标题： 谱方法在外边值和内边值椭圆问题匹配中的应用 显示英文标题

标题： Spectral method for matching exterior and interior elliptic problems

摘要： 描述了一种用于求解内部和外部区域耦合椭圆问题的谱方法。 该方法在二维内部泊松问题和外部拉普拉斯问题上进行公式化和测试，要求解及其法向导数在界面处连续。 在预处理步骤中计算了内部和外部区域对应的完整齐次解基，这些基对应于界面的所有可能Dirichlet边界值。 使用此基构建影响矩阵，以将耦合边界条件转换为内部问题的条件。 使用切比雪夫近似表示内部解和边界值。 使用标准切比雪夫谱方法计算内部解。 外部调和解通过自由空间格林函数与表面密度的卷积来计算；该表面密度本身是积分方程的解，当边界值以切比雪夫展开形式给出时，该积分方程具有解析解。 切比雪夫近似的性质保证了外部调和函数基均匀地表示外部近边界解。 通过计算矩形电荷分布产生的静电势来验证该方法。 所得的影响矩阵是良态的，并且随着分辨率的提高，解收敛得越来越快。 讨论了这种方法推广到三维问题的情况，特别是有限圆柱域内磁流体力学方程组在真空中的情况。 显示英文摘要

摘要： A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum.