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J. Li, J. Melenk, B. Wohlmuth, J. Zou:
"Optimal convergence of higher order finite element methods for elliptic interface problems";
in: "ASC Report 13/2008", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2008, ISBN: 978-3-902627-01-8.

Higher order finite element methods are applied to 2D and 3D second order elliptic interface
problems with smooth interfaces, and their convergence is analyzed in the H1- and L2-norm.
The error estimates are expressed explicitly in terms of the approximation order p and a parameter
δ that quantifies the mismatch between the smooth interface and the finite element
mesh. Optimal H1- and L2-norm convergence rates in the entire solution domain are established
when the mismatch between the interface and mesh is sufficiently small. Furthermore,
under weaker conditions on the mismatch between the interface and mesh, optimal estimates
are obtained in an H1-norm that excludes a thin tubular neighborhood of the interface. For
some typical cases of meshes where the interface is approximated by a spline, the mismatch δ
is expressed in terms of the order of the spline. The resulting error estimate is then explicit in
the approximation order and the order of the spline. Five numerical examples are presented
to illustrate and confirm the sharpness of the approximation theory.

Elliptic interface problems, blending finite element, higher order finite elements,

http://www.asc.tuwien.ac.at/preprint/2008/asc13x2008.pdf