@misc{Bornemann1991, 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}, year = {1991}, 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} } @misc{Bornemann1989, 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}, year = {1989}, 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{BornemannErdmannRoitzsch1991, 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}, year = {1991}, 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} } @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}, 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}, 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 = {Numerische Mathematik. II}, edition = {2}, publisher = {de Gruyter Lehrbuch. Berlin: de Gruyter}, year = {2002}, 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}, volume = {42}, journal = {Texts in Applied Mathematics}, publisher = {New York, NY: Springer}, year = {2002}, language = {en} }