zbMATH — the first resource for mathematics

Incremental unknowns for solving partial differential equations. (English) Zbl 0712.65103
Incremental unknowns were introduced as a means to approximate fractal attractors by using finite differences. In the case of linear elliptic problems, the utilization of incremental unknowns provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid methods.
In this article we describe the utilization of incremental unknowns for solving a Laplace operator in dimensions one and two. We provide some theoretical results concerning two-level approximations, and we present the numerical tests done with the multi-level approximations. The tests show that for this problem, the conjugate gradient method in conjunction with the incremental unknowns provides a method which has the efficiency comparable to the V-cycle multigrid method.
Reviewer: M.Chen

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI EuDML
[1] [AB] Axelsson, O., Barker, V.A.: Finite element solution of boundary value problem: Theory and computation. New York: Academic Press 1984 · Zbl 0537.65072
[2] [CT] Chen, M., Temam, R.: to appear
[3] [FMT] Foias, C., Manley, O., Temam, R.: Sur l’interaction des petits et grands tourbillons dans des ?coulements turbulents. C.R. Acad. Sc. Paris, Serie 1,305, 497-500 (1987) · Zbl 0624.76072
[4] Foias, C., Manley, O., Temam, R.: Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. Math. Mod. Numer. Anal. (M2AN)22, 93-114 (1988) · Zbl 0663.76054
[5] [FST] Foias, C., Sell, G.R., Temam, R.: Inertial Manifolds for nonlinear evolutionary equations. J. Diff. Equ.73, 309-353 (1988) · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[6] [G] Garcia, S.: to appear
[7] [H] Hackbusch, W.: Multi-Grid Methods and Applications. Berlin Heidelberg New York: Springer 1985 · Zbl 0595.65106
[8] [MT1] Marion, M., Temam, R.: Nonlinear Galerkin Methods. SIAM J. Numer. Anal.26, No. 5, 1139-1157 (1989) · Zbl 0683.65083 · doi:10.1137/0726063
[9] [MT2] Marion, M., Temam, R.: Nonlinear Galerkin Methods; The finite elements case. Numer. Math.57, 205-226 (1990) · Zbl 0702.65081 · doi:10.1007/BF01386407
[10] [T] Temam, R.: Inertial manifolds and multigrid methods. SIAM J. Math. Anal.21, No. 1, 154-178 (1990) · Zbl 0715.35039 · doi:10.1137/0521009
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.