the shape, alignment, and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh. An adaptive mesh is assessed by an overall quality measure defined as a weighted Lebesgue norm of a product of the three elementwise quality measures. It is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so-called roughness measure of the solution, defined as the ratio of two Lebesgue norms of a derivative of the solution. The definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh. As an immediate result, the task of mesh adaptation becomes to control the overall mesh quality. This idea is applied to variational mesh adaptation to develop two functionals, one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments. Numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality.