Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes

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• 作者：Gabriel R. Barrenechea ; Erik Burman ; Fotini Karakatsani
• 关键词：Mathematics Subject Classification65N30 ; 65N12
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：135
• 期：2
• 页码：521-545
• 全文大小：864KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

For the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.