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Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity

journal contribution
posted on 2023-06-08, 15:16 authored by Sören Bartels, Max Jensen, Rüdiger Müller
In this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into $L^2(\Omega)$, and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.

History

Publication status

  • Published

Journal

SIAM Journal on Numerical Analysis (SINUM)

ISSN

0036-1429

Publisher

Society for Industrial and Applied Mathematics

Issue

5

Volume

47

Page range

3720-3743

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2013-06-19

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