匹配外部和内部椭圆问题的谱方法
Spectral method for matching exterior and interior elliptic problems
AI总结：
提出一种谱方法求解内部和外部区域耦合椭圆问题，通过预处理构建影响矩阵转换边界条件，利用Chebyshev逼近实现指数收敛。
摘要：
描述了一种用于求解内部和外部区域上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了公式化和测试，要求这些问题的解及其法向导数在界面上连续。在预处理步骤中，计算了内部和外部区域对应于界面上所有可能Dirichlet边界值的完整齐次解基。该基用于构建影响矩阵，该矩阵将耦合边界条件转换为内部问题的条件。使用Chebyshev逼近来表示内部解和边界值。使用标准Chebyshev谱方法计算内部解。外部调和解计算为自由空间格林函数与面密度的卷积；当面密度以Chebyshev展开形式给出时，该面密度本身是积分方程的解，该积分方程具有解析解。Chebyshev逼近的性质确保了外部调和函数基均匀表示外部近边界解。通过计算矩形中电荷分布产生的静电势来测试该方法。所得影响矩阵条件良好，解随分辨率增加呈指数收敛。讨论了该方法向三维问题的推广，特别是真空包围的有限圆柱域中的磁流体动力学方程。
英文摘要：
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.
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