Abstract
A class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of these conditions. The theoretical results are illustrated with numerical examples.
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Ainsworth M., Rankin R., Technical Note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes, Numer. Methods Partial Differential Equations, 2012, 28(3), 1099–1104
Arnold D.N., Brezzi F., Cockburn B., Marini L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 2001/02, 39(5), 1749–1779
Berman A., Plemmons R.J., Nonnegative Matrices in the Mathematical Sciences, Comput. Sci. Appl. Math., Academic Press, New York-London, 1979
Ciarlet P.G., Discrete maximum principle for finite-difference operators, Aequationes Math., 1970, 4(3), 338–352
Ciarlet P.G., Raviart P.-A., Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 1973, 2, 17–31
Di Pietro D.A., Ern A., Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. (Berlin), 69, Springer, Heidelberg, 2012
Ern A., Guermond J.-L., Theory and Practice of Finite Elements, Appl. Math. Sci., 159, Springer, New York, 2004
Evans L.C., Partial Differential Equations, Grad. Stud. Math., 19, American Mathematical Society, Providence, 1997
Faragó I., Horváth R., A review of reliable numerical models for three-dimensional linear parabolic problems, Internat. J. Numer. Methods Engrg., 2007, 70(1), 25–45
Hannukainen A., Korotov S., Vejchodský T., On weakening conditions for discrete maximum principles for linear finite element schemes, In: Numerical Analysis and its Applications, Lozenetz, June 16–20, 2008, Lecture Notes in Comput. Sci., 5434, Springer, Berlin-Heidelberg, 2009, 297–304
Houston P., Süli E., Wihler T.P., A posteriori error analysis of hp-version discontinuous Galerkin finite-element methods for second-order quasi-linear elliptic PDEs, IMA J. Numer. Anal., 2008, 28(2), 245–273
Höhn W., Mittelmann H.-D., Some remarks on the discrete maximum-principle for finite elements of higher order, Computing, 1981, 27(2), 145–154
Mincsovics M.E., Horváth T.L., On the differences of the discrete weak and strong maximum principles for elliptic operators, In: Large-Scale Scientific Computing, Sozopol, June 6–10, 2011, Lecture Notes in Comput. Sci., 7116, Springer, Berlin-Heidelberg, 2012, 614–621
Rivière B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Frontiers Appl. Math., 35, Society for Industrial and Applied Mathematics, Philadelphia, 2008
Ruas Santos V., On the strong maximum principle for some piecewise linear finite element approximate problems of nonpositive type, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 1982, 29(2), 473–491
Vejchodský T., Discrete Maximum Principles, habilitation thesis, Institute of Mathematics of the Academy of Sciences and Faculty of Mathematics and Physics, Charles University, Prague, 2011
Vejchodský T., Šolín P., Discrete maximum principle for higher-order finite elements in 1D, Math. Comp., 2007, 76(260), 1833–1846
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Horváth, T.L., Mincsovics, M.E. Discrete maximum principle for interior penalty discontinuous Galerkin methods. centr.eur.j.math. 11, 664–679 (2013). https://doi.org/10.2478/s11533-012-0154-z
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DOI: https://doi.org/10.2478/s11533-012-0154-z