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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 $O\left({(1+log(H/h))}^2\right)$, independently of the coefficient jumps, where $H$ and $h$ denote...