Domain Decomposition Methods for Raviart - Thomas Vector Fields TR 2011 - 942

Raviart-Thomas finite elements are very useful for problems posed in H(div) since they are H(div)-conforming. We introduce two domain decomposition methods for solving vector field problems posed in H(div) discretized by RaviartThomas finite elements. A two-level overlapping Schwarz method is developed. The coarse part of the preconditioner is based on energy-minimizing extensions and the local parts consist of traditional solvers on overlapping subdomains. We prove that our method is scalable and that the condition number grows linearly with the logarithm of the number of degrees of freedom in the individual subdomains and linearly with the relative overlap between the overlapping subdomains. The condition number of the method is also independent of the values and jumps of the coefficients across the interface between subdomains. We provide numerical results to support our theory. We also consider a balancing domain decomposition by constraints (BDDC) method. The BDDC preconditioner consists of a coarse part involving primal constraints across the interface between subdomains and local parts related to the Schur complements corresponding to the local subdomain problems. We provide bounds of the condition number of the preconditioned linear system and suggest that the condition number has a polylogarithmic bound in terms of the number of degrees of freedom in the individual subdomains from our numerical experiments for arbitrary jumps of the coefficients across the subdomain interfaces.