Tailored finite point method for the interface problem

Zhongyi Huang; Zhongyi Huang

doi:10.3934/nhm.2009.4.91

Networks and Heterogeneous Media

2009, Volume 4, Issue 1: 91-106. doi: 10.3934/nhm.2009.4.91

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Tailored finite point method for the interface problem

Zhongyi Huang ^{1
,}

1.
Dept. of Mathematical Sciences, Tsinghua University, Beijing 100084

Received: 01 July 2008 Revised: 01 November 2008
Primary: 65N22, 65N35; Secondary: 35J25.

In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only $\O(N)$ for $N$ discrete points. We also extend our method to two-dimensional problems.
- Tailored finite point method,
- interface problem,
- elliptic equation,
- discontinuous coefficients,
- jump conditions
Citation: Zhongyi Huang. Tailored finite point method for the interface problem[J]. Networks and Heterogeneous Media, 2009, 4(1): 91-106. doi: 10.3934/nhm.2009.4.91

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Abstract

In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only $\O(N)$ for $N$ discrete points. We also extend our method to two-dimensional problems.
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