An Alternating-Direction Sinc–Galerkin method for elliptic problems

Abstract

a b s t r a c tA new Alternating-Direction Sinc-Galerkin (ADSG) method is developed and contrasted with classical Sinc-Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc-Galerkin method is used to approximate the solution of elliptic partial differential equations. We include parameter choices (derived from numerical experiments) that simplify existing alternating-direction algorithms. We compare the new scheme to a standard method employing Gaussian elimination on a system produced using the Kronecker product and Kronecker sum, as well as to a more efficient algorithm employing matrix diagonalization. We note that the ADSG method easily outperforms Gaussian elimination on the Kronecker sum and, while competitive with matrix diagonalization, does not require the computation of eigenvalues and eigenvectors.

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

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High-order blended compact difference schemes for the 3D elliptic partial differential equation with mixed derivatives and variable coefficients

“…The numerical solution to the elliptic PDE has important interest in numerical analysis. In the past few decades, a lot of numerical methods, which include meshless methods [3], spline collocation methods [4], finite element methods [5], fast domain de-composition methods [6], Sinc-Galerkin methods [7] and finite difference methods [2,, have been proposed by many authors. Among the methods above, the finite difference method has been widely used in scientific research and engineering practice because of its simple structure, it being easy to understand and needing only a small amount of calculation.…”

Error analysis of sinc-Galerkin method for time-dependent partial differential equations

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