Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

Abstract

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ R N (N = 2, 3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n ∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ω h before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω = Ω h , that is, the issues of domain perturbation. In particular, the approximation of n ∂Ω by n ∂Ω h makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator H 1 (Ω) N → H 1/2 (∂Ω); u → u·n ∂Ω . In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(h α + ǫ) and O(h 2α + ǫ) for the velocity in the H 1 -and L 2 -norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705-740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ǫ in the estimates.

A Strongly Mass Conservative Method for the Coupled Brinkman-Darcy Flow and Transport

“…We remark that the boundary condition employed for the Brinkman region mimics the slip boundary condition (cf. [23]).…”