An Asymptotic Expansion Method For Elliptic PDE And Application On Radiation Heat Conduction Equations

We often encounter some elliptic PDEs with jump ( or multi-scale characteristic) coefficients during the research of mathematical physics. The asymptotic expansion method is effective to solve such problems, whose main idea is how to decompose the PDEs with jump coefficients into several PDEs with smooth (or single-scale characteristic) coefficients. In this paper, we study the asymptotic expansion methods for elliptic PDEs and 2-D radiation heat conduction equations.Firstly, aiming at a kind of jump coefficient elliptic scalar equation with the general boundary condition, according to two different interfaces, we propose a linear finite element method based on the asymptotic expansion. By using the basic theory of FEM, we obtain the same order of error function as that of the classic linear FEM under L~2 norm. The numerical experiments verify the correctness of theoretical results. Secondly, for a kind of linear radiation heat conduction equations, we design and analyze a linear finite element method based on the asymptotic expansion. Furthermore, we give the numerical experiment results for a non-linear radiation heat conduction equation with single-temperature, which show that the asymptotic expansion method is effective.