Stabilized Finite Element Methods Based on Multiscale Enrichment for the Stokes Problem

Abstract

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov-Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods. Introduction.Finite element solution of the Stokes problem poses the basic problem of satisfying the discrete Babuska-Brezzi (or inf-sup) condition (see [24] and the references therein). This is indeed a restriction from the point of view of implementation since equal order velocity and pressure spaces do not satisfy this condition. On the other hand, the minimal space to imagine, namely continuous piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure, does not satisfy this condition either.Several solutions have been proposed to overcome this restriction, starting with that in [11] and the first consistent method in [28]. Moreover,in [23,27,29,34] the possibility of considering discontinuous spaces for the pressure was considered and justified. On the other hand, in [14,13], the idea from [16] has been used to propose a new kind of stabilized finite element methods, with stabilizing terms now containing only jump terms across the interelement boundaries. For an overview of stabilized finite element methods for the Stokes problem, see [19] and [5].On the other hand, the theoretical justification of stabilized methods has become a subject of interest in the last decade. In [2,3,4,31], the connection between stabilized finite element methods and Galerkin methods enriched with bubble functions has been used to propose new stabilized finite element methods for Stokes-like and linearized Navier-Stokes problems. Also, in [22] macro bubbles were used to derive a method analogous to the locally stabilized method from [29] containing jump terms across the interelement boundaries of the macroelements. In the resulting method, the stabilizing

Analysis of a finite volume element method for the Stokes problem

“…Moreover, the trial velocity space is enriched with a function that does not vanish on the element boundary, which is split into a bubble part and an harmonic extension of the boundary condition. An essential point in our analysis is based on one of the formulations presented in [2]. The main ingredients of that idea are included here for sake of completeness.…”

“…To overcome this difficulty, we include a stabilization correction similar to that introduced in [2]. In that paper, and differently than other stabilization techniques available, the stabilization parameter corresponding to the jump terms is known.…”

“…Further, let us denote H m (Ω) = H m (Ω) 2 , and in general M will denote the corresponding vectorial counterpart of the scalar space M . For a subset R ⊂ Ω, (·, ·) R denotes the L 2 (R)-inner product.…”

“…Then the enriched part of the solution is completely identified. A static condensation procedure (see the detailed development in [2]) allows to derive the following stabilized method:…”

“…9.4] and the references therein), in this paper we include a stabilization technique similar to the one introduced by Franca et al [16], in which a Petrov-Galerkin approach is used to enrich the trial space with bubble functions being solutions to a local problem involving the residual of the momentum equation, which can be solved analytically. As recently proposed by Araya et al [2], by enriching the velocity space using a multiscale approach combined with static condensation, the resulting FE method includes the classical GLS additional terms at the element level and a suitable jump term on the normal derivative of the velocity field at the element boundaries. For the latter, the stabilization parameter is known exactly.…”