The discrete maximum principle for Galerkin solutions of elliptic problems

Tomáš Vejchodský

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 25-43
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Tomáš Vejchodský. "The discrete maximum principle for Galerkin solutions of elliptic problems." Open Mathematics 10.1 (2012): 25-43. <http://eudml.org/doc/269224>.

@article{TomášVejchodský2012,
abstract = {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.},
author = {Tomáš Vejchodský},
journal = {Open Mathematics},
keywords = {Discrete maximum principle; Monotone methods; Galerkin solution; Finite element metho; discrete maximum principle; monotone methods; Galerkin solutions; finite element method; elliptic problem},
language = {eng},
number = {1},
pages = {25-43},
title = {The discrete maximum principle for Galerkin solutions of elliptic problems},
url = {http://eudml.org/doc/269224},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Tomáš Vejchodský
TI - The discrete maximum principle for Galerkin solutions of elliptic problems
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 25
EP - 43
AB - 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.
LA - eng
KW - Discrete maximum principle; Monotone methods; Galerkin solution; Finite element metho; discrete maximum principle; monotone methods; Galerkin solutions; finite element method; elliptic problem
UR - http://eudml.org/doc/269224
ER -

References

top
  1. [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 Zbl0124.33006
  2. [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 Zbl0126.32305
  3. [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 Zbl1112.51006
  4. [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 Zbl1164.65493
  5. [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 Zbl1154.65086
  6. [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 Zbl1172.51012
  7. [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 Zbl0194.12703
  8. [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 Zbl0198.14601
  9. [9] Ciarlet P.G., The Finite Element Method for Elliptic Problems, Stud. Math. Appl., 4, North-Holland, Amsterdam-New York-Oxford, 1978 Zbl0383.65058
  10. [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 Zbl0251.65069
  11. [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 Zbl0245.34012
  12. [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 Zbl1074.65129
  13. [13] Duffy D.G., Green’s Functions with Applications, Stud. Adv. Math., Chapman&Hall/CRC, Boca Raton, 2001 Zbl0983.35003
  14. [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 Zbl1054.65020
  15. [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 Zbl1130.65086
  16. [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 Zbl1194.80119
  17. [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 Zbl1070.65094
  18. [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 Zbl1262.35124
  19. [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. [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. [21] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin-New York, 1977 Zbl0361.35003
  22. [22] Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer Ser. Comput. Phys., Springer, New York, 1984 Zbl0536.65054
  23. [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. [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 Zbl1067.65127
  25. [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 Zbl1211.65147
  26. [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 Zbl1103.52008
  27. [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 Zbl0858.35008
  28. [28] Křížek M., Qun L., On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math., 1995, 3(1), 59–69 Zbl0824.65112
  29. [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 Zbl1163.65085
  30. [30] Nečas J., Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967 
  31. [31] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, 1967 Zbl0153.13602
  32. [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 Zbl1155.65087
  33. [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. [34] Šolín P., Segeth K., Doležel I., Higher-Order Finite Element Methods, Stud. Adv. Math., Chapman&Hall/CRC, Boca Raton, 2004 Zbl1032.65132
  35. [35] Stakgold I., Green’s Functions and Boundary Value Problems, 2nd ed., Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1998 
  36. [36] Szabó B., Babuška I., Finite Element Analysis, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1991 Zbl0792.73003
  37. [37] VanderZee E., Hirani A.N., Zharnitsky V., Guoy D., A dihedral acute triangulation of the cube, Comput. Geom., 2010, 43(5), 445–452 Zbl1185.65040
  38. [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 Zbl1066.65132
  39. [39] Varga R.S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962 
  40. [40] Varga R.S., On a discrete maximum principle, SIAM J. Numer. Anal., 1966, 3, 355–359 http://dx.doi.org/10.1137/0703029 Zbl0143.17603
  41. [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 Zbl1216.65163
  42. [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 Zbl1230.65088
  43. [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. [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 Zbl1172.65045
  45. [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 Zbl1125.65108
  46. [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 Zbl1262.65181
  47. [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 Zbl0931.65111

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.