## Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods.(English)Zbl 0833.65131

The mixed finite element discretization of second-order elliptic boundary value problems is considered. The efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations is studied. The authors present two multilevel solvers. The first one is based on ideas from the domain decomposition approach. A multilevel preconditioned conjugate gradient iteration acting on an appropriate subspace with a hierarchical type preconditioner that can be derived by subspace decomposition is presented.
The second algorithm is obtained from the mixed hybridization and an alternative adaptive multilevel method based on this technique is derived. The local refinement of the underlying triangulations is done by efficient and reliable a posteriori error estimators which can be obtained by a defect correction in higher order ansatz spaces or by taking into account the superconvergence results.
The performance of both algorithms is illustrated by two test examples. The solution of the first problem has a peak in an internal point while the solution of the second example has boundary layers.
Reviewer: K.Georgiev (Sofia)

### MSC:

 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 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 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations

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### References:

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