Método de Galerkin Descontínuo com Penalização de Fluxos para a Equação Reação-Difusão
DOI:
https://doi.org/10.5540/tema.2007.08.02.0287Abstract
Introduzimos um novo método de Galerkin descontínuo para problemas elípticos de segunda ordem com penalização simultânea nos saltos da solução e nos saltos dos fluxos da solução numérica. Efetuamos uma análise a priori do erro e demonstramos hp estimativas de convergência do método que são ótimas em h e quase-ótimas em p. As taxas de convergência demonstradas foram comprovados com uma série de experiências numéricas para uma solução suave do problema. Também estudamos a convergência no caso de uma solução irregular.References
[1] D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742–760.
I. Babuˇska, M. Suri, The hp version of the fnite element method with quasiuniform meshes, RAIRO, Math. Mod. Numer. Anal., 21 (1987), 199–238.
I. Babuˇska, M. Suri, The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 24 No. 4, 1987.
C.E. Baumann, “An H-P Adaptive Discontinous Finite Element Method for Computational Fluid Dinamics”, PhD thesis, The University of Texas at Austin, 1997.
E. Burman, A. Ern, Continuous interior penalty hp-finite element methods for transport operators, Math. Comp., Submitted, 2005.
E. Burman, P. Hansbo, Edge stabilization for galerkin approximations of convection-diffusion-reaction problems, Comp. Meth. Mech. Eng., 193 (2004), 1437–1453.
P. Houston, C. Schwab, E. S¨uli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM Journal of Numerical Analysis., 39 (2002), 2133–2163.
J. Nitsche, Uber ein Variationsprinzip zur L”osung von Dirichlet-Problemen bei der Verwendung von Teilr”aumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), 9–15.
J. T. Oden, I. Babuska, C. Baumann, The local discontinuous Galerkin finite element method for diffusion problems, J.Comput. Phys., 146 (1998), 491–519.
S. Prudhomme, F. Pascal, J.T. Oden, A. Romkes, Review of a priory error estimation for discontinuous Galerkin methods. Technical Report TICAM REPORT 00-27, Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, 2000.
B. Rivi`ere, Mary F. Wheeler, A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations, volume 11 of Lectures Notes in Computational Sciece and Engineering, pages 231–244. 1999.
B. Rivi`ere, M.F. Wheeler, V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. part 1, Computational Geosciences, 3 (1999), 337–360.
A. Romkes, J. T. Oden, S. Prudhomme, A priory error analyses of a stabilized discontinuous Galerkin methods, Computers and Mathematics with Applications, 46 (2003), 1289–1311.
C. Schwab, “p- and hp- Finite Element Methods. Theory and Applications to Solid and Fluid Mechanics”, Oxford University Press, 1998.
E. S¨uli, C. Schwab, P. Houston, hp-DGFEM for Partial Differential Equations with Nonegative Characteristic Form, volume 11 of Lectures Notes in Computational Science and Engineering, pages 221–230. Spriger, Berlin, 1999.
M.F. Wheeler, An elliptic collocation finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), 152–161.
Downloads
Published
How to Cite
Issue
Section
License
Copyright
Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.
Intellectual Property and Terms of Use
The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.
The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.
