TY - GEN A1 - Erdmann, Bodo A1 - Lang, Jens A1 - Matera, Sebastian A1 - Wilmanski, Krzysztof T1 - Adaptive Linearly Implicit Methods for Linear Poroelastic Equations N2 - Adaptive numerical methods in time and space are introduced and studied for linear poroelastic models in two and three space dimensions. We present equivalent models for linear poroelasticity and choose both the {\em displacement--pressure} and the {\em stress--pressure} formulation for our computations. Their discretizations are provided by means of linearly implicit schemes in time and linear finite elements in space. Our concept of adaptivity opens a way to a fast and reliable simulation of different loading cases defined by corresponding boundary conditions. We present some examples using our code {\sf Kardos} and show that the method works efficiently. In particular, it could be used in the simulation of some bone healing models. T3 - ZIB-Report - 06-37 KW - Poroelasticity KW - Biot's model KW - bone healing KW - adaptive finite elements KW - adaptive time integration KW - Rosenbrock methods Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9312 ER - TY - GEN A1 - Lang, Jens A1 - Erdmann, Bodo A1 - Roitzsch, Rainer T1 - Three-Dimensional Fully Adaptive Solution of Thermo-Diffusive Flame Propagation Problems N2 - In this paper we present a self--adaptive finite element method to solve flame propagation problems in 3D. An implicit time integrator of Rosenbrock type is coupled with a multilevel approach in space. The proposed method is applied to an unsteady thermo--diffusive combustion model to demonstrate its potential for the solution of complicated problems. T3 - ZIB-Report - SC-97-18 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2879 ER - TY - GEN A1 - Lang, Jens T1 - Adaptive FEM for Reaction-Diffusion Equations N2 - An integrated time--space adaptive finite element method for solving mixed systems of nonlinear parabolic, elliptic, and differential algebraic equations is presented. The approach is independent of the spatial dimension. For the discretization in time we use singly diagonally linearly implicit Runge--Kutta methods of Rosenbrock type. Local time errors for the step size control are defined by an embedded strategy. A multilevel finite element Galerkin method is subsequently applied for the discretization in space. A posteriori estimates of local spatial discretization errors are obtained solving local problems with higher order approximation. Superconvergence arguments allow to simplify the required computations. Two different strategies to obtain the start grid of the multilevel process are compared. The devised method is applied to a solid--solid combustion problem. T3 - ZIB-Report - SC-96-28 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2393 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Lang, Jens T1 - Twodimensional Cascadic Finite Element Computations of Combustion Problems N2 - We present an integrated time--space adaptive finite element method for solving systems of twodimensional nonlinear parabolic systems in complex geometry. The partial differential system is first discretized in time using a singly linearly implicit Runge--Kutta method of order three. Local time errors for the step size control are defined by an embedding strategy. These errors are used to propose a new time step by a PI controller algorithm. A multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space. The local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation. This two--fold adaptivity successfully ensures an a priori prescribed tolerance of the solution. The devised method is applied to laminar gaseous combustion and to solid--solid alloying reactions. We demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm. T3 - ZIB-Report - SC-96-05 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2167 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Lang, Jens A1 - Roitzsch, Rainer T1 - Selfadaptive Finite Element Computations with Smooth Time Controller and Anisotropic Refinement N2 - We present Multilevel Finite Element computations for twodimensional reaction-diffusion systems modelling laminar flames. These systems are prototypes for extreme stiffness in time and space. The first of these two rather general features is accounted for by an improved control mechanism for the time step. The second one is reflected through very thin travelling reaction fronts for which we propose an anisotropic discretization by local directional refinement. T3 - ZIB-Report - SC-96-16 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2270 ER - TY - GEN A1 - Deuflhard, Peter A1 - Lang, Jens A1 - Nowak, Ulrich T1 - Adaptive Algorithms in Dynamical Process Simulation N2 - Dynamical simulation of industrially relevant processes strongly advises the use of algorithms, which are {\em adaptive} both in time and in space discretization. The paper presents two alternatives: (a) a fully adaptive method of lines approach, which is based on finite difference methods and essentially applicable to 1D problems; (b) a fully adaptive Rothe method, which is based on a fast multilevel finite element method and applicable to 1D up to 3D. T3 - ZIB-Report - SC-95-16 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1823 ER - TY - GEN A1 - Lang, Jens T1 - Two-Dimensional Fully Adaptive Solutions of Reaction-Diffusion Equations N2 - We present an adaptive Rothe method for two--dimensional problems combining an embedded Runge--Kutta scheme in time and a multilevel finite element discretization in space. The spatial discretization error is controlled by a posteriori error estimates based on interpolation techniques. A computational example for a thermodiffusive flame propagation model illustrates the high accuracy that is possible with the proposed method. T3 - ZIB-Report - SC-94-34 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1627 ER - TY - GEN A1 - Lang, Jens T1 - High-Resolution Selfadapative Computations on Chemical Reaction-Diffusion Problems with Internal Boundaries N2 - Large chemical computations show the need for full adaptivity supporting the development of robust and highly efficient programs. For solutions possessing sharp moving spatial transitions, as travelling wavefronts or emerging boundary and internal layers, an automatic adjustment of both the space and the time stepsize is generally accepted to be more successful in efficient resolving critical regions of high spatial and temporal activity. In contrast to the widespread discretization sequence first space then time the reversed sequence first time then space is employed. Full adaptivity of the proposed algorithm is realized by combining embedded time discretization and multilevel finite element space discretization. In this paper the algorithm is described for one--dimensional problems. The numerical results show the significantly new perspectives opened by this approach. T3 - ZIB-Report - SC-94-09 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1395 ER - TY - GEN A1 - Lang, Jens A1 - Walter, Artur T1 - A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems. N2 - Large scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction- diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means, we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods whereas for the space discretization a finite element method is used. The different behaviour of the proposed approaches are demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics and combustion theory. {\bf Keywords:} initial boundary value problem, Rothe- method, adaptive Runge-Kutta method, finite elements, mesh refinement. {\bf AMS CLASSIFICATION:} 65J15, 65M30, 65M50. T3 - ZIB-Report - SC-92-05 KW - initial boundary value problem KW - Rothe method KW - adaptive Runge-Kutta method KW - finite elements KW - mesh refinement Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-755 ER - TY - GEN A1 - Lang, Jens A1 - Walter, Artur T1 - An Adaptive Discontinuous Finite Element Method for the Transport Equation. N2 - In this paper we introduce a discontinuous finite element method. In our approach, it is possible to combine the advantages of finite element and finite difference methods. The main ingredients are numerical flux approximation and local orthogonal basis functions. The scheme is defined on arbitrary triangulations and can be easily extended to nonlinear problems. Two different error indicators are derived. Especially the second one is closely connected to our approach and able to handle arbitrary variing flow directions. Numerical results are given for boundary value problems in two dimensions. They demonstrate the performance of the scheme, combined with the two error indicators. {\bf Key words:} neutron transport equation, discontinuous finite element, adaptive grid refinement. {\bf Subject classifications:} AMS(MOS) 65N30, 65M15. T3 - ZIB-Report - SC-91-07 KW - neutron transport equation KW - discontinuous finite element KW - adaptive grid refinement Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-579 ER -