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A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems.

Please always quote using this URN: urn:nbn:de:0297-zib-755
  • 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.

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Metadaten
Author:Jens Lang, Artur Walter
Document Type:ZIB-Report
Tag:Rothe method; adaptive Runge-Kutta method; finite elements; initial boundary value problem; mesh refinement
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J15 Equations with nonlinear operators (do not use 65Hxx)
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M30 Improperly posed problems
65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M50 Mesh generation and refinement
Date of first Publication:1992/03/31
Series (Serial Number):ZIB-Report (SC-92-05)
ZIB-Reportnumber:SC-92-05
Published in:Appeared in: IMPACT Comput. Sci. Engrg. 4, pp. 269-314 (1992)
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