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**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.

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

### MSC:

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 |

### Keywords:

incremental unknowns; multi-level discretization; conjugate gradient; elliptic equations; finite differences; fractal attractors; Laplace operator; numerical tests; V-cycle multigrid method
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\textit{M. Chen} and \textit{R. Temam}, Numer. Math. 59, No. 3, 255--271 (1991; Zbl 0712.65103)

### References:

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