A discontinuous skeletal method for the viscosity-dependent Stokes problem. (English) Zbl 1436.76022
Summary: We devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree \(k \geq 0\), and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order \((k + 1)\). The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an \(L^2\)-pressure estimate of order \((k + 1)\) and an \(L^2\)-velocity estimate of order \((k + 2)\), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
Stokes problem; mixed methods; curl-free forces; higher-order reconstruction; hybrid discontinuous Galerkin method; static condensationReferences:
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