# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2020; 57(5): 1239-1266

Published online September 1, 2020 https://doi.org/10.4134/JKMS.j190628

## Minimal locally stabilized \textit{Q1-Q0} schemes for the generalized Stokes problem

Alima Chibani, Nasserdine Kechkar

University Fr\{e}res Mentouri; University Fr\{e}res Mentouri

### Abstract

In this paper, some novel discrete formulations for stabilizing the mixed finite element method \textit{Q1-Q0} (bilinear velocity and constant pressure approximations) are introduced and discussed for the generalized Stokes problem. These are based on stabilizing discontinuous pressure approximations via local jump operators. The developing idea consists in a reduction of terms in the local jump formulation, introduced earlier, in such a way that stability and convergence properties are preserved. The computer implementation aspects and numerical evaluation of these stabilized discrete formulations are also considered. For illustrating the numerical performance of the proposed approaches and comparing the three versions of the local jump methods alongside with the global jump setting, some obtained results for two test generalized Stokes problems are presented. Numerical tests confirm the stability and accuracy characteristics of the resulting approximations.

Keywords: Finite elements, mixed methods, generalized Stokes problem, stabilization

MSC numbers: Primary 65N30, 65N12, 65N15, 76D07