This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys the state of the art in the field of the discrete maximum principle and provides new generalizations of several results.
@article{bwmeta1.element.doi-10_2478_s11533-011-0085-0,
author = {Tom\'a\v s Vejchodsk\'y},
title = {The discrete maximum principle for Galerkin solutions of elliptic problems},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {25-43},
zbl = {1247.65151},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0085-0}
}
Tomáš Vejchodský. The discrete maximum principle for Galerkin solutions of elliptic problems. Open Mathematics, Tome 10 (2012) pp. 25-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0085-0/
[1] Bramble J.H., Hubbard B.E., New monotone type approximations for elliptic problems, Math. Comp., 1964, 18, 349–367 http://dx.doi.org/10.1090/S0025-5718-1964-0165702-X | Zbl 0124.33006
[2] Bramble J.H., Hubbard B.E., On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type, Journal of Mathematics and Physics, 1964, 43, 117–132 | Zbl 0126.32305
[3] Brandts J.H., Korotov S., Křížek M., Dissection of the path-simplex in ℝn into n path-subsimplices, Linear Algebra Appl., 2007, 421(2–3), 382–393 http://dx.doi.org/10.1016/j.laa.2006.10.010 | Zbl 1112.51006
[4] Brandts J.H., Korotov S., Křížek M., Simplicial finite elements in higher dimensions, Appl. Math., 2007, 52(3), 251–265 http://dx.doi.org/10.1007/s10492-007-0013-6 | Zbl 1164.65493
[5] Brandts J.H., Korotov S., Křížek M., The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem, Linear Algebra Appl., 2008, 429(10), 2344–2357 http://dx.doi.org/10.1016/j.laa.2008.06.011 | Zbl 1154.65086
[6] Brandts J., Korotov S., Křížek M., Šolc J., On nonobtuse simplicial partitions, SIAM Rev., 2009, 51(2), 317–335 http://dx.doi.org/10.1137/060669073 | Zbl 1172.51012
[7] Ciarlet P.G., Discrete variational Green’s function. I, Aequationes Math., 1970, 4(1–2), 74–82 http://dx.doi.org/10.1007/BF01817748 | Zbl 0194.12703
[8] Ciarlet P.G., Discrete maximum principle for finite-difference operators, Aequationes Math., 1970, 4(3), 338–352 http://dx.doi.org/10.1007/BF01844166 | Zbl 0198.14601
[9] Ciarlet P.G., The Finite Element Method for Elliptic Problems, Stud. Math. Appl., 4, North-Holland, Amsterdam-New York-Oxford, 1978 | Zbl 0383.65058
[10] Ciarlet P.G., Raviart P.-A., Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 1973, 2(1), 17–31 http://dx.doi.org/10.1016/0045-7825(73)90019-4 | Zbl 0251.65069
[11] Ciarlet P.G., Varga R.S., Discrete variational Green’s function. II. One dimensional problem, Numer. Math., 1970, 16(2), 115–128 http://dx.doi.org/10.1007/BF02308864 | Zbl 0245.34012
[12] Drăgănescu A., Dupont T.F., Scott L.R., Failure of the discrete maximum principle for an elliptic finite element problem, Math. Comp., 2005, 74(249), 1–23 http://dx.doi.org/10.1090/S0025-5718-04-01651-5 | Zbl 1074.65129
[13] Duffy D.G., Green’s Functions with Applications, Stud. Adv. Math., Chapman&Hall/CRC, Boca Raton, 2001 | Zbl 0983.35003
[14] Eppstein D., Sullivan J.M., Üngör A., Tiling space and slabs with acute tetrahedra, Comput. Geom., 2004, 27(3), 237–255 http://dx.doi.org/10.1016/j.comgeo.2003.11.003 | Zbl 1054.65020
[15] Faragó I., Horváth R., Discrete maximum principle and adequate discretizations of linear parabolic problems, SIAM J. Sci. Comput., 2006, 28(6), 2313–2336 http://dx.doi.org/10.1137/050627241 | Zbl 1130.65086
[16] 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 http://dx.doi.org/10.1002/nme.1863 | Zbl 1194.80119
[17] Faragó I., Horváth R., Korotov S., Discrete maximum principle for linear parabolic problems solved on hybrid meshes, Appl. Numer. Math., 2005, 53(2–4), 249–264 http://dx.doi.org/10.1016/j.apnum.2004.09.001 | Zbl 1070.65094
[18] Faragó I., Korotov S., Szabó T., On modifications of continuous and discrete maximum principles for reaction-diffusion problems, Adv. Appl. Math. Mech., 2011, 3(1), 109–120 | Zbl 1262.35124
[19] Fiedler M., Special Matrices and their Applications in Numerical Mathematics, Martinus Nijhoff, Dordrecht, 1986 http://dx.doi.org/10.1007/978-94-009-4335-3
[20] Fujii H., Some remarks on finite element analysis of time-dependent field problems, In: Theory and Practice in Finite Element Structural Analysis, Univ. Tokyo Press, Tokyo, 1973, 91–106
[21] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin-New York, 1977 | Zbl 0361.35003
[22] Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer Ser. Comput. Phys., Springer, New York, 1984 | Zbl 0536.65054
[23] Ikeda T., Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Lecture Notes Numer. Appl. Anal., 4, Kinokuniya Book Store, Tokyo, 1983
[24] Karátson J., Korotov S., Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math., 2005, 99(4), 669–698 http://dx.doi.org/10.1007/s00211-004-0559-0 | Zbl 1067.65127
[25] Knobloch P., Tobiska L., On the stability of finite-element discretizations of convection-diffusion-reaction equations, IMA J. Numer. Anal., 2011, 31(1), 147–164 http://dx.doi.org/10.1093/imanum/drp020 | Zbl 1211.65147
[26] Křížek M., There is no face-to-face partition of R5 into acute simplices, Discrete Comput. Geom., 2006, 36(2), 381–390 http://dx.doi.org/10.1007/s00454-006-1244-0 | Zbl 1103.52008
[27] Křížek M., Liu L., On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type, Appl. Math. (Warsaw), 1996, 24(1), 97–107 | Zbl 0858.35008
[28] Křížek M., Qun L., On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math., 1995, 3(1), 59–69 | Zbl 0824.65112
[29] Kuzmin D., Shashkov M.J., Svyatskiy D., A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems, J. Comput. Phys., 2009, 228(9), 3448–3463 http://dx.doi.org/10.1016/j.jcp.2009.01.031 | Zbl 1163.65085
[30] Nečas J., Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967
[31] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, 1967 | Zbl 0153.13602
[32] Roos H.-G., Stynes M., Tobiska L., Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., Springer Ser. Comput. Math., 24, Springer, Berlin, 2008 | Zbl 1155.65087
[33] Schatz A.H., A weak discrete maximum principle and stability of the finite element method in L ∞ on plane polygonal domains. I, Math. Comp., 1980, 34(149), 77-91
[34] Šolín P., Segeth K., Doležel I., Higher-Order Finite Element Methods, Stud. Adv. Math., Chapman&Hall/CRC, Boca Raton, 2004 | Zbl 1032.65132
[35] Stakgold I., Green’s Functions and Boundary Value Problems, 2nd ed., Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1998
[36] Szabó B., Babuška I., Finite Element Analysis, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1991 | Zbl 0792.73003
[37] VanderZee E., Hirani A.N., Zharnitsky V., Guoy D., A dihedral acute triangulation of the cube, Comput. Geom., 2010, 43(5), 445–452 | Zbl 1185.65040
[38] Vanselow R., About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation, Appl. Math., 2001, 46(1), 13–28 http://dx.doi.org/10.1023/A:1013775420323 | Zbl 1066.65132
[39] Varga R.S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962
[40] Varga R.S., On a discrete maximum principle, SIAM J. Numer. Anal., 1966, 3, 355–359 http://dx.doi.org/10.1137/0703029 | Zbl 0143.17603
[41] Vejchodský T., Angle conditions for discrete maximum principles in higher-order FEM, In: Numerical Mathematics and Advanced Applications, ENUMATH 2009, Uppsala, June 29–July 3, 2009, Springer, Berlin, 2010, 901–909 | Zbl 1216.65163
[42] Vejchodský T., Higher-order discrete maximum principle for 1D diffusion-reaction problems, Appl. Numer. Math., 2010, 60(4), 486–500 http://dx.doi.org/10.1016/j.apnum.2009.10.009 | Zbl 1230.65088
[43] Vejchodský T., Šolín P., Discrete Green’s function and maximum principles, In: Programs and Algorithms of Numerical Mathematics, 13, Institute of Mathematics, Academy of Sciences, Czech Republic, 2006, 247–252, available at http://www.math.cas.cz/~panm13
[44] Vejchodský T., Šolín P., Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM, J. Numer. Math., 2007, 15(3), 233–243 http://dx.doi.org/10.1515/jnma.2007.011 | Zbl 1172.65045
[45] Vejchodský T., Šolín P., Discrete maximum principle for higher-order finite elements in 1D, Math. Comp., 2007, 76(260), 1833–1846 http://dx.doi.org/10.1090/S0025-5718-07-02022-4 | Zbl 1125.65108
[46] Vejchodský T., Šolín P., Discrete maximum principle for Poisson equation with mixed boundary conditions solved by hp-FEM, Adv. Appl. Math. Mech., 2009, 1(2), 201–214 | Zbl 1262.65181
[47] Xu J., Zikatanov L., A monotone finite element scheme for convection-diffusion equations, Math. Comp., 1999, 68(228), 1429–1446 http://dx.doi.org/10.1090/S0025-5718-99-01148-5 | Zbl 0931.65111