×

A finite element time relaxation method. (English) Zbl 1305.76052

Summary: We discuss a finite element time-relaxation method for high Reynolds number flows. The method uses local projections on polynomials defined on macroelements of each pair of two elements sharing a face. We prove that this method shares the optimal stability and convergence properties of the continuous interior penalty (CIP) method. We give the formulation both for the scalar convection-diffusion equation and the time-dependent incompressible Euler equations and the associated convergence results. This note finishes with some numerical illustrations.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, N. A.; Stolz, S., A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178, 391-426 (2002) · Zbl 1139.76319
[2] Becker, R.; Braack, M., A two-level stabilization scheme for the Navier-Stokes equations, (Numerical Mathematics and Advanced Applications (2004), Springer: Springer Berlin), 123-130 · Zbl 1198.76062
[3] Burman, E., Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation, Comput. Methods Appl. Mech. Engrg., 196, 4045-4058 (2007) · Zbl 1173.76332
[4] Burman, E.; Fernández, M. A., Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: Space discretization and convergence, Numer. Math., 107, 39-77 (2007) · Zbl 1117.76032
[5] Burman, E.; Guzmàn, J.; Leykekhman, D., Weighted error estimates of the continuous interior penalty method for singularly perturbed problems, IMA J. Numer. Anal., 29, 284-314 (2009) · Zbl 1166.65054
[6] Burman, E.; Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems, Comput. Methods Appl. Mech. Engrg., 193, 1437-1453 (2004) · Zbl 1085.76033
[7] Connors, J.; Layton, W., On the accuracy of the finite element method plus time relaxation, Math. Comp., 79, 619-648 (2010) · Zbl 1201.65168
[8] Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modeling, Modél. Math. Anal. Numér., 33, 1293-1316 (1999) · Zbl 0946.65112
[9] Hansbo, P.; Szepessy, A., A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 84, 175-192 (1990) · Zbl 0716.76048
[10] Principe, J.; Codina, R.; Henke, F., The dissipative structure of variational multiscale methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 199, 791-801 (2010) · Zbl 1406.76034
[11] Smagorinsky, J., Some historical remarks on the use of nonlinear viscosities, (Large Eddy Simulation of Complex Engineering and Geophysical Flows (1993), Cambridge Univ. Press: Cambridge Univ. Press New York), 3-36
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.