scholarly journals Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition

2019 ◽  
Vol 40 (4) ◽  
pp. 2696-2716
Author(s):  
Changjie Fang ◽  
Kenneth Czuprynski ◽  
Weimin Han ◽  
Xiaoliang Cheng ◽  
Xiaoxia Dai
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Directional Derivative ◽  
Slip Boundary Condition ◽  
Hemivariational Inequality ◽  
Slip Boundary ◽  
Element Method

Abstract This paper is devoted to the study of a hemivariational inequality problem for the stationary Stokes equations with a nonlinear slip boundary condition. The hemivariational inequality is formulated with the use of the generalized directional derivative and generalized gradient in the sense of Clarke. We provide an existence and uniqueness result for the hemivariational inequality. Then we apply the finite element method to solve the hemivariational inequality. The incompressibility constraint is treated through a mixed formulation. Error estimates are derived for numerical solutions. Numerical simulation results are reported to illustrate the theoretically predicted convergence orders.

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

2019 ◽  
Vol 53 (3) ◽  
pp. 869-891
Author(s):  
Takahito Kashiwabara ◽  
Issei Oikawa ◽  
Guanyu Zhou
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Trace Operator ◽  
Essential Boundary Condition ◽  
Slip Boundary ◽  
Polyhedral Domain ◽  
Discrete Counterpart ◽  
Original Domain

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ 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., continuous right 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) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.

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