Two-Level Additive Schwarz Methods Using Rough Polyharmonic Splines Based Coarse Spaces

This thesis focuses on two-level additive Schwarz preconditioner for a class of elliptic equations with rough coefficients. We discuss how to choose a proper coarse space for the two-level additive Schwarz preconditioner and introduce two novel methods to construct the coarse space. One method is based on global rough poly harmonic spline, the other is based on local rough poly harmonic s-plines. At the same time, we introduce some basic results related to domain decomposition and two-level additive Schwarz preconditioner. We compare the methods based on rough polyharmonic splines with other two methods based on different coarse space construction:one is based on classical linear finite element and another is based on the multi-scale finite element. We use 9 bench-mark problems to compare the efficiency of the four two-level additive Schwarz preconditioners. Results show that the method based on global rough polyhar-monic splines (G-RPS) performs the best for most cases, and the method based on local rough polyharmonic splines (L-RPS) performs better than the method based on multi-scale finite element for several cases. For the case of coarse space construction based on local rough polyharmonic splines, we also investigate the relationship between the performance of preconditioner and the region of the support of the local rough polyharmonic splines. Results show that the larger the support of the local rough polyharmonic spline, the better the performance of the L-RPS based preconditioner for most cases. For most cases, there exist-s an optimal support for the local rough polyharmonic splines which balances accuracy and efficiency.