Generating Equidistributed Meshes in 2D via Domain Decomposition

There are many occasions when the use of a uniform spatial grid would be prohibitively expensive for the numerical solution of partial differential equations (PDEs). In such situations, a popular strategy is to generate an adaptive mesh by either varying the number of mesh points, the order of the numerical method, or the location of mesh points throughout the domain, in order to best resolve the solution. It is the latter of these options, known as moving mesh methods, which is our focus. In this case the physical PDE of interest is coupled with equations which adjust the position of mesh points to best “equidistribute” a particular measure of numerical error. This coupled system of equations is solved to generate the solution and the corresponding mesh simultaneously, see [7] for a recent overview. A simple method for adaptive grid generation in two spatial dimensions is outlined in [8] by Huang and Sloan, in which a finite difference two dimensional adaptive mesh method is developed by applying a variation of de Boor’s equidistribution principle (EP) [1, 2]. The equidistribution principle states that an appropriately chosen mesh should equally distribute some measure of the solution variation or computational error over the entire domain. Mackenzie [9] extends upon the work of [8] by presenting a finite volume discretization of the mesh equations, as well as an efficient iterative approach for solving these equations, referred to as “an alternating line Gauss-Seidel relaxation approach”. In this paper, we propose a parallel domain decomposition (DD) solution of the 2D adaptive method of [8]. In Section 2 we review the derivation of the mesh PDEs of [8] and discuss possible boundary conditions. In Sections 3 and 4 we present classical and optimized Schwarz methods for the generation of 2D equidistributed meshes, and in Section 5 we describe the numerical implementation of this approach and provide numerical results.