In this paper, we discuss the vertex space decomposition method (VSDDM) for solving the algebraic system of equations which arise from the discretization of symmetric and positive definite elliptic boundary value problems via finite element methods on general unstructured meshes. Our theory does not require that the subdomains are shape regular and coarse grid is nested to the fine 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 fine and coarse triangulations, the VSDDM on the unstructured meshes has the same optimal convergence rate as the usual VSDDM on strucutred meshes. We also estimate the condition number of the VSDDM for elliptic problems with highly jumps coefficients across the boundaries of subdomains.

AMS subject classifications: 65N20, 65F10.