A discontinuous skeletal method for the viscosity-dependent Stokes problem

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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik

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 ≥ 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 divergencepreserving velocity econstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-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.

Stokes problem, mixed methods, curl-free forces, higher-order reconstruction, superconvergence, hybrid discontinuous Galerkin method, static condensation.
Ern, A., Di Pietro, D., Linke, A., & Schieweck, F. (2015). A discontinuous skeletal method for the viscosity-dependent Stokes problem (Version publishedVersion, Vol. 2195). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.