TY - GEN A1 - Bornemann, Folkmar A. T1 - An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions. N2 - 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. T3 - ZIB-Report - TR-91-07 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4821 ER - TY - GEN A1 - Bornemann, Folkmar A. T1 - Adaptive multilevel discretization in time and space for parabolic partial differential equations. N2 - 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. T3 - ZIB-Report - TR-89-07 Y1 - 1989 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4638 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Erdmann, Bodo A1 - Roitzsch, Rainer T1 - KASKADE - Numerical Experiments. N2 - 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. T3 - ZIB-Report - TR-91-01 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4762 ER - TY - BOOK A1 - Deuflhard, Peter A1 - Bornemann, Folkmar A. T1 - Numerische Mathematik. II Y1 - 1994 PB - De Gruyter Lehrbuch. Berlin: de Gruyter ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Deuflhard, Peter T1 - The cascadic multigrid method for elliptic problems JF - Numer. Math. Y1 - 1996 VL - 75 SP - 135 EP - 152 ER - TY - CHAP A1 - Bornemann, Folkmar A. A1 - Deuflhard, Peter ED - Glowinski, J. R. ED - Widlund, Olof T1 - Cascadic Multigrid Methods T2 - Domain Decomposition Methods in Sciences and Engineering Y1 - 1996 SP - 205 EP - 212 PB - John Wiley & Sons Ltd ER - TY - BOOK A1 - Deuflhard, Peter A1 - Bornemann, Folkmar A. T1 - Numerische Mathematik. II Y1 - 2002 PB - de Gruyter Lehrbuch. Berlin: de Gruyter ET - 2 ER - TY - BOOK A1 - Deuflhard, Peter A1 - Bornemann, Folkmar A. T1 - Scientific computing with ordinary differential equations. Transl. from the German by Werner C. Rheinboldt T3 - Texts in Applied Mathematics Y1 - 2002 VL - 42 PB - New York, NY: Springer ER -