BAIL 2006 Book of Abstracts - Institut für Numerische und ...

P. KNOBLOCH: On methods diminishing spurious oscillations in finite element
solutions of convection-diffusion equations
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to be the methods of do Carmo and Galeão [5], of Almeida and Silva [1] and slight modifications of the
methods of Codina [6] and of Burman and Ern [3], see also John and Knobloch [8].
Our aim is to continue the discussion of properties of the various methods diminishing spurious
oscillations in finite element solutions of (1) mentioned above. We would like to concentrate on three
aspects:
• properties of methods of Mizukami–Hughes type satisfying the discrete maximum principle where
we mention the improved Mizukami–Hughes method of [9] for P1 finite elements and will discuss
its generalization to other types of finite elements;
• comparison of the above–mentioned discontinuity–capturing methods for higher order finite elements;
• choice of stabilization parameters where we will present a novel approach to the choice of the
SUPG stabilization parameter in outflow boundary layers.
The above described discussion will be performed for the two–dimensional case and the results obtained
in three–dimensions will be mentioned only briefly.
References
[1] R.C. Almeida, R.S. Silva, A stable Petrov–Galerkin method for convection–dominated problems,
Comput. Methods Appl. Mech. Eng. 140 (1997) 291–304.
[2] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection
dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput.
Methods Appl. Mech. Eng. 32 (1982) 199–259.
[3] E. Burman, A. Ern, Nonlinear diffusion and discrete maximum principle for stabilized Galerkin
approximations of the convection–diffusion–reaction equation, Comput. Methods Appl. Mech.
Eng. 191 (2002) 3833–3855.
[4] E.G.D. do Carmo, G.B. Alvarez, A new stabilized finite element formulation for scalar convection–
diffusion problems: The streamline and approximate upwind/Petrov–Galerkin method, Comput.
Methods Appl. Mech. Eng. 192 (2003) 3379–3396.
[5] E.G.D. do Carmo, A.C. Galeão, Feedback Petrov–Galerkin methods for convection–dominated
problems, Comput. Methods Appl. Mech. Eng. 88 (1991) 1–16.
[6] R. Codina, A discontinuity–capturing crosswind–dissipation for the finite element solution of the
convection–diffusion equation, Comput. Methods Appl. Mech. Eng. 110 (1993) 325–342.
[7] V. John, P. Knobloch, A comparison of spurious oscillations at layers diminishing (SOLD) methods
for convection–diffusion equations: Part I, Preprint Nr. 156, FR 6.1 – Mathematik, Universität des
Saarlandes, Saarbrücken, 2005.
[8] V. John, P. Knobloch, On discontinuity–capturing methods for convection–diffusion equations, submitted
to the Proceedings of the conference ENUMATH 2005, Santiago de Compostela, July 18–22,
2005.
[9] P. Knobloch, Improvements of the Mizukami–Hughes method for convection–diffusion equations,
Preprint No. MATH–knm–2005/6, Faculty of Mathematics and Physics, Charles University,
Prague, 2005.
[10] Knobloch, P.: Numerical solution of convection–diffusion equations using upwinding techniques
satisfying the discrete maximum principle. Submitted to the Proceedings of the Czech–Japanese
Seminar in Applied Mathematics 2005, Kuju Training Center, Oita, Japan, September 15–18, 2005.
Speaker: KNOBLOCH, P. 103 BAIL 2006
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