FLUX BALANCED APPROXIMATION WITH LEAST-SQUARES GRADIENT FOR DIFFUSION EQUATION ON POLYHEDRAL MESH

PETER FROLKOVIC; KAROL MIKULA; JOOYOUNG HAHN; DIRK MARTIN; BRANISLAV BASARA

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FLUX BALANCED APPROXIMATION WITH LEAST-SQUARES GRADIENT FOR DIFFUSION EQUATION ON POLYHEDRAL MESH

机译：用最小二乘梯度在多面体网上扩散方程的磁通平衡近似

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摘要

A numerical method for solving diffusion problems on polyhedralmeshes is presented. It is based on a finite volume approximation with thedegrees of freedom located in the centers of computational cells. A numericalgradient is defined by a least-squares minimization for each cell, where wesuggest a restricted form in the case of discontinuous diffusion coefficient. Theux balanced approximation is proposed without numerically computing thegradient itself at the faces of computational cells in order to find a normaldiffusive ux. To apply the method for parallel computations with a 1-ringneighborhood, we use an iterative method to solve the obtained system ofalgebraic equations. Several numerical examples illustrate some advantages ofthe proposed method.

机译：一种求解多面体扩散问题的数值方法呈现网格。它基于有限体积近似位于计算单元中心的自由度。一个数字梯度由每个单元格的最小平方最小化定义，我们在不连续扩散系数的情况下提出限制形式。这提出了UX平衡近似，而无需数值计算梯度本身在计算单元的面上，以找到正常的扩散UX。用1圈应用一个平行计算的方法邻居，我们使用迭代方法来解决所获得的系统代数方程。几个数字示例说明了一些优点所提出的方法。

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来源
《Discrete and Continuous Dynamical Systems,Series S》 |2021年第3期|865-879|共15页
作者
PETER FROLKOVIC; KAROL MIKULA; JOOYOUNG HAHN; DIRK MARTIN; BRANISLAV BASARA;
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Faculty of Civil Engineering Slovak University of TechnologyDepartment of Mathematics and Descriptive GeometryRadlinskeho 11 810 05 Bratislava Slovak Republic;

Faculty of Civil Engineering Slovak University of TechnologyDepartment of Mathematics and Descriptive GeometryRadlinskeho 11 810 05 Bratislava Slovak Republic;

Advanced Simulation Technologies AVL List GmbHHans-List Platz 1 8010 Graz Austria;

Advanced Simulation Technologies AVL List GmbHHans-List Platz 1 8010 Graz Austria;

Advanced Simulation Technologies AVL List GmbHHans-List Platz 1 8010 Graz Austria;

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正文语种 eng
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关键词
Finite volume methods; polyhedral mesh; least-squares gradient; deferred correction; cross diffusion;

机译：有限体积方法;多面体网状;最小二乘梯度;延期纠正;交叉扩散;

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专利

1. Approximations with piecewise constant fluxes for diffusion equations on polyhedral meshes: algorithms and applications [J] . O. BOYARKIN, E. KIKINZON, Yu. A. KUZNETSOV, Russian Journal of Numerical Analysis and Mathematical Modelling . 2012,第4期

机译：多面体网格上扩散方程的分段恒定通量近似值：算法和应用
2. Quasi-optimal preconditioners for finite element approximations of diffusion dominated convection-diffusion equations on (nearly) equilateral triangle meshes [J] . Russo Alessandro, Serra-Capizzano Stefano, Tablino-Possio Cristina Numerical linear algebra with applications . 2015,第1期

机译：（近）等边三角形网格上扩散占主导地位的对流扩散方程的有限元逼近的拟最佳预处理器
3. A vertex-centered and positivity-preserving scheme for anisotropic diffusion equations on general polyhedral meshes [J] . Su Shuai, Dong Qiannan, Wu Jiming Mathematical Methods in the Applied Sciences . 2019,第1期

机译：一般多面体网眼各向异性扩散方程的顶点居中和正阳性保存方案
4. Grid Approximations of the Solution and Diffusion Flux for Singularly Perturbed Equations with Neumann Boundary Conditions [C] . Grigorii I.Shishkin International workshop on numerical analysis and its applications . 1997

机译：用于尼姆农边界条件的单个扰动方程的溶液和扩散通量的栅极近似
5. Numerical Approximations for the Fractional Laplacian in Space-Fractional Reaction-Diffusion Equations [D] . ?Alzahrani, Saham 2020

机译：空间 - 分数反应扩散方程分数拉普拉斯的数值近似
6. Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes [O] . Gabriel R. Barrenechea, Erik Burman, Fotini Karakatsani -1

机译：对流扩散方程有限元逼近的基于边缘的非线性扩散及其与代数通量校正方案的关系
7. Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes [O] . Barrenechea, Gabriel, Burman, Erik, Karakatsani, Fotini 2016

机译：对流扩散方程有限元逼近的基于边缘的非线性扩散及其与代数通量校正方案的关系
8. Investigation of discretization errors for mesh centered finite difference approximations to the transport equation using a spherical harmonic expansion of the flux [R] . Fletcher, J. K. 1997

机译：利用磁通量的球谐函数展开研究网格中心有限差分近似与传输方程的离散化误差

FLUX BALANCED APPROXIMATION WITH LEAST-SQUARES GRADIENT FOR DIFFUSION EQUATION ON POLYHEDRAL MESH

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