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In this paper the Stokes system in an unbounded domain is solved by the artificial boundary method. The novelty lies in an operator form of the exact Dirichlet-to-Neumann (DtN) mapping. With the help of the Chebyshev rational approximation of the square root function, we derive a highly accurate approximate DtN mapping, which can be numerically implemented without resorting to the eigen-decomposition in terms of the vectorial spherical harmonics. In addition, we develop an efficient block preconditioner for the augmented truncated saddle point problem. Numerical experiments demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a fast numerical algorithm for the multi-dimensional heat equation on unbounded domains with strips tails. The artificial boundary method is used to confine the computational domains. By applying BDF2 for the time discretization and performing the Z transform, we derive an exact semi-discrete artificial boundary condition which contains the discrete temporal convolution and the surface Laplacian operator. Based on the best relative Chebyshev approximation of the square-root function, we design a fast algorithm to approximate and localize the exact semi-discrete artificial boundary condition. The spatial discretization is realized by the Galerkin finite element method with a special boundary treatment. A complete error estimate for the fully discrete problems is performed, and numerical tests are presented to demonstrate the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]