Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems

TLDR

The moving mesh PDE (MMPDE) strategy, previously developed for one‑dimensional problems, is extended here to two dimensions. This paper introduces a moving mesh method for solving partial differential equations in two dimensions. The MMPDE is derived from a gradient‑flow equation based on a mesh‑adaptation functional inspired by harmonic maps, and its geometric interpretation, basic properties, and numerical examples for mesh generation and time‑dependent PDEs are presented. Numerical results show that the mesh movement strategy concentrates points around problem features while maintaining mesh orthogonality. See Huang et al., SIAM J.

Abstract

In this paper we introduce a moving mesh method for solving PDEs in two dimensions. It can be viewed as a higher-dimensional generalization of the moving mesh PDE (MMPDE) strategy developed in our previous work for one-dimensional problems [W. Huang, Y. Ren, and R. D. Russell, SIAM J. Numer. Det., 31 (1994), pp. 709--730]. The MMPDE is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps. Geometrical interpretations are given for the gradient equation and functional, and basic properties of this MMPDE are discussed. Numerical examples are presented where the method is used both for mesh generation and for solving time-dependent PDEs. The results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality.