The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems

Abstract

In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $Ω$ which is a union of $N$ disjoint polygonal subdomains $Ω_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces $\partial Ω_i$. In each $Ω_i$ a conforming quasiuniform triangulation with parameters $h_i$ is constructed. We assume that the resulting triangulation in $Ω$ is also conforming, i.e., the meshes are assumed to match across the subdomain interfaces. On the fine triangulation the problem is discretized by a DG method. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by $C(1 + \max_i \log H_i/h_i)^2$ with a constant $C$ independent of $h_i$, $H_i$ and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. This result is an extension, to the case of full fine-grid DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013), pp.~400--422] where it was considered a conforming finite element method inside the subdomains and a discontinuous Galerkin method only across the subdomain interfaces. Numerical results are presented to validate the theory.