Discrete maximum principle for higher-order finite elements in 1D
Authors:
Tomás Vejchodsky and Pavel Solín
Journal:
Math. Comp. 76 (2007), 1833-1846
MSC (2000):
Primary 65N30; Secondary 35B50
DOI:
https://doi.org/10.1090/S0025-5718-07-02022-4
Published electronically:
April 30, 2007
MathSciNet review:
2336270
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the -FEM. The DMP holds if a relative length of every element
in the mesh is bounded by a value
, where
is the polynomial degree of the element
. The values
are calculated for
.
- 1. I. Babuška, G. Caloz, J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31 (1994), pp. 945-981. MR 1286212 (95g:65146)
- 2. E. Burman, A. Ern, Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes, C. R. Math. Acad. Sci. Paris 338 (2004), 641-646. MR 2056474
- 3. P.G. Ciarlet, Discrete maximum principle for finite difference operators, Aequationes Math. 4 (1970), 338-352. MR 0292317 (45:1404)
- 4. P.G. Ciarlet, P.A. Raviart, Maximum principle and uniform convergence for the finite element method, Computer Methods Appl. Mech. Engrg. 2 (1973), 17-31. MR 0375802 (51:11992)
- 5. A. Draganescu, T.F. Dupont, L.R. Scott, Failure of the discrete maximum principle for an elliptic finite element problem, Math. Comp. 74 (2005), 1-23 (electronic). MR 2085400 (2005f:65148)
- 6. M. Fiedler, Special matrices and their applications in numerical mathematics, Martinus Nijhoff Publishers, Dordrecht, 1986. MR 1105955 (92b:15003)
- 7. W. Höhn, H.D. Mittelmann, Some remarks on the discrete maximum principle for finite elements of higher-order, Computing 27 (1981), 145-154. MR 632125 (83a:65109)
- 8. A. Jüngel, A. Unterreiter, Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations, Numer. Math. 99 (2005), 485-508. MR 2117736 (2005m:65269)
- 9. J. Karátson, S. Korotov, Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math. 99 (2005), 669-698. MR 2121074 (2005k:65253)
- 10. S. Korotov, M. Krízek, P. Neittaanmäki, Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle, Math. Comp. 70 (2000), 107-119. MR 1803125 (2001i:65126)
- 11.
A.H. Schatz, A weak discrete maximum principle and stability of the finite element method in
on plane polygonal domains. I, Math. Comp. 34 (1980), 77-91. MR 551291 (81e:65063)
- 12. P. Šolín, Partial differential equations and the finite element method, J. Wiley & Sons, 2005. MR 2180081 (2006f:35004)
- 13. P. Šolín, K. Segeth, I. Dolezel, Higher-order finite element methods, Chapman & Hall/CRC Press, Boca Raton, 2003.
- 14.
P. Šolín, T. Vejchodský, A weak discrete maximum principle for
-FEM, J. Comput. Appl. Math., 2006 (to appear).
- 15. B. Szabó, I. Babuška, Finite element analysis, John Wiley & Sons, New York, 1991. MR 1164869 (93f:73001)
- 16. R.S. Varga, Matrix iterative analysis, Englewood Cliffs, New Jersey, Prentice-Hall, 1962. MR 0158502 (28:1725)
- 17. T. Vejchodský, On the nonnegativity conservation in semidiscrete parabolic problems. In: M. Krízek, P. Neittaanmäki, R. Glowinski, S. Korotov (Eds.), Conjugate gradients algorithms and finite element methods, Berlin, Springer-Verlag, 2004, pp. 197-210. MR 2082563 (2005i:65135)
- 18. T. Vejchodský, Method of lines and conservation of nonnegativity. In: Proc. of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyväskylä, Finland, 2004.
- 19. T. Vejchodský, P. Šolín, Discrete Maximum Principle for Mixed Boundary Conditions in 1D, Research Report No. 2006-09, Department of Math. Sciences, University of Texas at El Paso, July 2006.
- 20. J. Xu, L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999), 1429-1446. MR 1654022 (99m:65225)
- 21. E.G. Yanik, Sufficient conditions for a discrete maximum principle for high-order collocation methods, Comput. Math. Appl. 17 (1989), 1431-1434. MR 999250 (90c:65106)
Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 35B50
Retrieve articles in all journals with MSC (2000): 65N30, 35B50
Additional Information
Tomás Vejchodsky
Affiliation:
Mathematical Institute, Academy of Sciences, Žitná 25, Praha 1, CZ-115 67, Czech Republic
Email:
vejchod@math.cas.cz
Pavel Solín
Affiliation:
Institute of Thermomechanics, Academy of Sciences, Dolejškova 5, Praha 8, CZ-182 00, Czech Republic
Address at time of publication:
Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514
Email:
solin@utep.edu
DOI:
https://doi.org/10.1090/S0025-5718-07-02022-4
Keywords:
Discrete maximum principle,
discrete Green's function,
higher-order elements,
$hp$-FEM,
Poisson equation.
Received by editor(s):
January 31, 2006
Received by editor(s) in revised form:
July 25, 2006
Published electronically:
April 30, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.