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**The incremental unknown method. I.**
*(English)*
Zbl 0726.65133

Summary: Incremental unknowns were introduced by the second author [SIAM J. Math. Anal. 21, No.1, 154-178 (1990; Zbl 0715.35039)] as a means to approximate fractal attractors by using finite differences. However, incremental unknowns also provide a new way for solving linear elliptic problems using several levels of discretization; the method is similar but different from the classical multigrid methods. It is efficient and easy to implement. We also expect the method to be suitable for problems for which the utilization of the standard multigrid methods is difficult.

In this article we describe the utilization of incremental unknowns for solving Laplace operator in dimension two. We provide some theoretical results concerning two-level approximations and we present the numerical tests done with the multi-level approximations. The numerical tests show that for this problem, the efficiency of the incremental unknown method is comparable to the V-cycle multigrid method.

[For part II see ibid. 4, No.3, 77-88 (1991; reviewed below)].

In this article we describe the utilization of incremental unknowns for solving Laplace operator in dimension two. We provide some theoretical results concerning two-level approximations and we present the numerical tests done with the multi-level approximations. The numerical tests show that for this problem, the efficiency of the incremental unknown method is comparable to the V-cycle multigrid method.

[For part II see ibid. 4, No.3, 77-88 (1991; reviewed below)].

### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

### Keywords:

Laplace equation; multigrid methods; numerical tests; multi-level approximations; incremental unknown method
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\textit{M. Chen} and \textit{R. Temam}, Appl. Math. Lett. 4, No. 3, 73--76 (1991; Zbl 0726.65133)

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### References:

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[2] | Brandt, A., Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics (1984), Gesellschaft für Mathematik und Datenverarbeitung: Gesellschaft für Mathematik und Datenverarbeitung St. Augustin, GMD-Studien Nr. 85 · Zbl 0581.76033 |

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[4] | Chen(a), M. and R. Temam, Incremental unknowns for solving partial differential equations, (to appear).; Chen(a), M. and R. Temam, Incremental unknowns for solving partial differential equations, (to appear). · Zbl 0712.65103 |

[5] | Chen(b), M. and R. Temam, The Incremental Unknown Method II, Appl. Math. Letters (this volume) (to appear).; Chen(b), M. and R. Temam, The Incremental Unknown Method II, Appl. Math. Letters (this volume) (to appear). · Zbl 0726.65134 |

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[7] | Hackbusch, W., Multi-Grid Methods and Applications (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0585.65030 |

[8] | (McCormick, S. F., Multigrid Methods (1987), SIAM Philadelphia: SIAM Philadelphia PA) · Zbl 0659.65094 |

[9] | Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM J. of Num. Anal., 26, 1139-1157 (1989) · Zbl 0683.65083 |

[10] | Temam, R., Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21, 154-178 (1990) · Zbl 0715.35039 |

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