Trace Averaging Domain Decomposition Method with Nonconforming Finite Elements

Abstract

We consider, in this paper, the trace averaging domain decomposition method for the second order self-adjoint elliptic problems discretized by a class of nonconforming finite elements, which is only continuous at the nodes of the quasi-uniform mesh. We show its geometric convergence and present the dependence of the convergence factor on the relaxation factor, the subdomain diameter $H$ and the mesh parameter $h$. In essence, this method is equivalent to the simple iterative method for the preconditioned capacitance equation. The preconditioner implied in this iteration is easily invertible and can be applied to preconditioning the capacitance matrix with the condition number no more than $O\bigl ( (1+\ln {H\over h}) max(1+H^{-2}, 1+\ln {H\over h}) \bigr )$.