×

Explicit iterations with monotonicity for finite element approximations applied to a system of nonlinear elliptic equations. (English) Zbl 0599.65074

For a nonlinear elliptic Dirichlet problem explicit Gauss-Seidel type iteration schemes for triangular finite element approximations are presented. Starting from known bounds on the solution, sequences are constructed, which bracket the finite element solution and converge monotonically to it. Numerical examples are given.
Reviewer: H.Matthies

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablow, C. M.; Perry, C. L., Iterative solutions of the Dirichlet problem for \(Δu = u^2\), SIAM J. Appl. Math., 7, 459-467 (1959) · Zbl 0104.35003
[2] Ciarlet, P. G.; Raviart, P. A., Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2, 17-31 (1973) · Zbl 0251.65069
[3] Collatz, L., Functional Analysis and Numerical Mathematics (1966), Academic Press: Academic Press New York · Zbl 0221.65088
[4] Dankwerts, P. V., Gas Liquid Reactions (1970), McGraw-Hill: McGraw-Hill New York
[5] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0691.35001
[6] Ishihara, K., On finite element schemes of the Dirichlet problem for a system of nonlinear elliptic equations, Numer. Funct. Anal. Optim., 3, 105-136 (1981) · Zbl 0469.65071
[7] Ishihara, K., Finite element approximations applied to the nonlinear boundary value problem \(Δu = bu^2\), Publ. Res. Inst. Math. Sci., 18, 17-34 (1982) · Zbl 0492.65062
[8] Ishihara, K., Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem, Numer. Math., 43, 419-437 (1984) · Zbl 0531.65061
[9] Kahane, C. S., On a system of nonlinear parabolic equations arising in chemical engineering, J. Math. Anal. Appl., 53, 343-358 (1976) · Zbl 0326.35044
[10] McAllister, G. T., Difference methods for a nonlinear elliptic system of partial differential equations, Quart. Appl. Math., 23, 355-360 (1966) · Zbl 0131.09902
[11] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046
[12] Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, N. J · Zbl 0133.08602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.