2-Dimensional primal domain decomposition theory in detail

We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is O ((1 + log( H / h )) 2 ), independently of the coefficient jumps, where H and h denote the discretization parameters of the coarse and fine triangulations, respectively. Although this preconditioner and its analysis date back to the pioneering work J.H.Bramble, J. E.Pasciak, A.H. Schatz (1986), and it was revisited and extended by many authors including M.Dryja, O.B.Widlund (1990) and A.Toselli, O.B.Widlund (2005), the theory is hard to understand and some details, to our best knowledge, have never been published. In this paper we present all the proofs in detail by means of fundamental calculus.