A Novel Decomposition as a Fast Finite Difference Method for Second Derivatives

초록

In this paper, we propose a fast solver for the linear system-induced from applying the finite difference method. For this purpose, we provide a new decomposition of the matrix consisting of a second-order central finite difference matrix and a small condition number matrix. We analyze the fourth-order finite difference matrix using this decomposition and the good properties of the two decomposed matrices. In addition, we compare the upper bounds of the condition numbers of the three matrices, which are closely related to the number of iterations. In terms of computational cost, we show the superiority of the proposed solver by solving the one-dimensional Poisson's equations. We also demonstrated the efficiency of using the proposed solver by numerically observing the factors, such as condition number and spectral radius.

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