The Vertex Space Domain Decomposition Method for Elliptic Problems with Discontinuous Coefficient on Unstructured Meshes

In this paper, we discuss the vertex space domain decomposition method (VSDDM) for solving the algebraic system of equations which arise from the discretization of symmetric and positive de nite elliptic boundary value problems via nite element methods on general unstructured meshes. Our theory does not require that the subdomains are shape regular and coarse grid is nested to the ne grid. Furthermore, the vertices of subdomains are not necessary to belong to the grids of coarse mesh. We have shown that, with only shape regular assumption on the elements of ne and coarse triangulations, the VSDDM on the unstructured meshes has the same optimal convergence rate as the usual VSDDM on the structured meshes. We also estimate the condition number of the VSDDM for the elliptic problems with highly jumps coe cients across the boundaries of subdomains.