任意凸四边形区域上二阶变系数椭圆边值问题有效的谱Galerkin逼近

Efficient Spectral Galerkin Approximation for Second Order Elliptic Boundary ValueProblems with Variable Coefficient on Arbitrary Convex Quadrilateral Domain

摘要: 本文提出了在任意凸四边形区域上二阶变系数椭圆边值问题的一种有效谱Galerkin逼近。 首先，通过双线性等参变换和坐标变换将任意四边形区域转换到D˜= [−1, 1]²，并建立其在D˜ 的弱形式及相应的离散格式。 其次，我们证明了弱解的存在唯一性。 另外，利用Legendre 正交多项式构 造了逼近空间中一组有效的基函数，推导出离散格式的矩阵形式。 最后通过数值实验，验证了 谱Galerkin逼近任意凸四边形区域上二阶变系数椭圆边值问题的谱收敛。

Abstract: In this paper, an efficient spectral Galerkin approximation for second-order elliptic boundary value problems with variable coefficients on an arbitrary convex quadrilat-eral region is proposed. Firstly, any quadrilateral region is converted to D˜= [−1, 1]² by bilinear isoparametric transformation and coordinate transformation, and its weak form and corresponding discrete format on D˜ are established. Secondly, we prove the existence and uniqueness of the weak solution. In addition, the Legendre orthogonal polynomial is used to construct a set of effective basis functions in the approximation space, and the matrix form of the discrete scheme is derived. Finally, the spectral convergence of spectral Galerkin approximation to the second-order elliptic boundary value problem with variable coefficients on arbitrary convex quadrilateral region is verified by numerical experiments.