This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad|$\acute{\textrm{e}}$| approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method.

中文翻译：

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无限域中三维泊松方程的快速算法

本文涉及无限域中三维泊松方程的快速有限元方法。既考虑了外部问题，也考虑了带尾问题。导出精确的Dirichlet-to-Neumann（DtN）型人工边界条件（ABC），以将原始无限域问题简化为合适的截断域问题。基于平方根函数的最佳相对Chebyshev近似，开发了一种快速算法来近似精确的ABC。一个显着的优点是无需在人工边界上计算与表面拉普拉斯算子相关的完整特征系统。另外，与模态展开法和基于Pad | $ \ acute {\ textrm {e}} $ |的方法相比对于平方根函数的近似，DtN映射的计算成本进一步降低。进行了误差分析，并通过数值算例验证了该方法的有效性。