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

Li, Hengguang ; Mazzucato, Anna ; Nistor, Victor

Electronic transactions on numerical analysis, Tome 37 (2010), pp. 41-69

Résumé

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 $\Omega $that may have cracks or vertices that touch the boundary. We consider in particular the equation - $div(A\nabla u) = f \in $Hm - $1(\Omega )$ 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 + \sigma $, into a function ureg with better decay at the vertices and a function $\sigma $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 $\geq 1$. Several numerical tests are included.

Détail

Classification : 65N30, 35J25, 46E35, 65N12
Keywords: Neumann-Neumann vertex, transmission problem, augmented weighted Sobolev space, finite element method, graded mesh, optimal rate of convergence