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

• Autores: Jingzhi Li, Melenk Jens Markus, Barbara I. Wohlmuth, Jun Zou
• Localización: Applied numerical mathematics, ISSN-e 0168-9274, Vol. 60, Nº. 1-2, 2010, págs. 19-37
• Idioma: inglés
• DOI: 10.1016/j.apnum.2009.08.005
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• Resumen

  • We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the L2- and H1-norm are expressed in terms of the approximation order p and a parameter ? that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the H1(?)-norm is only achieved under stringent assumptions on ?, namely, ?=O(h2p). Under weaker conditions on ?, optimal a priori estimates can be established in the L2- and in the H1(??)-norm, where ?? is a subdomain that excludes a tubular neighborhood of the interface of width O(?). 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 L2(?)- and the H1(??)-norm can be expected but not order p for the approximation in the H1(?)-norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results.