Geodesic
$L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes
Applications of Mathematics, Tome 56 (2011) no. 2, pp. 177-206.

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An $L^2$-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.
DOI : 10.1007/s10492-011-0002-7
Classification : 35J25, 65D05, 65N15, 65N30, 65N50
Keywords: elliptic boundary value problem; a priori error estimates; interpolation of non-smooth functions; finite element error; non-convex domains; edge singularities; anisotropic mesh grading; Dirichlet and a Neumann boundary value problem 
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     title = {$L^2$-error estimates for {Dirichlet} and {Neumann} problems on anisotropic finite element meshes},
     journal = {Applications of Mathematics},
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Apel, Thomas; Sirch, Dieter. $L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes. Applications of Mathematics, Tome 56 (2011) no. 2, pp. 177-206. doi : 10.1007/s10492-011-0002-7. http://geodesic.mathdoc.fr/articles/10.1007/s10492-011-0002-7/

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