@misc{BornemannErdmannKornhuber, 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}, 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{Bornemann, author = {Bornemann, Folkmar A.}, title = {Adaptive Solution of One-Dimensional Scalar Conservation Laws with Convex Flux.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-880}, number = {SC-92-18}, abstract = {A new adaptive approach for one-dimensional scalar conservation laws with convex flux is proposed. The initial data are approximated on an adaptive grid by a problem dependent, monotone interpolation procedure in such a way, that the multivalued problem of characteristic transport can be easily and explicitly solved. The unique entropy solution is chosen by means of a selection criterion due to LAX. For arbitrary times, the solutions is represented by an adaptive monotone spline interpolation. The spatial approximation is controlled by local \$L^1\$-error estimated. As a distinctive feature of the approach, there is no discretization in time. The method is monotone on fixed grids. Numerical examples are included, to demonstrate the predicted behavior. {\bf Key words.} method of characteristics, adaptive grids, monotone interpolation, \$L^1\$-error estimates {\bf AMS(MOS) subject classification.} 65M15, 65M25, 65M50.}, language = {en} } @misc{BornemannYserentant, author = {Bornemann, Folkmar A. and Yserentant, Harry}, title = {A Basic Norm Equivalence for the Theory of Multilevel Methods.}, doi = {10.1007/BF01388699}, number = {SC-92-01}, abstract = {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.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations III.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-519}, number = {SC-91-01}, abstract = {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.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {A Sharpened Condition Number Estimate for the BPX Preconditioner of Elliptic Finite Element Problems on Highly Nonuniform Triangulations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-596}, number = {SC-91-09}, abstract = {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.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations I. General Theory \& 1D-Implementation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-320}, number = {SC-90-04}, abstract = {A new adaptive multilevel approach for parabolic PDE's is presented. Full adaptivity of the algorithm is realized by combining multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization. In the theoretical part of the paper the existence of asymptotic expansions in terms of time-steps for single-step methods in Hilbert space is established. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The theoretical presentation is independent of space dimension. In this part I of the paper details of the algorithm and numerical examples are given for the 1D case only. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {On the Convergence of Cascadic Iterations for Elliptic Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1389}, number = {SC-94-08}, abstract = {We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them {\em cascadic iterations}. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of \$H^2\$-regular two-dimensional problems with quasi-uniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional \$H^{1+\alpha}\$-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method.}, language = {en} } @misc{BornemannErdmannKornhuber, 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}, 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{Bornemann, author = {Bornemann, Folkmar A.}, title = {Interpolation Spaces and Optimal Multilevel Preconditioners.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1285}, number = {SC-93-33}, abstract = {This paper throws light on the connection between the optimal condition number estimate for the BPX method and constructive approximation theory. We provide a machinery, which allows to understand the optimality as a consequence of an approximation property and an inverse inequality in \$H^{1+\epsilon}\$, \$\epsilon > 0\$. This machinery constructs so-called {\em approximation spaces}, which characterize a certain rate of approximation by finite elements and relates them with interpolation spaces, which characterize a certain smoothness.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {Homogenization of Highly Oscillatory Hamiltonian Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2050}, number = {SC-95-39}, abstract = {The paper studies Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale. In particular, we are interested in the limit situation where the size \$\epsilon\$ of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of order \${\rm O}(1)\$. The process of establishing an effective initial value problem for the limit positions will be called {\em homogenization} of the Hamiltonian system. This problem occurs in mechanics as the problem of realization of holonomic constraints, in plasma physics as the problem of guiding center motion, in the simulation of biomolecules as the so called smoothing problem. We suggest the systematic use of the notion of {\em weak convergence} in order to approach this problem. This methodology helps to establish unified and short proofs of the known results which throw light on the inherent structure of the problem. Moreover, we give a careful and critical review of the literature.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {Homogenization in Time II: Mechanical Systems Subject to Friction and Gyroscopic Forces}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3348}, number = {SC-97-65}, abstract = {In our previous work [Preprint SC 97-48] we have studied natural mechanical systems on Riemannian manifolds with a strong constraining potential. These systems establish fast nonlinear oscillations around some equilibrium manifold. Important in applications, the problem of elimination of the fast degrees of freedom, or {\em homogenization in time}, leads to determine the singular limit of infinite strength of the constraining potential. In the present paper we extend this study to systems which are subject to external forces that are non-potential, depending in a mixed way on positions {\em and}\/ velocities. We will argue that the method of weak convergence used in [1997] covers such forces if and only if they result from viscous friction and gyroscopic terms. All the results of [1997] directly extend if there is no friction transversal to the equilibrium manifold; elsewise we show that instructive modifications apply.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {A Mathematical Investigation of the Car-Parrinello Method}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2302}, number = {SC-96-19}, abstract = {The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the {\em quantitative} influence of the involved parameter \$\mu\$, the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order \${\rm O}(\mu^{1/2})\$ --- provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general. Finally, we show how to use the gained quantitative insight for an automatic control of the unphysical ``fake'' kinetic energy of the method.}, language = {en} } @misc{SchuetteBornemann, author = {Sch{\"u}tte, Christof and Bornemann, Folkmar A.}, title = {Homogenization Approach to Smoothed Molecular Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2410}, number = {SC-96-31}, abstract = {{\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.}}, language = {en} } @misc{BornemannSchemann, author = {Bornemann, Folkmar A. and Schemann, Martin}, title = {Adaptive Rothe's Method for the 2D Wave Equation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2492}, number = {SC-96-39}, abstract = {The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved {\em virtually} using an adaptive state-of-the-art integrator. The {\em actual} realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus {\em perturbing} the virtual function space method. The admissible size of that perturbation can be computed {\em a priori} and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4821}, number = {TR-91-07}, abstract = {A new adaptive multilevel approach for linear partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A Multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy.}, language = {en} } @book{DeuflhardBornemann1994, author = {Deuflhard, Peter and Bornemann, Folkmar A.}, title = {Numerische Mathematik. II}, publisher = {De Gruyter Lehrbuch. Berlin: de Gruyter}, year = {1994}, language = {en} } @article{BornemannDeuflhard1996, author = {Bornemann, Folkmar A. and Deuflhard, Peter}, title = {The cascadic multigrid method for elliptic problems}, series = {Numer. Math.}, volume = {75}, journal = {Numer. Math.}, pages = {135 -- 152}, year = {1996}, language = {en} } @inproceedings{BornemannDeuflhard1996, author = {Bornemann, Folkmar A. and Deuflhard, Peter}, title = {Cascadic Multigrid Methods}, series = {Domain Decomposition Methods in Sciences and Engineering}, booktitle = {Domain Decomposition Methods in Sciences and Engineering}, editor = {Glowinski, J. R. and Widlund, Olof}, publisher = {John Wiley \& Sons Ltd}, pages = {205 -- 212}, year = {1996}, language = {en} } @book{DeuflhardBornemann2002, author = {Deuflhard, Peter and Bornemann, Folkmar A.}, title = {Scientific computing with ordinary differential equations. Transl. from the German by Werner C. Rheinboldt}, series = {Texts in Applied Mathematics}, volume = {42}, journal = {Texts in Applied Mathematics}, publisher = {New York, NY: Springer}, year = {2002}, language = {en} } @book{DeuflhardBornemann2002, author = {Deuflhard, Peter and Bornemann, Folkmar A.}, title = {Numerische Mathematik. II}, edition = {2}, publisher = {de Gruyter Lehrbuch. Berlin: de Gruyter}, year = {2002}, language = {en} } @article{NettesheimBornemannSchmidtetal.1996, author = {Nettesheim, Peter and Bornemann, Folkmar A. and Schmidt, Burkhard and Sch{\"u}tte, Christof}, title = {An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics}, series = {Chem. Phys. Lett.}, volume = {256}, journal = {Chem. Phys. Lett.}, number = {6}, doi = {10.1016/0009-2614(96)00471-X}, pages = {581 -- 588}, year = {1996}, language = {en} } @article{BornemannNettesheimSchuette1996, author = {Bornemann, Folkmar A. and Nettesheim, Peter and Sch{\"u}tte, Christof}, title = {Quantum-classical molecular dynamics as an approximation to full quantum dynamics}, series = {J. Chem. Phys.}, volume = {105}, journal = {J. Chem. Phys.}, number = {3}, doi = {10.1063/1.471952}, pages = {1074 -- 1083}, year = {1996}, language = {en} } @article{BornemannSchuette1999, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {On the Singular Limit of the Quantum-Classical Molecular Dynamics Model}, series = {J. Appl. Math.}, volume = {59}, journal = {J. Appl. Math.}, number = {4}, doi = {10.1137/S0036139997318834}, pages = {1208 -- 1224}, year = {1999}, language = {en} } @article{BornemannSchuette1998, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {A mathematical investigation of the Car-Parrinello Method}, series = {Num. Math.}, volume = {78}, journal = {Num. Math.}, number = {3}, doi = {10.1007/s002110050316}, pages = {359 -- 376}, year = {1998}, language = {en} } @article{SchuetteBornemann1997, author = {Sch{\"u}tte, Christof and Bornemann, Folkmar A.}, title = {Homogenization Approach to Smoothed Molecular Dynamics}, series = {Nonlinear Analysis}, volume = {30}, journal = {Nonlinear Analysis}, number = {3}, doi = {10.1016/S0362-546X(97)00216-2}, pages = {1805 -- 1814}, year = {1997}, language = {en} } @article{BornemannSchuette1997, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {Homogenization of Hamiltonian Systems with a Strong Constraining Potential}, series = {Physica D}, volume = {102}, journal = {Physica D}, number = {1-2}, doi = {10.1016/S0167-2789(96)00245-X}, pages = {57 -- 77}, year = {1997}, language = {en} } @inproceedings{SchuetteBornemann1999, author = {Sch{\"u}tte, Christof and Bornemann, Folkmar A.}, title = {Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model}, series = {Computational Molecular Dynamics}, volume = {4}, booktitle = {Computational Molecular Dynamics}, editor = {Deuflhard, Peter and Hermans, J. and Leimkuhler, Benedict and Marks, A. and Reich, Sebastian and Skeel, R.}, publisher = {Springer}, pages = {380 -- 395}, year = {1999}, language = {en} } @article{BornemannSchuette1999, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {Adaptive Accuracy Control for Car-Parrinello Simulations}, series = {Num. Math.}, volume = {83}, journal = {Num. Math.}, number = {2}, doi = {10.1007/s002110050445}, pages = {179 -- 186}, year = {1999}, language = {en} }