@misc{SanderKlapprothYouettetal.2012, author = {Sander, Oliver and Klapproth, Corinna and Youett, Jonathan and Kornhuber, Ralf and Deuflhard, Peter}, title = {Towards an Efficient Numerical Simulation of Complex 3D Knee Joint Motion}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14512}, number = {12-06}, year = {2012}, abstract = {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.}, language = {en} } @misc{Kornhuber1996, author = {Kornhuber, Ralf}, title = {On Robust Multigrid Methods for Piecewise Smooth Variational Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2487}, number = {SC-96-38}, year = {1996}, abstract = {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.}, language = {en} } @misc{Kornhuber1993, author = {Kornhuber, Ralf}, title = {Monotone Multigrid Methods for Elliptic Variational Inequalities I.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1147}, number = {SC-93-18}, year = {1993}, abstract = {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.}, language = {en} } @misc{Kornhuber1993, author = {Kornhuber, Ralf}, title = {Monotone Multigrid Methods for Elliptic Variational Inequalities II.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1151}, number = {SC-93-19}, year = {1993}, abstract = {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.}, language = {en} } @misc{KornhuberYserentant1993, author = {Kornhuber, Ralf and Yserentant, Harry}, title = {Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1170}, number = {SC-93-21}, year = {1993}, abstract = {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.}, language = {en} } @misc{BornemannErdmannKornhuber1993, author = {Bornemann, Folkmar A. and Erdmann, Bodo and Kornhuber, Ralf}, title = {A Posteriori Error Estimates for Elliptic Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1257}, number = {SC-93-29}, year = {1993}, abstract = {{\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.}}, language = {en} } @misc{ErdmannHoppeKornhuber1993, author = {Erdmann, Bodo and Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Adaptive Multilevel-Methods for Obstacle Problems in Three Space Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1049}, number = {SC-93-08}, year = {1993}, abstract = {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.}, language = {en} } @misc{BornemannErdmannKornhuber1992, author = {Bornemann, Folkmar A. and Erdmann, Bodo and Kornhuber, Ralf}, title = {Adaptive Multilevel-Methods in 3-Space Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-843}, number = {SC-92-14}, year = {1992}, abstract = {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.}, language = {en} } @misc{HoppeKornhuber1991, author = {Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Adaptive Multilevel - Methods for Obstacle Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-667}, number = {SC-91-16}, year = {1991}, abstract = {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.}, language = {en} } @misc{HoppeKornhuber1988, author = {Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Multi-Grid Solution of Two Coupled Stefan Equations Arising in Induction Heating of Large Steel Slabs.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-160}, number = {SC-88-08}, year = {1988}, abstract = {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.}, language = {en} }