Volume 37, pp. 41-69, 2010.

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

Hengguang Li, Anna Mazzucato, and Victor Nistor

Abstract

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 $\mathrm{\Omega }$ that may have cracks or vertices that touch the boundary. We consider in particular the equation $-\text{div}(A\mathrm{\nabla }u)=f\in H^{m-1}(\mathrm{\Omega })$ 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 $u=u_{\mathrm{r}\mathrm{e}\mathrm{g}}+\sigma$, into a function $u_{\mathrm{r}\mathrm{e}\mathrm{g}}$ 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 $m\ge 1$. Several numerical tests are included.

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Key words

Neumann-Neumann vertex, transmission problem, augmented weighted Sobolev space, finite element method, graded mesh, optimal rate of convergence

AMS subject classifications

65N30, 35J25, 46E35, 65N12