@misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {Adaptive multilevel discretization in time and space for parabolic partial differential equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4638}, number = {TR-89-07}, abstract = {The present paper developes an adaptive multilevel approach for parabolic PDE's - as a first step, for one linear scalar equation. Full adaptivity of the algorithm is conceptually realized by simultaneous multilevel discretization in both time and space. Thus the approach combines multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization such as the hierarchical basis method. The algorithmic approach is theoretically backed by careful application of fundamental results from semigroup theory. These results help to establish the existence of asymptotic expansions (in terms of time-steps) in Hilbert space. 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 arising space grids are not required to satisfy any quasi- uniformity assumption. Even though the theoretical presentation is independent of space dimension details of the algorithm and numerical examples are given for the 1-D case only. For the 1-D elliptic solver, which is used, an error estimator is established, which works uniformly well for a family of elliptic problems. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations II.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-410}, number = {SC-90-13}, abstract = {In continuation of part I this paper develops a variable-order time discretization in Hilbert space based on a multiplicative error correction. Matching of time and space errors as explained in part I allows to construct an adaptive multilevel discretization of the parabolic problem. In contrast to the extrapolation method in time, which has been used in part I, the new time discretization allows to separate space and time errors and further to solve fewer elliptic subproblems with less effort, which is essential in view of the application to space dimension greater than one. Numerical examples for space dimension one are included which clearly indicate the improvement.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {Adaptive Accuracy Control for Microcanonical Car-Parrinello Simulations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2311}, number = {SC-96-20}, abstract = {The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter \$\mu\$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP--solution from the BO-solution {\em decreases} like \$\mu^{1/2}\$ asymptotically. Since the computational effort {\em increases} like \$\mu^{-1/2}\$, the choice of \$\mu\$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls \$\mu\$: the parameter \$\mu\$ is repeatedly adapted during the simulation by choosing \$\mu\$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example.}, language = {en} } @misc{BornemannKrause, author = {Bornemann, Folkmar A. and Krause, Rolf}, title = {Classical and Cascadic Multigrid - A Methodical Comparison}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2368}, number = {SC-96-25}, abstract = {Using the full multigrid method {\em without} any coarse grid correction steps but with an a posteriori control of the number of smoothing iterations was shown by Bornemann and Deuflhard [1996] to be an optimal iteration method with respect to the energy norm. They named this new kind of multigrid iteration the {\em cascadic multigrid method}. However, numerical examples with {\em linear} finite elements raised serious doubts whether the cascadic multigrid method can be made optimal with respect to the {\em \$L^2\$-norm}. In this paper we prove that the cascadic multigrid method cannot be optimal for linear finite elements and show that the case might be different for higher order elements. We present a careful analysis of the two grid variant of the cascadic multigrid method providing a setting where one can understand the methodical difference between the cascadic multigrid method and the classical multigrid \$V\$-cycle almost immediately. As a rule of thumb we get that whenever the cascadic multigrid works the classical multigrid will work too but not vice versa.}, language = {en} } @misc{BornemannErdmannRoitzsch, author = {Bornemann, Folkmar A. and Erdmann, Bodo and Roitzsch, Rainer}, title = {KASKADE - Numerical Experiments.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4762}, number = {TR-91-01}, abstract = {The C-implementation of KASKADE, an adaptive solver for linear elliptic differential equations in 2D, is object of a set of numerical experiments to analyze the use of resources (time and memory) with respect to numerical accuracy. We study the dependency of the reliability, robustness, and efficiency of the program from the parameters controlling the algorithm.}, language = {en} } @misc{BornemannNettesheimSchuette, author = {Bornemann, Folkmar A. and Nettesheim, Peter and Sch{\"u}tte, Christof}, title = {Quantum-Classical Molecular Dynamics as an Approximation to Full Quantum Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1922}, number = {SC-95-26}, abstract = {This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a {\em partial} classical limit of the full Schr{\"o}dinger equation. This limit is achieved in two steps: separation of the full wavefunction and short wave asymptotics for its ``classical'' part. Both steps can be rigorously justified under certain smallness assumptions. Moreover, the results imply that neither the time-dependent self-consistent field method nor mixed quantum-semi-classical models lead to better approximations than QCMD since they depend on the separation step, too. On the other hand, the theory leads to a characterization of the critical situations in which the models are in danger of largely deviating from the solution of the full Schr{\"o}dinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of an Argon atom with a harmonic quantum oscillator.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {A Mathematical Approach to Smoothed Molecular Dynamics: Correcting Potentials for Freezing Bond Angles}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1960}, number = {SC-95-30}, abstract = {The interaction potential of molecular systems which are typically used in molecular dynamics can be split into two parts of essentially different stiffness. The strong part of the potential forces 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). However, the naive way of doing this via holonomic constraints is bound to produce incorrect results. The paper presents a mathematically rigorous discussion of the limit situation in which the stiffness of the strong part of the potential is increased to infinity. 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 and its significant contribution is demonstrated in an illustrative example. It appears that this correcting potential is definitely not identical with the Fixman-potential as was repeatedly assumed in the literature.}, language = {en} } @phdthesis{Bornemann, author = {Bornemann, Folkmar A.}, title = {Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3170}, number = {SC-97-48}, abstract = {We present a particular method for the explicit elimination of rapidly oscillating micro-scales in certain singularly perturbed conservative mechanical systems. Non-linear effects call for a non-trivial averaging procedure that we call {\em homogenization in time.} This method is based on energy principles and weak convergence techniques. Since non-linear functionals are in general {\em not} weakly sequentially continuous, we have to study {\em simultaneously} the weak limits of all those non-linear quantities of the rapidly oscillating components which are of importance for the underlying problem. Using the physically motivated concepts of {\em virial theorems}, {\em adiabatic invariants}, and {\em resonances}, we will be able to establish sufficiently many relations between all these weak limits, allowing to calculate them explicitly. Our approach will be {\em paradigmatical} rather than aiming at the largest possible generality. This way, we can show most clearly how concepts and notions from the physical background of the underlying mathematical problem enter and help to determine relations between weak limit quantities. In detail we will discuss natural mechanical systems with a strong constraining potential on Riemannian manifolds, the questions of realization of holonomic constraints, and singular limits of mixed quantum-classical coupling models. This latter class of problems also leads to a new proof for the adiabatic theorem of quantum mechanics. The strength of our methodology will be illustrated by applications to problems from plasma physics, molecular dynamics and quantum chemistry.}, language = {en} } @misc{SchuetteBornemann, author = {Sch{\"u}tte, Christof and Bornemann, Folkmar A.}, title = {Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3102}, number = {SC-97-41}, abstract = {In molecular dynamics applications there is a growing interest in including quantum effects for simulations of larger molecules. This paper is concerned with {\em mixed quantum-classical} models which are currently discussed: the so-called QCMD model with variants and the time-dependent Born-Oppenheimer approximation. All these models are known to approximate the full quantum dynamical evolution---under different assumptions, however. We review the meaning of these assumptions and the scope of the approximation. In particular, we characterize those typical problematic situations where a mixed model might largely deviate from the full quantum evolution. One such situation of specific interest, a non-adiabatic excitation at certain energy level crossings, can promisingly be dealt with by a modification of the QCMD model that we suggest.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {On the Singular Limit of the Quantum-Classical Molecular Dynamics Model}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2761}, number = {SC-97-07}, abstract = {\noindent In molecular dynamics applications there is a growing interest in so-called {\em mixed quantum-classical} models. These models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of quantum mechanics. A particularly extensively used model, the QCMD model, consists of a {\em singularly perturbed}\/ Schr{\"o}dinger equation nonlinearly coupled to a classical Newtonian equation of motion. This paper studies the singular limit of the QCMD model for finite dimensional Hilbert spaces. The main result states that this limit is given by the time-dependent Born-Oppenheimer model of quantum theory---provided the Hamiltonian under consideration has a smooth spectral decomposition. This result is strongly related to the {\em quantum adiabatic theorem}. The proof uses the method of {\em weak convergence} by directly discussing the density matrix instead of the wave functions. This technique avoids the discussion of highly oscillatory phases. On the other hand, the limit of the QCMD model is of a different nature if the spectral decomposition of the Hamiltonian happens not to be smooth. We will present a generic example for which the limit set is not a unique trajectory of a limit dynamical system but rather a {\em funnel} consisting of infinitely many trajectories.}, language = {en} } @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. 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Math.}, volume = {83}, journal = {Num. Math.}, number = {2}, doi = {10.1007/s002110050445}, pages = {179 -- 186}, year = {1999}, language = {en} }