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 ΕHm-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 = ureg + 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.
|Original language||English (US)|
|Number of pages||29|
|Journal||Electronic Transactions on Numerical Analysis|
|State||Published - Dec 1 2010|
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