Measuring Mesh Qualities and Application to Variational Mesh Adaptation

Abstract

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.Key words. mesh quality, mesh adaptation, variational mesh adaptation, error estimate AMS subject classifications. 65M50, 65M60, 65L50, 65L60 DOI. 10.1137/S10648275034294051. Introduction. In the last two decades, variational mesh adaptation has received considerable attention from scientists and engineers; see [8,9,13,15,18,19,21,23,24,29,32] and the books [14,22,28,31] and references therein. With a variational method, adaptive meshes are generated as images of a reference mesh under the coordinate transformation determined by a so-called adaptation functional. Such a functional is commonly designed to measure the difficulty in the numerical approximation of the physical solution. It often involves mesh properties and employs a monitor function to control mesh concentration. The development of variational mesh adaptation has so far focused on the design of the adaptation functional (e.g., see [9,13,18,24]), and there is little work on assessment of an existing mesh for a given solution. Mesh assessment is not without importance, especially since many variational methods generate a mesh of unknown quality. A good understanding of the effects of mesh qualities on the solution error can in turn help with the design of a better adaptation functional. Moreover, studies of mesh quality may lead to rigorous error analysis on adaptive meshes, which is much needed in the context of variational mesh adaptation.Mesh assessment has been extensively studied in the context of finite elements; e.g., see the recent review paper [3] and references therein. For example, the minimum angle [36], the maximum angle [6,20,26,30], and the aspect ratio [11] have been widely used to characte...

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