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Convergence of a mixed method for a semi-stationary compressible Stokes system

Authors: Kenneth H. Karlsen and Trygve K. Karper
Journal: Math. Comp. 80 (2011), 1459-1498
MSC (2010): Primary 35Q30, 74S05; Secondary 65M12
Published electronically: December 6, 2010
MathSciNet review: 2785465
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Abstract: We propose and analyze a finite element method for a semi- stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nédélec spaces of the first kind, while the continuity equation is approximated by a piecewise constant upwind discontinuous Galerkin method. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only $\operatorname {div}$ or $\operatorname {curl}$ conforming. (ii) To prove the strong convergence of the discontinuous Galerkin approximations, which is required in view of a nonlinear pressure function. Some tools involved in the analysis include a higher space-time integrability estimate for the discontinuous Galerkin approximations, an equation for the effective viscous flux, various renormalized formulations of the discontinuous Galerkin method, and weak convergence arguments.

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Additional Information

Kenneth H. Karlsen
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway and Center for Biomedical Computing, Simula Research Laboratory, P.O. Box 134, N–1325 Lysaker, Norway

Trygve K. Karper
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Keywords: Semi-stationary Stokes system, compressible fluid flow, Navier-slip boundary condition, mixed finite element method, discontinuous Galerkin method, convergence
Received by editor(s): April 4, 2009
Received by editor(s) in revised form: April 28, 2010
Published electronically: December 6, 2010
Additional Notes: The authors thank the anonymous referees for many valuable comments leading to improvements in the paper. This work was supported by the Research Council of Norway through an Outstanding Young Investigators Award (K. H. Karlsen). This article was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.