A new DG method for a pure–stress formulation of the Brinkman problem with strong symmetry

Salim Meddahi; Ricardo Ruiz-Baier; Salim Meddahi; Ricardo Ruiz-Baier

doi:10.3934/nhm.2022031

Networks and Heterogeneous Media

2022, Volume 17, Issue 6: 893-916. doi: 10.3934/nhm.2022031

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A new DG method for a pure–stress formulation of the Brinkman problem with strong symmetry

Salim Meddahi ^{1
,
,},
Ricardo Ruiz-Baier ^{2,3,4
,}

1.
Facultad de Ciencias, Universidad de Oviedo, Federico García Lorca, 18, 33007-Oviedo, Spain
2.
School of Mathematics, Monash University, 9 Rainforest Walk, Clayton, Victoria 3800, Australia
3.
World-Class Research Center "Digital biodesign and personalized healthcare", Sechenov First Moscow State Medical University, Moscow, Russia
4.
Universidad Adventista de Chile, Casilla 7-D Chillán, Chile

Received: 01 April 2022 Revised: 01 August 2022 Published: 25 August 2022
65N30, 65M12, 65M15, 74H15

A strongly symmetric stress approximation is proposed for the Brinkman equations with mixed boundary conditions. The resulting formulation solves for the Cauchy stress using a symmetric interior penalty discontinuous Galerkin method. Pressure and velocity are readily post-processed from stress, and a second post-process is shown to produce exactly divergence-free discrete velocities. We demonstrate the stability of the method with respect to a DG-energy norm and obtain error estimates that are explicit with respect to the coefficients of the problem. We derive optimal rates of convergence for the stress and for the post-processed variables. Moreover, under appropriate assumptions on the mesh, we prove optimal $ L^2 $-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D.
- Mixed finite elements,
- Brinkman problem,
- discontinuous Galerkin method,
- parameter-robust error estimates
Citation: Salim Meddahi, Ricardo Ruiz-Baier. A new DG method for a pure–stress formulation of the Brinkman problem with strong symmetry[J]. Networks and Heterogeneous Media, 2022, 17(6): 893-916. doi: 10.3934/nhm.2022031

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Abstract

A strongly symmetric stress approximation is proposed for the Brinkman equations with mixed boundary conditions. The resulting formulation solves for the Cauchy stress using a symmetric interior penalty discontinuous Galerkin method. Pressure and velocity are readily post-processed from stress, and a second post-process is shown to produce exactly divergence-free discrete velocities. We demonstrate the stability of the method with respect to a DG-energy norm and obtain error estimates that are explicit with respect to the coefficients of the problem. We derive optimal rates of convergence for the stress and for the post-processed variables. Moreover, under appropriate assumptions on the mesh, we prove optimal $ L^2 $-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D.

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