Mathematical and Numerical Analysis of a Robust and Efficient Grid Deformation Method in the Finite Element Context

Abstract

Among a variety of grid deformation methods, the method proposed by Bochev, Liao, and de la Pena [Methods Partial Differential Equations, 12 (1996), pp. 489–506], Cai et al. [Comput. Math. Appl., 48 (2004), pp. 1077–1086], and Liao and Anderson [Appl. Anal., 44 (1992), pp. 285–298] is one of the most favorable, because it prevents mesh tangling and offers precise control over the element volumes. Its numerical realization requires only solving a Poisson problem and a system of fully decoupled initial value problems. Many other deformation methods, in contrast, involve the solution of complicated nonlinear partial differential equations (PDEs). In this article, we introduce a generalization of Liao's method, which allows for generating a desired mesh size distribution for quite arbitrary grids without giving rise to mesh tangling. We elaborate on its numerical realization and prove the convergence of our method. Our results are confirmed by numerical experiments.