2018/09/25 by Takahito Kashiwabara, Issei Oikawa, Kashiwabara, Takahito +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1809.09464
openalex publication_date 2018/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω⊂ \mathbb RN (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 Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator H1(Ω)N → H1/2(∂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(hα+ ε) and O(h2α + ε) for the velocity in the H1- and L2-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 ε in the estimates.