Discrete analysis of domain decomposition approaches for mesh generation via the equidistribution principle

Abstract

Moving mesh methods based on the equidistribution principle are powerful techniques for the space-time adaptive solution of evolution problems. Solving the resulting coupled system of equations, namely the original PDE and the mesh PDE, however, is challenging in parallel. Recently several Schwarz domain decomposition algorithms were proposed for this task and analyzed at the continuous level. However, after discretization, the resulting problems may not even be well posed, so the discrete algorithms requires a different analysis, which is the subject of this paper. We prove that when the number of grid points is large enough, the classical parallel and alternating Schwarz methods converge to the unique monodomain solution. Thus, such methods can be used in place of Newton's method, which can suffer from convergence difficulties for challenging problems. The analysis for the nonlinear domain decomposition algorithms is based on M-function theory and is valid for an arbitrary number of subdomains. An asymptotic convergence rate is provided and numerical experiments illustrate the results.

A new functional for variational mesh generation and adaptation based on equidistribution and alignment conditions

“…For 0 < θ ≤ 1 2 , dp ≥ 2, and p ≥ 1, the functional is coercive and polyconvex and thus has a minimizer [25]. It has been shown in [16] that the MMPDE mesh equation (cf. Sect.…”

“…In particular, the mesh trajectory of the discrete MMPDE stays nonsingular if it is so initially provided that the meshing functional under consideration satisfies a coercivity condition (cf. (16) below). To our best knowledge, this is the only nonsingularity result at the discrete level available in the context of variational mesh generation and adaptation and mesh movement.…”

Unstructured mesh generation based on Parallel Virtual Machine in cyber-physical system

“…However, there exist several aspects to be improved in Lohner's methods, such as optimization of sub-domain boundary and grid smoothness; at the same time, there are too many communication overheads [17][18][19] in the method of Okusanya's Delaunay triangulation. Hence, it is necessary to further improve the parallel grid generation technique [20][21][22] to solve flow problem with complex geometries [23][24][25]. It is well known that CFD is becoming increasingly sophisticated: grids define highly complex geometries and flows are simulated involving very different length and time scales [26,27].…”

Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources

“…The moving mesh, which continuously adjusts the locations of mesh points to resolve the solution as well as possible with a fixed total number of mesh points, can be generated adaptively by the moving mesh partial differential equations (MMPDEs) derived from the equidistribution principle [15,16]. Based on the Schwarz domain decomposition method, one of the overlapping domain decomposition methods, the parallel generation of moving mesh has been studied in recent years [17][18][19].…”