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

journal contribution
posted on 2023-06-08, 06:55 authored by Erik Burman, Alexandre Ern
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.

History

Publication status

  • Published

Journal

Computer Methods in Applied Mechanics and Engineering

ISSN

0045-7825

Publisher

Elsevier

Issue

35

Volume

191

Page range

3833-3855

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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