zbMATH — the first resource for mathematics

Monotone enclosure for nonlinear PDEs using multigrid techniques. (English) Zbl 0747.65091
The author investigates two possibilities to include numerical solutions of weakly nonlinear elliptic boundary value problems. For the case of a single equation an algorithm combining the including properties of nonlinear monotone-including iteration processes with the efficiency of FAS is proposed.
The second way of solving nonlinear differential equations consists in the following steps: linearize the given equations by a conventional monotone-including method and solve the resulting linear equations by a linear multigrid method. It is demonstrated that this way is very convenient if a weakly coupled system of equations is given.
The efficiency of the proposed algorithms is illustrated by numerical examples.
Reviewer: M.Jung (Chemnitz)

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
[1] Blat, J.; Brown, K.J., Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. roy. soc. Edinburgh, 97A, 21-34, (1984) · Zbl 0554.92012
[2] Braess, D.; Hackbusch, W., A new convergence proof for the multigrid method including the V-cycle, SIAM J. numer. anal., 20, 967-975, (1983) · Zbl 0521.65079
[3] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. comp., 31, 333-390, (1977) · Zbl 0373.65054
[4] Dancer, E.N., On positive solutions of some pairs of differential equations, Trans. amer. math. soc., 284, 729-743, (1984) · Zbl 0524.35056
[5] Förster, H.; Witsch, K., Multigrid software for the solution of elliptic problems on rectangular domains: MG00, (), 427-460, Lecture Notes in Math.
[6] Förster, H.; Witsch, K., On efficient multigrid software for elliptic problems on rectangular domains, Math. comput. simulation, 23, 293-298, (1981)
[7] Hackbusch, W., Multigrid convergence theory, (), 177-221, Lecture Notes in Math.
[8] Hackbusch, W.; Trottenberg, U., Multigrid methods, Lecture notes in math., 960, (1982), Springer-Verlag Berlin · Zbl 0497.00015
[9] Hackbusch, W.; Trottenberg, U., Multigrid methods II, Lecture notes in math., 1228, (1986), Springer-Verlag Berlin · Zbl 0596.00016
[10] Huy, C.U.; McKenna, P.J.; Walter, W., Finite-difference approximations to the Dirichlet problem for elliptic systems, Numer. math., 49, 227-237, (1986) · Zbl 0602.65068
[11] Kaspar, B., Overrelaxation in monotonically convergent iteration methods, (), 80-87, Lecture Notes in Math. · Zbl 0494.65026
[12] Korman, P.; Leung, A.W., A general monotone scheme for elliptic systems with applications to ecological models, Proc. roy. soc. Edinburgh, 102A, 315-325, (1986) · Zbl 0606.35034
[13] Korman, P.; Leung, A.W., On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Appl. anal., 26, 145-160, (1987) · Zbl 0639.35026
[14] Krasnoselski, M., Positive solutions of operator equations, (1964), Nordhoff Groningen
[15] McKenna, P.J.; Walter, W., On the Dirichlet problem for elliptic systems, Appl. anal., 21, 207-224, (1986) · Zbl 0593.35042
[16] Ortega, J.M.; Rheinboldt, W.C., Iterative solutions of nonlinear equations in several variables, (1970), Academic New York · Zbl 0241.65046
[17] Potra, F.A., Newton-like methods with monotone convergence for solving nonlinear operator equations, Nonlinear anal., 11, 697-717, (1987) · Zbl 0633.65050
[18] Stüben, K.; Trottenberg, U., Multigrid methods: fundamental algorithms, model problem analysis and applications, (), 1-176, Lecture Notes in Math.
[19] Törnig, W., Monoton einschließende konvergente iterationsprozesse vom gauß-Seidel-typ zur Lösung nichtlinearer gleichungssyteme im rn und anwendunge n, Math. methods appl. sci., 2, 489-503, (1980) · Zbl 0459.65034
[20] Voller, R.L., Monotonieeigenschaften von abbildungen und einschließung von Lösungen nichtlinearer operatorgleichungen, Habilitationsschrift, (1989), Düsseldorf · Zbl 0734.47044
[21] Voller, R.L., A monotone including multigrid method, Computing, 45, 377-382, (1990) · Zbl 0719.65085
[22] R.L. Voller, Monoton einschließende Mehrgitterverfahren, Z. Angew. Math. Mech. 91, to appear.
[23] Wesseling, P., A robust and efficient multigrid method, (), 614-630, Lecture Notes in Math. · Zbl 0505.65052
[24] Jun, Zou, A new fast solver—monotone MG method (MMG), J. comput. math., 5, 325-335, (1987) · Zbl 0648.65069
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.