一类变系数二阶椭圆交面问题有效的谱元法

An Efficient Spectral Element Method for A Class of Second Order Elliptic Interface Problems with Variable Coefficients

摘要: 本文针对圆域上一类变系数二阶椭圆交面问题提出了一种有效的谱元法。首先，根据极坐标变换公式，极条件以及傅里叶基函数展开，原问题被分解为一系列关于径向变量的相互独立的一维二阶问题，并建立了弱形式和相应的离散格式。其次，我们根据变系数的正则性，构造了基于分片高阶多项式逼近的一种谱元方法，再通过利用勒让德多项式的正交性质，构造了一组适当的基函数，使得离散变分形式中的系数矩阵在变系数为分片多项式条件下是分块对角的稀疏矩阵。最后，我们呈现了一些数值例子，通过数值结果验证了我们提出的算法是收敛的和高精度的。

Abstract: In this paper, we put forward an efficient spectral element method for a class of second order ellip-tic interface problem with variable coefficients in a circular region. Firstly, because of the polarity transformation, pole condition and fourier basis function expansion, the original problem resolve into a succession of independent one-dimensional second-order problems about radial variables. Furthermore, the weak form and relevant discrete scheme are established. Secondly, according to the regularity of variable coefficients, we construct a spectral element method based on piecewise high-order polynomial approximation, and then by taking advantage of the orthogonal property of Legendre polynomials, we construct a set of appropriate basis functions, so that the coefficient ma-trix in the discrete scheme is sparse diagonal matrix in the case that the variable coefficients are piecewise polynomials. Finally, some numerical examples are presented, and the numerical results show that the proposed algorithm is convergent and high-precision.