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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]

Abstract: An error estimate is presented for the numerical approximation to an exterior problem of the Poisson equation when a coupled finite–infinite element method is employed in the computation. This estimate involves not only the size h and the order p of the quasiuniform finite element approximation, but also the radius R of the computational domain covered with finite elements and the number N of the radial terms of infinite elements. Numerical tests are performed, and the results partly validate the theoretical analysis. [Copyright &y& Elsevier]

Abstract: The numerical approximation of one-dimensional cubic nonlinear Schrödinger equations on the whole real axis is studied in this paper. Based on the work of A. Boutet de Monvel, A.S. Fokas and D. Shepelsky [Lett. Math. Phys., 65(3): 199–212, 2003], a kind of exact nonreflecting boundary conditions are derived on the artificially introduced boundary points. The related numerical issues are discussed in detail. Several numerical tests are performed to demonstrate the behaviour of the proposed scheme. [Copyright &y& Elsevier]