Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods
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.
second-order elliptic boundary value problems
multilevel preconditioned conjugate gradient iteration
scientific article; zbMATH DE number 806349
scientific article