A Schwarz Waveform Moving Mesh Method

An r-refinement (moving mesh) method is considered for solving time dependent partial differential equations (PDEs). The resulting coupled system, consisting of the physical PDE and a moving mesh PDE, is solved by a Schwarz waveform relaxation method. In particular, the computational space-time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of PDEs on each subdomain. Dirichlet boundary conditions are used to pass solution information between neighboring regions. The efficacy of this approach is demonstrated for some model problems. For problems where the solutions evolve on disparate time scales in different regions of the spatial domain, this approach demonstrates the significant savings in computational time and effort which are possible.