TY - GEN A1 - Sander, Oliver A1 - Klapproth, Corinna A1 - Youett, Jonathan A1 - Kornhuber, Ralf A1 - Deuflhard, Peter T1 - Towards an Efficient Numerical Simulation of Complex 3D Knee Joint Motion N2 - We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity. Its efficient numerical simulation requires advanced numerical algorithms that have been developed just recently. Up to now, the model comprises bones, cartilage, and the major ligaments (patella and menisci are still missing). Bones (femur, tibia, and fibula) are modelled by linear elastic materials, cartilage by viscoelastic materials, ligaments by one-dimensional so-called Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity in the presence of strict contact inequalities. For the total spatio-temporal discretization we apply a method of layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts, we apply a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet-Neumann method, and the arising large algebraic systems are solved by a recent fast multigrid solver, the truncated non-smooth Newton multigrid method. T3 - ZIB-Report - 12-06 KW - contact problem KW - contact-stabilized Newmark method KW - domain decomposition KW - energy-momentum method KW - geodesic finite elements KW - knee model Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14512 SN - 1438-0064 ER - TY - GEN A1 - Kornhuber, Ralf T1 - On Robust Multigrid Methods for Piecewise Smooth Variational Problems N2 - We consider the fast solution of large, piecewise smooth minimization problems as resulting from the approximation of elliptic free boundary problems. The most delicate question in constructing a multigrid method for a nonlinear, non--smooth problem is how to represent the nonlinearity on the coarse grids. This process usually involves some kind of linearization. The basic idea of monotone multigrid methods to be presented here is first to select a neighborhood of the actual smoothed iterate in which a linearization is possible and then to constrain the coarse grid correction to this neighborhood. Such a local linearization allows to control the local corrections at each coarse grid node in such a way that the energy functional is monotonically decreasing. This approach leads to globally convergent schemes which are robust with respect to local singularities of the given problem. The numerical performance is illustrated by approximating the well-known Barenblatt solution of the porous medium equation. T3 - ZIB-Report - SC-96-38 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2487 ER - TY - GEN A1 - Kornhuber, Ralf T1 - Monotone Multigrid Methods for Elliptic Variational Inequalities I. N2 - Extending well--known linear concepts of successive subspace correction, we arrive at extended relaxation methods for elliptic variational inequalities. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, which are called symmetric and unsymmetric multigrid methods, respectively. While the asymptotic convergence rates of the symmetric method suffer from insufficient coarse--grid transport, it turns out in our numerical experiments that reasonable application of the unsymmetric multigrid method may lead to the same efficiency as in the linear, unconstrained case. T3 - ZIB-Report - SC-93-18 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1147 ER - TY - GEN A1 - Kornhuber, Ralf T1 - Monotone Multigrid Methods for Elliptic Variational Inequalities II. N2 - We derive fast solvers for discrete elliptic variational inequalities of the second kind as resulting from the approximation by piecewise linear finite elements. Following the first part of this paper, monotone multigrid methods are considered as extended underrelaxations. Again, the coarse grid corrections are localized by suitable constraints, which in this case are fixed by fine grid smoothing. We consider the standard monotone multigrid method induced by the multilevel nodal basis and a truncated version. Global convergence results and asymptotic estimates for the convergence rates are given. The numerical results indicate a significant improvement in efficiency compared with previous multigrid approaches. T3 - ZIB-Report - SC-93-19 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1151 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 - GEN A1 - Bornemann, Folkmar A. A1 - Erdmann, Bodo A1 - Kornhuber, Ralf T1 - A Posteriori Error Estimates for Elliptic Problems. N2 - {\def\enorm {\mathop{\mbox{\boldmath{$|\!|$}}}\nolimits} Let $u \in H$ be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some $\tilde{u} \in {\cal S}$, $\cal S$ being a suitable finite element space. Efficient and reliable a posteriori estimates of the error $\enorm u - \tilde{u}\enorm $, measuring the (local) quality of $\tilde{u}$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.} T3 - ZIB-Report - SC-93-29 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1257 ER - TY - GEN A1 - Erdmann, Bodo A1 - Hoppe, Ronald H. W. A1 - Kornhuber, Ralf T1 - Adaptive Multilevel-Methods for Obstacle Problems in Three Space Dimensions. N2 - We consider the discretization of obstacle problems for second order elliptic differential operators in three space dimensions by piecewise linear finite elements. Linearizing the discrete problems by suitable active set strategies, the resulting linear sub--problems are solved iteratively by preconditioned cg--iterations. We propose a variant of the BPX preconditioner and prove an $O(j)$ estimate for the resulting condition number. To allow for local mesh refinement we derive semi--local and local a posteriori error estimates. The theoretical results are illustrated by numerical computations. T3 - ZIB-Report - SC-93-08 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1049 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Erdmann, Bodo A1 - Kornhuber, Ralf T1 - Adaptive Multilevel-Methods in 3-Space Dimensions. N2 - We consider the approximate solution of selfadjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. as the performance of hierarchical bases preconditioners deteriorate in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a-posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy. T3 - ZIB-Report - SC-92-14 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-843 ER - TY - GEN A1 - Hoppe, Ronald H. W. A1 - Kornhuber, Ralf T1 - Adaptive Multilevel - Methods for Obstacle Problems. N2 - We consider the discretization of obstacle problems for the Laplacian by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned c-g iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semi-local and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations. T3 - ZIB-Report - SC-91-16 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-667 ER - TY - GEN A1 - Hoppe, Ronald H. W. A1 - Kornhuber, Ralf T1 - Multi-Grid Solution of Two Coupled Stefan Equations Arising in Induction Heating of Large Steel Slabs. N2 - Induction heating of large steel slabs can be described by a coupled system of nonlinear evolution equations of Stefan type representing the temporal and spatial distribution of the induced magnetic field and the generated temperature within the slab. Discretizing these equations implicitly in time and by finite differences in space, at each time step the solution of a system of difference inclusions is required. For the solution of that system two multi-grid algorithms are given which combined with a nested iteration type continuation strategy to proceed in time result in computationally highly efficient schemes for the numerical simulation of the induction heating process. {\bf Keywords:} induction heating, system of two coupled Stefan equations, multi-grid algorithms. {\bf Subject Classification:} AMS(MOS): 35K60, 35R35, 65H10, 65N05, 65N20, 78A25, 78A55. T3 - ZIB-Report - SC-88-08 KW - induction heating KW - system of two coupled Stefan equations KW - multi-grid algorithms Y1 - 1988 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-160 ER - TY - GEN A1 - Kornhuber, Ralf A1 - Roitzsch, Rainer T1 - Adaptive Finite-Elemet- Methoden für konvektionsdominierte Randwertprobleme bei partiellen Differentialgleichungen. N2 - Ausgangspunkt bei der Behandlung konvektiv dominierter, elliptischer Probleme sind die bekannten hierarchischen Finite-Element-Methoden für den rein elliptischen Fall. Als stabile Erweiterung des Standard-Galerkin-Verfahrens wird das Stromlinien-Diffusions-Verfahren durch physikalische Überlegungen motiviert und kurz diskutiert. Anschließend zeigen wir, daß diese Methode erst in Verbindung mit einer hier erstmals vorgestellten lokalen Ausrichtung der Kanten wirksam eingesetzt werden kann. Zusammen mit einer ebenfalls neu entwickelten richtungsorientierten Verfeinerungsstrategie erhält man eine erheblich stabilere, genauere und schnellere Auflösung von Grenzschichten als mit herkömmlichen Methoden. T3 - ZIB-Report - SC-88-09 Y1 - 1989 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-170 ER - TY - GEN A1 - Kornhuber, Ralf A1 - Wittum, Gabriel T1 - Discretization and Iterative Solution of Convection Diffusion Equations. N2 - We propose an extended box method which turns out to be a variant of standard finite element methods in the case of pure diffusion and an extension of backward differencing to irregular grids if only convective transport is present. Together with the adaptive orientation proposed in a recent paper and a streamline ordering of the unknowns, this discretization leads to a highly efficient adaptive method for the approximation of internal layers in the case of large local Peclet numbers. T3 - ZIB-Report - SC-92-06 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-762 ER - TY - GEN A1 - Kornhuber, Ralf A1 - Roitzsch, Rainer T1 - Self Adaptive Computation of the Breakdown Voltage of Planar pn-Junctions with Multistep Field Plates. N2 - The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm. T3 - ZIB-Report - SC-91-02 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-525 ER - TY - GEN A1 - Hoppe, Ronald H. W. A1 - Kornhuber, Ralf T1 - Multilevel Preconditioned CG-Iterations for Variational Inequalities. N2 - We consider such variational inequalities which either describe obstacle problems or result from an implicit time discretization of moving boundary problems of two phase Stefan type. Based on a discretization in space by means of continuous, piecewise linear finite elements with respect to a nested hierarchy of triangulations, in both cases we use iterative processes consisting of inner and outer iterations. The outer iterations are either active set strategies or generalized Newton methods while the inner iterations are preconditioned cg- iterations with multilevel preconditioners. T3 - ZIB-Report - SC-91-06 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-561 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 - Kornhuber, Ralf A1 - Roitzsch, Rainer T1 - Self Adaptive FEM Simulation of Reverse Biased pn-Junctions. N2 - The potential distribution of reverse biased pn-junctions can be described by a double obstacle problem for the Laplacian. This problem is solved by a self adaptive Finite Element Method involving automatic termination criteria for the iterative solver, local error estimation and local mesh refinement. Special attention is paid to the efficient resolution of the geometries typically arising in semiconductor device simulation. The algorithm is applied to a reverse biased pn- junction with multi-step field plate and stop- electrode to illustrate its efficiency and reliability. T3 - ZIB-Report - SC-90-10 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-389 ER - TY - GEN A1 - Kornhuber, Ralf T1 - Globally Convergent Multigrid Methods for Porous Medium Type Problems N2 - We consider the fast solution of large, piecewise smooth minimization problems as typically arising from the finite element discretization of porous media flow. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with {\em constrained} Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give logarithmic upper bounds for the asymptotic convergence rates. T3 - ZIB-Report - SC-97-45 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3142 ER - TY - GEN A1 - Kornhuber, Ralf T1 - Monotone Iterations for Elliptic Variational Inequalities N2 - A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications. T3 - ZIB-Report - SC-98-10 KW - finite elements KW - multigrid methods KW - variational inequalities Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3539 ER - TY - GEN A1 - Roitzsch, Rainer A1 - Kornhuber, Ralf T1 - BOXES - a Program to Generate Triangulations from a Rectangular Domain Description. N2 - BOXES computes a triangulation from a 2D domain description which consists of an arbitrary set of rectangles. Each rectangle may have attributes to control the triangulating process, define subdomain classes, or specify boundary conditions. The output of the program can be used as a coarse grid for KASKADE or one of its variants. Additional features are extensive checking of the user input, graphical display, and simple editing. T3 - ZIB-Report - TR-90-09 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4738 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 - TY - GEN A1 - Deuflhard, Peter A1 - Kornhuber, Ralf A1 - Sander, Oliver A1 - Schiela, Anton A1 - Weiser, Martin ED - Deuflhard, Peter ED - Grötschel, Martin ED - Hömberg, Dietmar ED - Horst, Ulrich ED - Kramer, Jürg ED - Mehrmann, Volker ED - Polthier, Konrad ED - Schmidt, Frank ED - Schütte, Christof ED - Skutella, Martin ED - Sprekels, Jürgen T1 - Mathematics cures virtual patients T2 - MATHEON-Mathematics for Key Technologies Y1 - 2014 VL - 1 SP - 7 EP - 25 PB - European Mathematical Society ER - TY - JOUR A1 - Gelß, Patrick A1 - Issagali, Aizhan A1 - Kornhuber, Ralf T1 - Fredholm integral equations for function approximation and the training of neural networks JF - SIAM Journal on Mathematics of Data Science Y1 - 2024 U6 - https://doi.org/10.1137/23M156642X VL - 6 IS - 4 ER -