Covolume-upwind finite volume approximations for linear elliptic partial differential equations |
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Authors: | Lili Ju Li Tian Xiao Xiao Weidong Zhao |
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Institution: | 1. Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;2. Department of Mathematics, Pennsylvania State University, State College, PA 16801, USA;3. School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
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Abstract: | In this paper, we study covolume-upwind finite volume methods on rectangular meshes for solving linear elliptic partial differential equations with mixed boundary conditions. To avoid non-physical numerical oscillations for convection-dominated problems, nonstandard control volumes (covolumes) are generated based on local Peclet’s numbers and the upwind principle for finite volume approximations. Two types of discretization schemes with mass lumping are developed with use of bilinear or biquadratic basis functions as the trial space respectively. Some stability analyses of the schemes are presented for the model problem with constant coefficients. Various examples are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods. |
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