Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures

Journal of Computational and Applied Mathematics

Volumn 192, Issue 1, 2006, Pages 75-88

  Karatson J ^a   Korotov, S ^b  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

Discrete maximum principle; Finite element method; Mixed boundary condition; Nonlinear elliptic problem; Quadratures

Indexed keywords

BOUNDARY CONDITIONS; FINITE ELEMENT METHOD; MATRIX ALGEBRA; NONLINEAR SYSTEMS;

DISCRETE MAXIMUM PRINCIPLE; MIXED BOUNDARY CONDITION; NONLINEAR ELLIPTIC PROBLEM; QUADRATURES;

PROBLEM SOLVING;

EID: 33646115200     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.04.050     Document Type: Article

References (26)

1. Axelsson O., and Maubach J. On the updating and assembly of the Hessian matrix in finite element methods. Comput. Methods Appl. Mech. Engrg. 71 (1988) 41-67
2. J. Brandts, S. Korotov, M. Křížek, N-simplicial finite element partitions, Preprint of University of Amsterdam, 2005.
3. Burman E., and Ern A. Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. C. R. Math. Acad. Sci. Paris 338 (2004) 641-646
4. Christie I., and Hall C. The maximum principle for bilinear elements. Internat. J. Numer. Methods Engrg. 20 (1984) 549-553
5. Ciarlet P.G. Discrete maximum principle for finite-difference operators. Aequationes Math. 4 (1970) 338-352
6. Ciarlet P.G., and Raviart P.-A. Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2 (1973) 17-31
7. Drǎgǎnescu A., Dupont T.F., and Scott L.R. Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp. 74 (2005) 1-23
8. Engels H. Numerical Quadrature and Cubature (1980), Academical Press, London
9. I. Faragó, J. Karátson, Numerical solution of nonlinear elliptic problems via preconditioning operators. Theory and applications, Advances in Computation, vol. 11, NOVA Science Publishers, New York, 2002.
10. H. Fujii, Some remarks on finite element analysis of time-dependent field problems, Theory and Practice in Finite element Structural Analysis, University of Tokyo Press, Tokyo, 1973, pp. 91-106.
11. Höhn W., and Mittelmann H.-D. Some remarks on the discrete maximum-principle for finite elements of higher order. Computing 27 (1981) 145-154
12. Ishihara K. Strong and weak discrete maximum principles for matrices associated with elliptic problems. Linear Algebra Appl. 88/89 (1987) 431-448
13. J. Karátson, S. Korotov, On discrete maximum principles for finite element solutions of nonlinear elliptic problems, Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS-2004), Jyväskylä, Finland, 2004, pp. 1-13.
14. Karátson J., and Korotov S. Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99 (2005) 669-698
15. Korotov S., Křížek M., and Neittaanmäki P. Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comp. 70 (2001) 107-119
16. Křížek M., and Lin Q. On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3 (1995) 59-69
17. Křížek M., and Liu L. On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type. Appl. Math. (Warsaw) 24 (1996) 97-107
18. Křížek M., and Liu L. On the maximum and comparison principles for a steady-state nonlinear heat conduction problem. ZAMM Z. Angew. Math. Mech. 83 (2003) 559-563
19. Křížek M., and Neittaanmäki P. Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications (1996), Kluwer Academic Publishers, Dordrecht
20. Ladyzhenskaya O., and Ural'tseva N.N. Linear and Quasilinear Elliptic Equations (1968), Leon Ehrenpreis Academic Press, New York, London
21. Protter M.H., and Weinberger H.F. Maximum Principles in Differential Equations (1984), Springer, New York
22. Ruas Santos V. On the strong maximum principle for some piecewise linear finite element approximate problems of non-positive type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 473-491
23. P. Šolin, T. Vejchodský, On a discrete maximum principle for one-dimensional hp-FEM, submitted for publication.
24. Strang G., and Fix G.J. An Analysis of the Finite Element Method (1973), Prentice-Hall, Englewood Cliffs, NJ
25. Unterreiter A., and Juengel A. Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations. Numer. Math. 99 (2005) 485-508
26. Varga R. Matrix Iterative Analysis (1962), Prentice-Hall, New Jersey