Volume 11, Issue 3
Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems

Junliang Lv, Guangwei Yuan & Jingyan Yue

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 413-436.

Published online: 2018-11

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  • Abstract

In the numerical simulation for nonlinear diffusion problems with degenerate diffusion coefficients, some classical methods are often invalid since they involve a harmonic mean of diffusion coefficients on the adjacent cells. To avoid such problem, we consider using linear finite element method to solve a class of 1D degenerate nonlinear parabolic equations. This method can effectively capture the profile of true solution, but at the front it generates some nonphysical numerical oscillations, even brings forth negative values in numerical solution for approximating nonnegative physical quantities. In order to preserve the nonnegativity of true solution, we discuss three repair techniques for finite element solutions based on a posteriori corrections. The first one is a zero-setting method, in which we directly set those negative values to be zero. The second one is a local approach, in which any negative energy associate to some node is absorbed by the positive values around the current node. The third one is a global strategy, in which the total negative energy is redistributed to all positive values with a one-time effort. Numerical examples show that the numerical solution profile is improved remarkably by using the repair techniques for degenerate parabolic equations.

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@Article{NMTMA-11-413, author = {}, title = {Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {3}, pages = {413--436}, abstract = {

In the numerical simulation for nonlinear diffusion problems with degenerate diffusion coefficients, some classical methods are often invalid since they involve a harmonic mean of diffusion coefficients on the adjacent cells. To avoid such problem, we consider using linear finite element method to solve a class of 1D degenerate nonlinear parabolic equations. This method can effectively capture the profile of true solution, but at the front it generates some nonphysical numerical oscillations, even brings forth negative values in numerical solution for approximating nonnegative physical quantities. In order to preserve the nonnegativity of true solution, we discuss three repair techniques for finite element solutions based on a posteriori corrections. The first one is a zero-setting method, in which we directly set those negative values to be zero. The second one is a local approach, in which any negative energy associate to some node is absorbed by the positive values around the current node. The third one is a global strategy, in which the total negative energy is redistributed to all positive values with a one-time effort. Numerical examples show that the numerical solution profile is improved remarkably by using the repair techniques for degenerate parabolic equations.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017-OA-0125}, url = {http://global-sci.org/intro/article_detail/nmtma/12437.html} }
TY - JOUR T1 - Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 413 EP - 436 PY - 2018 DA - 2018/11 SN - 11 DO - http://doi.org/10.4208/nmtma.2017-OA-0125 UR - https://global-sci.org/intro/article_detail/nmtma/12437.html KW - AB -

In the numerical simulation for nonlinear diffusion problems with degenerate diffusion coefficients, some classical methods are often invalid since they involve a harmonic mean of diffusion coefficients on the adjacent cells. To avoid such problem, we consider using linear finite element method to solve a class of 1D degenerate nonlinear parabolic equations. This method can effectively capture the profile of true solution, but at the front it generates some nonphysical numerical oscillations, even brings forth negative values in numerical solution for approximating nonnegative physical quantities. In order to preserve the nonnegativity of true solution, we discuss three repair techniques for finite element solutions based on a posteriori corrections. The first one is a zero-setting method, in which we directly set those negative values to be zero. The second one is a local approach, in which any negative energy associate to some node is absorbed by the positive values around the current node. The third one is a global strategy, in which the total negative energy is redistributed to all positive values with a one-time effort. Numerical examples show that the numerical solution profile is improved remarkably by using the repair techniques for degenerate parabolic equations.

Junliang Lv, Guangwei Yuan & Jingyan Yue. (2020). Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems. Numerical Mathematics: Theory, Methods and Applications. 11 (3). 413-436. doi:10.4208/nmtma.2017-OA-0125
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