@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}, 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 = {Adaptive Accuracy Control for Car-Parrinello Simulations}, volume = {83}, journal = {Num. Math.}, number = {2}, doi = {10.1007/s002110050445}, pages = {179 -- 186}, year = {1999}, 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}, 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 = {On the Singular Limit of the Quantum-Classical Molecular Dynamics Model}, 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}, 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}, 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}, volume = {102}, journal = {Physica D}, number = {1-2}, doi = {10.1016/S0167-2789(96)00245-X}, pages = {57 -- 77}, year = {1997}, 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}, volume = {256}, journal = {Chem. Phys. Lett.}, number = {6}, doi = {10.1016/0009-2614(96)00471-X}, pages = {581 -- 588}, year = {1996}, language = {en} } @misc{BornemannSchuette1996, 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}, year = {1996}, 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{BornemannKrause1996, 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}, year = {1996}, 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} }