Hokkaido Mathematical Journal

KIM Seokchan, CAI Zhiqiang, PYO Jae-Hong, KONG Sooryoun,

A finite element method using singular functions: interface problems.

Hokkaido Mathematical Journal, 36 (2007) pp.815-836

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Abstract

The solution of the interface problem is only in $H^{1+\alpha}(\Omega)$ with $\alpha>0$ possibly close to zero and, hence, it is difficult to be approximated accurately. This paper studies an accurate numerical method on quasi-uniform grids for two-dimensional interface problems. The method makes use of a singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors. Using continuous piecewise linear elements on quasi-uniform grids, our finite element approximation is shown to be optimal, $O(h)$, accurate in the $H^1$ norm. This is confirmed by numerical experiments for interface problems with $\alpha < 0.1$. An $O(h^{1+\alpha})$ error bound in the $L^2$ norm is also established by the standard duality argument. For small $\alpha$, this improvement over the $H^1$ error bound is negligible. However, numerical tests presented in this paper indicate that the $L^2$ norm accuracy is much better than the theoretical error bound.

MSC(Primary)65F30
MSC(Secondary)65F10
Uncontrolled Keywordsinterface singularity, finite element, singular function, stress intensity factor