## Hokkaido Mathematical Journal

### A finite element method using singular functions: interface problems

#### 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.

#### Article information

Source
Hokkaido Math. J., Volume 36, Number 4 (2007), 815-836.

Dates
First available in Project Euclid: 3 May 2010

https://projecteuclid.org/euclid.hokmj/1272848035

Digital Object Identifier
doi:10.14492/hokmj/1272848035

Mathematical Reviews number (MathSciNet)
MR2378293

Zentralblatt MATH identifier
1140.65084

Subjects
Primary: 65F30: Other matrix algorithms