Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains

We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain λ that may have cracks or vertices that touch the boundary. We consider in particular the equation-div(Aδu) = f ΕH^m-1λ with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = u_reg + φ, into a function u _reg with better decay at the vertices and a function that is locally constant near the vertices, thus proving well-posedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi-)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included.