Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes

Abstract

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\mathbf{H}(\mathrm{div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\mathbf{H}(\mathbf{curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.

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Cited by in corpus (8)

• A new family of semi-implicit Finite Volume / Virtual Element methods for incompressible flows on unstructured meshes
• A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations
• An exterior calculus framework for polytopal methods
• Cohomology of the discrete de Rham complex on domains of general topology
• A Reynolds-semi-robust method with hybrid velocity and pressure for the unsteady incompressible Navier--Stokes equations
• A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations
• An arbitrary-order fully discrete Stokes complex on general polyhedral meshes
• A pressure-robust HHO method for the solution of the incompressible Navier-Stokes equations on general meshes