A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

Abstract

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete unknowns, we reconstruct inside each element a Stokes velocity one degree higher than face unknowns, and a Darcy velocity in the Raviart-Thomas-Nédélec space. These reconstructed velocities are respectively used to formulate the discrete versions of the Stokes and Darcy terms in the momentum equation, along with suitably designed penalty contributions. The proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local (Stokes-or Darcydominated) regimes, as identified by a dimensionless number which can be interpreted as a friction coefficient. The singular limit corresponding to the Darcy equation is also fully supported by the method. Numerical examples corroborate the theoretical results. This paper also contains two contributions whose interest goes beyond the specific method and application treated in this work: an investigation of the dependence of the constant in the second Korn inequality on star-shaped domains and its application to the study of the approximation properties of the strain projector in general Sobolev seminorms.

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

“…(ii) Stability. We proceed as in [27,Lemma 17]. From the surjectivity of the continuous divergence operator from H 1 0 (Ω) d to L 2 0 (Ω) (see, e.g., [55, Section 2.2]), we infer the existence of v q h ∈ H 1 0 (Ω) d such that −∇·v q h = q h and v q h H 1 (Ω) d q h , with hidden constant depending only on Ω.…”

“…For a study of the relations among HDG, HHO, and VEM in the context of scalar diffusion problems, we refer the reader to [23,24]. Finally, as regards HHO methods, applications to incompressible flows have been considered in [25,26,20,27].…”

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

“…Recently, some novel methods have been proposed to solve the Brinkman problem on general quadrilateral and polygonal meshes, which is tricky and usually requires special treatment in order to deliver robust results with respect to the rough grids. The development of numerical methods for Brinkman problem on general meshes is still in its infancy and the existing methods that have been successfully designed on general meshes for Brinkman problem include the weak Galerkin method, the virtual element method and the hybrid high-order methods [35,12,37,7].…”

Bricks for the mixed high-order virtual element method: Projectors and differential operators

“…(29)), f h , g (cf. (26)) the global version of the div-matrix and discrete right-hand side respectively. Denoting with GNDoF the total amount of global velocity DoFs, the velocity and pressure solutions can be expressed in terms of the global bases…”

“…The Virtual Element Method has been developed successfully for a large range of mathematical and engineering problems, we mention, as sample, the very brief list of papers [10,28,32,18,46,6,33,7], while for the specific topic of implementation aspects related to the VEM we mention [43,25,14,24,4,8,42,35,45,19]. Concernig the mixed PDEs we refer to [41,30,29,37,31] as a sample of VEM papers dealing with such kind of problem, and to [40,34,26,36] as a representative list of papers treating the same topic with different polytopal technologies.…”