Abstract

The goal of this work is to construct and study overlapping Schwarz preconditioners for the linear systems resulting from the quasi-static Biot model. By introducing an intermediate variable, the Biot model is reformulated into a 3-by-3 block tri-diagonal system. The two Schur complements are approximated in an analytic way by using a Fourier analysis approach. Then, based on the approximations, two types of overlapping Schwarz preconditioners are designed: a) Block triangular and block diagonal precodnitioners with overlapping additive Schwarz approximations for the elliptic operators involved in. b) Indefinite overlapping Schwarz preconditioners applied directly to the mixed formulation of linear elasticity operator and combined with an overlapping Schwarz approximation for the last Schur complement operator. The properties of the overlapping Schwarz preconditioners are analyzed and investigated. Numerical results show that the proposed overlapping Schwarz preconditioners are scalable, optimal in the number of unknowns across individual subdomains.

Short bio

Dr. Cai is an associate professor of Mathematics from Morgan State University. His research is in the field of numerical analysis with focuses on numerical methods for partial differential equations, computational biomechanics, and fast algorithms for data science. His main interest is to develop fast solvers for partial differential equations and their applications in biomechanics, material science, environmental science and geoscience.

Dr. Cai graduated from Hong Kong University of Science and Technology in 2008 and joined Morgan State University in 2015. He has around 30 publication and received several major grants from NSF and NIH.