Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle

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. We propose in this paper several Schwarz domain decomposition algorithms for this task. We then study in detail the convergence properties of these algorithms applied to the nonlinear mesh PDE in one spatial dimension. We prove convergence for classical transmission conditions, and optimal and optimized variants for the generation of steady equidistributing grids. A classical, parallel, Schwarz algorithm is presented and analysed for the generation of time dependent (moving) equidistributing grids. We conclude our study with numerical experiments.