Optimal a priori estimates for higher order finite elements for elliptic interface problems

Authors:
Jingzhi Li;Jens Markus Melenk;Barbara Wohlmuth;Jun Zou
Affiliations:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong;Institut für Analysis und Scientific Computing, Technische Universität Wien, Austria;Institut für Angewandte Analysis und Numerische Simulation (IANS), Universität Stuttgart, 70569 Stuttgart, Germany;Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Venue:
Applied Numerical Mathematics
Year:
2010

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Abstract

We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the L^2- and H^1-norm are expressed in terms of the approximation order p and a parameter @d that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the H^1(@W)-norm is only achieved under stringent assumptions on @d, namely, @d=O(h^2^p). Under weaker conditions on @d, optimal a priori estimates can be established in the L^2- and in the H^1(@W"@d)-norm, where @W"@d is a subdomain that excludes a tubular neighborhood of the interface of width O(@d). In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders p+1 and p for the approximation in the L^2(@W)- and the H^1(@W"@d)-norm can be expected but not order p for the approximation in the H^1(@W)-norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results.