Abstract
An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with postitive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2-dimensional or 3-dimensional problem and whether the elliptic equations are scalar equations or belong to a system.
Zusammenfassung
Es wird ein algebraisches Mehrgitterverfahren vorgestellt, das zur Vorkonditionierung von positiv definiten Matrizen geeignet ist. Es wurde entworfen für Gleichungssysteme, die aus der Diskretisierung von elliptischen Differentialgleichungen stammen. Alle Information wird aus der Matrix herausgezogen. Man braucht nur zu wissen, ob ein zwei- oder dreidimensionales Problem und ob eine skalare Gleichung oder ein System zugrunde liegt.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bank, R. E.: An algorithm for coarsening unstructured meshes. (submitted)
Braess, D., Biebighäuser, M., Grassberger, P., Leuverink, P.: Multi-grid method for steady state diffusion in random media. J. Comput. Physics107, 118–123 (1993).
Braess, D., König, Ch.: A preconditioning technique for the fast solution of 3-D groundwater problems (in preparation).
Chatelin, F., Miranker, W. L.: Acceleration by aggregation of successive approximation methods. LAA43, 17–47 (1982).
Hackbusch, W.: Multi-grid methods and applications. Berlin Heidelberg New York Tokyo: Springer 1985.
Jung, M.: Konvergenzfaktoren von Mehrgitterverfahren für Probleme der ebenen, linearen Elastizitätstheorie. ZAMM67, 165–173 (1987).
Jung, M., Langer, U.: Applications of multilevel methods to pratical problems. Surv. Math. Ind.1, 217–257 (1991).
Kettler, R., Wesseling, P.: Aspects of multigrid methods for problems in three dimensions. Appl. Math. Comp.19, 159–168 (1986).
Ruge, J. W., Stüben, K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Multigrid methods for integral and differential equations (Paddon, D. J., Holsten, H., eds.), pp. 169–212. Oxford: Clarendon Press 1985.
Ruge, J. W., Stüben, K.: Algebraic multigrid (AMG). In: Multigrid methods (Mc Cormick, St. ed.). Philadelphia: SIAM 1986 (Frontiers in Applied Mathematics, Vol. 5).
Stüben, K.: Algebraic multigrid (AMG): Experience and comparisons. Appl. Math. Comput.13, 419–452 (1983).
Wesseling, P.: An introduction to multigrid methods. Chichester-New York: J. Wiley (1992).
de Zeeuw, P. M.: Matrix-dependent prolongations and restrictions in a black-box multigrid solver. J. Comp. Appl. Math.33, 1–27 (1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Braess, D. Towards algebraic multigrid for elliptic problems of second order. Computing 55, 379–393 (1995). https://doi.org/10.1007/BF02238488
Issue Date:
DOI: https://doi.org/10.1007/BF02238488