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November 2007 A finite element method using singular functions: interface problems
Seokchan KIM, Zhiqiang CAI, Jae-Hong PYO, Sooryoun KONG
Hokkaido Math. J. 36(4): 815-836 (November 2007). DOI: 10.14492/hokmj/1272848035


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.


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Seokchan KIM. Zhiqiang CAI. Jae-Hong PYO. Sooryoun KONG. "A finite element method using singular functions: interface problems." Hokkaido Math. J. 36 (4) 815 - 836, November 2007.


Published: November 2007
First available in Project Euclid: 3 May 2010

zbMATH: 1140.65084
MathSciNet: MR2378293
Digital Object Identifier: 10.14492/hokmj/1272848035

Primary: 65F30
Secondary: 65F10

Rights: Copyright © 2007 Hokkaido University, Department of Mathematics


Vol.36 • No. 4 • November 2007
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