Recent Advances in Schwarz Waveform Moving Mesh Methods – A New Moving Subdomain Method

It is well accepted that the efficient solution of complex partial differential equations (PDEs) often requires methods which are adaptive in both space and time. In this paper we are interested in a class of spatially adaptive moving mesh (r-refinement) methods introduced in [9, 10, 12]. Our purpose is to introduce and explore a natural coupling of domain decomposition, Schwarz waveform relaxation (SWR) [ 4], and spatially adaptive moving mesh PDE (MMPDE) methods for time dependent PDEs. SWR allows the focus of computational energy to evolve to the changing behaviour of the solution locally in regions or subdomains of the space-time domain. In particular, this will enable different time steps and indeed integration methods in each subdomain. The spatial mesh, provided by the MMPDE, will react to the local solution dynamics, providing distinct advantages for problems with evolving regions of interesting features. In this paper we detail and compare approaches which couple SWR with moving meshes. Section 2 provides a brief review of the r-refinement method. We contrast the related approaches introduced in [6, 7] with a new moving subdomain method in Sect. 3. We conclude in Sect. 4 with a brief presentation of numerical results to demonstrate the moving subdomain method.