An Overlapping Schwarz Algorithm for Raviart-Thomas Vector Fields with Discontinuous Coefficients

Overlapping Schwarz methods form one of two major families of domain decomposition methods. We consider a two-level overlapping Schwarz method for Raviart-Thomas vector fields. The coarse part of the preconditioner is based on the energy-minimizing extensions and the local parts are based on traditional solvers on overlapping subdomains. We show 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. Numerical results for 2D and 3D problems, which support the theory, are also presented.