TY - GEN A1 - Oevermann, Michael A1 - Scharfenberg, Carsten A1 - Klein, Rupert T1 - A sharp interface finite volume method for elliptic equations on Cartesian grids N2 - We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm. T3 - ZIB-Report - 08-38 KW - elliptische Gleichungen KW - Finite Volumen KW - eingebettete Ränder KW - variable und diskontinuierliche Koeffizienten KW - diskontinuierliche Lösung KW - Elliptic equations KW - finite volume methods KW - embedded interface KW - variable and discontinuous coefficients KW - discontinuous solution Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10900 SN - 1438-0064 ER - TY - GEN A1 - Oevermann, Michael A1 - Klein, Rupert T1 - A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces N2 - We present a finite volume method for the solution of the two-dimensional Poisson equation $ \nabla\cdot( \beta( {\mbox{\boldmath $x$}}) \nabla u({\mbox{\boldmath $x$}})) = f(\mbox{\boldmath $x$}) $ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution $u({\mbox{\boldmath $x$})$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the $L^\infty$ and $L^2$ norm. T3 - ZIB-Report - 06-05 KW - Poisson equation KW - finite volume methods KW - embedded interface KW - ariable and discontinuous coefficients KW - discontinuous solution Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8996 ER - 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 - Bornemann, Folkmar A. T1 - An Adaptive Multilevel Approach to Parabolic Equations III. N2 - Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30. T3 - ZIB-Report - SC-91-01 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-519 ER - TY - GEN A1 - Lang, Jens A1 - Walter, Artur T1 - An Adaptive Discontinuous Finite Element Method for the Transport Equation. N2 - In this paper we introduce a discontinuous finite element method. In our approach, it is possible to combine the advantages of finite element and finite difference methods. The main ingredients are numerical flux approximation and local orthogonal basis functions. The scheme is defined on arbitrary triangulations and can be easily extended to nonlinear problems. Two different error indicators are derived. Especially the second one is closely connected to our approach and able to handle arbitrary variing flow directions. Numerical results are given for boundary value problems in two dimensions. They demonstrate the performance of the scheme, combined with the two error indicators. {\bf Key words:} neutron transport equation, discontinuous finite element, adaptive grid refinement. {\bf Subject classifications:} AMS(MOS) 65N30, 65M15. T3 - ZIB-Report - SC-91-07 KW - neutron transport equation KW - discontinuous finite element KW - adaptive grid refinement Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-579 ER - TY - GEN A1 - Bornemann, Folkmar A. T1 - A Sharpened Condition Number Estimate for the BPX Preconditioner of Elliptic Finite Element Problems on Highly Nonuniform Triangulations. N2 - In this paper it is shown that for highly nonuniformly refined triangulations the condition number of the BPX preconditioner for elliptic finite element problems grows at most linearly in the depth of refinement. This is achieved by viewing the computational available version of the BPX preconditioner as an abstract additive Schwarz method with exact solvers. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65N20, 65N30. T3 - ZIB-Report - SC-91-09 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-596 ER - TY - GEN A1 - Kornhuber, Ralf A1 - Roitzsch, Rainer T1 - On Adaptive Grid Refinement in the Presence of Internal or Boundary Layers. N2 - We propose an anisotropic refinement strategy which is specially designed for the efficient numerical resolution of internal and boundary layers. This strategy is based on the directed refinement of single triangles together with adaptive multilevel grid orientation. It is demonstrated by several numerical examples that compared to usual methods, the new anisotropic refinement ends up in more stable and more accurate solutions at much less computational cost. {\bf Keywords:} Adaptive finite elements, directed refinement, adaptive grid orientation, convection diffusion equation, internal and boundary layers. T3 - ZIB-Report - SC-89-05 KW - adaptive finite elements KW - directed refinement KW - adaptive grid orientation KW - convection diffusion equation KW - internal and boundary layers Y1 - 1989 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-230 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 - Walter, Artur T1 - Improvement of Incomplete Factorizations by a Sparse Secant Method. N2 - In the present paper, the improvement of an incomplete factorization of a non-symmetric matrix A is discussed. Starting from the ideas of sparsity preserving quasi-Newton methods, an algorithm is developed which improves the approximation of A by the incomplete factorization maintaining the sparsity structure of the matrices. No renumbering of the unknowns or the admittance of additional fill-in is necessary. The linear convergence of the algorithm is proved under the assumption, that $ L $ and $ U $* have the same sparsity structure and an incomplete factorization with some reasonable approximation property exits. In combination with this algorithm, the method of incomplete factorization and its several modifications are applicable to a wider class of problems with improved convergence qualities. This is shown by a numerical example. {\bf Key Words:} non-symmetric linear system, sparse secant method, incomplete factorization. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20, 65N30. T3 - ZIB-Report - SC-90-12 KW - non-symmetric linear systems KW - sparse secant method KW - incomplete factorization Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-401 ER -