Parsing All of C by Taming the Preprocessor
Authors: Duk-Soon Oh
Title: Domain Decomposition Methods for Raviart-Thomas Vector Fields
Abstract
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
Raviart-Thomas 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.