Numerical Mathematics and Advanced Applications pp 512-519 | Cite as
A Nonconforming Finite Element Method with Face Penalty for Advection—Diffusion Equations
Conference paper
Abstract
We present a nonconforming finite element method with face penalty to approximate advection-diffusion-reaction equations. The a priori error analysis leads to (quasi-)optimal error estimates in the mesh-size keeping the Péclet number fixed. The a posteriori error analysis yields residual-type error indicators that are semi-robust in the sense that the lower and upper bounds of the error differ by a factor bounded by the square root of the Péclet number. Finally, to illustrate the theory, numerical results including adaptively generated meshes are presented.
Keywords
Posteriori Error Posteriori Error Estimate Galerkin Approximation Error Indicator Posteriori Error Estimator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Brezzi, F., Russo, A.: Choosing bubbles for advection-diffusion problems. Math. Models Meth. Appl. Sci. 4, 571–587 (1994)MATHCrossRefMathSciNetGoogle Scholar
- 2.Brooks, A., Hughes, T.: Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32, 199–259 (1982)MATHCrossRefMathSciNetGoogle Scholar
- 3.Burman, E.: A unified analysis for conforming and non-conforming stabilized finite element methods using interior penalty. to appear in SIAM, J. Numer. Anal. (2005)Google Scholar
- 4.Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193, 1437–1453 (2004)MATHCrossRefMathSciNetGoogle Scholar
- 5.Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming mixed finite element methods for solving the stationary Stokes equations I. RAIRO Modél Math. Anal. Numér. 3, 33–75 (1973)MathSciNetGoogle Scholar
- 6.El Alaoui, L., Ern, A., Burman E.: A priori and a posteriori error analysis of nonconforming finite elements with face penalty for advection-diffusion-reaction equations. Submitted (2005) [CERMICS Technical Report 2005–289]Google Scholar
- 7.Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Springer, New York, 2004MATHGoogle Scholar
- 8.Guermond, J.-L.: Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA, Journal of Numerical Analysis. 21, 165–197 (2001)MATHCrossRefMathSciNetGoogle Scholar
- 9.Johnson, C. and Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)MATHCrossRefGoogle Scholar
- 10.Matthies, G. and Tobiska, L.: The streamline-diffusion method for conforming and nonconforming finite elements of lowest order applied to convection- diffusion problems. Computing. 66, 343–364 (2001)MATHCrossRefMathSciNetGoogle Scholar
- 11.Sangalli, G.: On robust a posteriori estimators for the advection-diffusionreaction problem. Technical Report 04–55, ICES, (2004)Google Scholar
- 12.Verfürth, R.: A posteriori error estimators for convection-diffusion equations. Numer. Math. 80, 641–663 (1998)MATHCrossRefMathSciNetGoogle Scholar
- 13.Verfürth, R.: Robust a posteriori error estimates for stationary convection- diffusion equations. SIAM, J. Numer. Anal., (2004) (submitted)Google Scholar
Copyright information
© Springer 2006