Domain decomposition adaptivity for the Richards equation model

Abstract

This paper presents a study on efficient and economical domain decomposition adaptivity for Richards equation problems. Many real world applications of the Richards equation model typically involve solving systems of linear equations of huge dimensions. Multi-thread methods are therefore often preferred in order to reduce the required computation time. Multi-thread (parallel) execution is typically achieved by domain decomposition methods. In the case of non-homogeneous materials, the problem conditioning can be significantly improved if the computational domain is split efficiently, as each subdomain can cover only a certain material set within some defined parameter range. For linear problems, e.g. heat conduction, it is very easy to split the domain in this way. A problem arises for the nonlinear Richards equation, where the values of the constitutive functions, even over a homogeneous material, can vary within several orders of magnitude, see e.g. Kuraz et al. (Appl Math Comput 2012, in press). If the Rothe method is considered for time integration, a robust algorithm will be obtained if the domain is split adaptively over the time integration levels. The domain decomposition technique considered here is the standard multiplicative Schwarz method with coarse level, see e.g. Toselli and Widlund (Domain decomposition: methods algorithms and theory, Springer, Berlin, 2005). A method of the domain decomposition adaptivity will be studied in this paper.