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**Concepts of an adaptive hierarchical finite element code.**
*(English)*
Zbl 0706.65111

The paper describes the basic mathematical concept of the adaptive hierarchical triangular finite element code KASKADE for solving linear scalar 2nd-order 2D symmetric elliptic boundary value problems. The algorithm used in the code is based on the nested iteration principle combined with an efficient nested solver and adaptive techniques for constructing the triangulation hierarchy. The hierarchical preconditioned conjugate gradient method of the third author [Numer. Math. 49, 379-412 (1986; Zbl 0608.65065)] is used as nested solver.

The adaptive technique is based on a comparison of the linear finite element solution with the quadratic finite element solution at the nested level. However, the solution of the hierarchical quadratic finite element problem is replaced by a much simpler problem the system matrix of which is near to the hierarchical quadratic one in the spectral sense. Because of this spectral equivalence, the errors can be compared.

This result provides the foundation for a simple, but efficient adaptive technique for controlling the discretization error and the number of nested iterations which must be in balance. Three numerical examples with typical singularities confirm the efficiency of this adaptive approach.

The adaptive technique is based on a comparison of the linear finite element solution with the quadratic finite element solution at the nested level. However, the solution of the hierarchical quadratic finite element problem is replaced by a much simpler problem the system matrix of which is near to the hierarchical quadratic one in the spectral sense. Because of this spectral equivalence, the errors can be compared.

This result provides the foundation for a simple, but efficient adaptive technique for controlling the discretization error and the number of nested iterations which must be in balance. Three numerical examples with typical singularities confirm the efficiency of this adaptive approach.

Reviewer: U.Langer

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |