Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation

By Erik Burman, Alexandre Ern. 2002

104 citations recorded by the source corpus.

We investigate stabilized Galerkin approximations of linear and nonlinear convectiondiffusion-reaction equations. We derive nonlinear streamline and cross-wind diffusion methods that guarantee a discrete maximum principle for strictly acute meshes and first order polynomial interpolation. For pure convectiondiffusion problems, the discrete maximum principle is achieved using a nonlinear cross-wind diffusion factor that depends on the angle between the discrete solution and the flow velocity. For convectiondiffusion-reaction problems, two methods are considered: residual based, isotropic diffusion and the previous nonlinear cross-wind diffusion factor supplemented by additional isotropic diffusion scaling as the square of the mesh size. Practical versions of the present methods suitable for numerical implementation are compared to previous discontinuity capturing schemes lacking theoretical justification. Numerical results are investigated in terms of both solution quality (violation of maximum principle, smearing of internal layers) and computational costs.

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/276a28373261a491b838582fe3127ff148adf716

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In Computational Mechanics

1. Petr Knobloch
2. Volker John
3. Julia Novo
4. Timo Heister
5. Thomas Richter
6. Gerd Rapin
7. Malte Braack

In Computational Theory and Mathematics

1. Petr Knobloch
2. Volker John

Citations in the corpus

• On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review (2007, methodology): “Burman and Ern [12] proposed to use the SOLD term (36) with WKðuhÞ defined by WKðuhÞjE ¼ Cjbj½diamðKÞ 2j½jruhj Ej 8E oK; ð38Þ where C is a suitable constant.”
• On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II – Analysis for P1 and Q1 finite elements (2008, methodology): “In contrast, the methods of do Carmo and Galeão, (5) and (6), and of Burman and Ern, (8) and (13), whose parameters correspond to the dash-dot line, should produce solutions with larger oscillations at the interior layer.”