TY - GEN A1 - Deuflhard, Peter A1 - Leinen, P. A1 - Yserentant, Harry T1 - Concepts of an Adaptive Hierarchical Finite Element Code. N2 - The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods. T3 - ZIB-Report - SC-88-05 KW - finite elements KW - hierarchical basis KW - adaptive mesh refinement KW - preconditioned conjugate gradient methods Y1 - 1988 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-131 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Yserentant, Harry T1 - A Basic Norm Equivalence for the Theory of Multilevel Methods. N2 - Subspace decompositions of finite element spaces based on $L2$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the $H^1$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of $ K$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50. T3 - ZIB-Report - SC-92-01 KW - multilevel methods KW - nonuniform meshes KW - optimal convergence rates Y1 - 1992 U6 - https://doi.org/10.1007/BF01388699 N1 - No preprint available ER - TY - GEN A1 - Yserentant, Harry T1 - Two Preconditioners Based on the Multi-Level Splitting of Finite Element Spaces. N2 - The hierarchical basis preconditioner and the recent preconditioner of BRAMBLE, PASCIAK and XU are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; our theory is based exclusively on local properties like the shape regularity of the finite elements. T3 - ZIB-Report - SC-89-09 Y1 - 1989 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-274 ER - TY - GEN A1 - Bank, Randolph A1 - Dupont, Todd F. A1 - Yserentant, Harry T1 - The Hierarchical Basis Multigrid Method. N2 - We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß- Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. T3 - ZIB-Report - SC-87-02 KW - hierarchical basis KW - multigrid KW - finite elements KW - adaptive mesh refinement KW - preconditioned conjugate gradient methods KW - symmetric Gauss-Seidel Y1 - 1987 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-44 ER - TY - GEN A1 - Yserentant, Harry T1 - Preconditioning Indefinite Discretization Matrices. N2 - The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can be used also for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case. {\bf Keywords: }Preconditioned conjugate gradient methods, finite elements. {\bf Subject Classification: } AMS(MOS):65F10, 65N20, 65N30. T3 - ZIB-Report - SC-87-06 KW - preconditioned conjugate gradient methods KW - finite elements Y1 - 1987 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-81 ER - TY - GEN A1 - Kornhuber, Ralf A1 - Yserentant, Harry T1 - Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. N2 - Elliptic boundary value problems are frequently posed on complicated domains which cannot be covered by a simple coarse initial grid as it is needed for multigrid like iterative methods. In the present article, this problem is resolved for selfadjoint second order problems and Dirichlet boundary conditions. The idea is to construct appropriate subspace decompositions of the corresponding finite element spaces by way of an embedding of the domain under consideration into a simpler domain like a square or a cube. Then the general theory of subspace correction methods can be applied. T3 - ZIB-Report - SC-93-21 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1170 ER - TY - CHAP A1 - Polthier, Konrad A1 - Bobenko, Alexander A1 - Hildebrandt, Klaus A1 - Kornhuber, Ralf A1 - Tycowicz, Christoph von A1 - Yserentant, Harry A1 - Ziegler, Günter M. T1 - Geometry processing T2 - MATHEON - Mathematics for Key Technologies Y1 - 2014 SN - 978-3-03719-137-8 U6 - https://doi.org/10.4171/137 VL - 1 SP - 341 EP - 355 PB - EMS Publishing House ER -