A finite element method adaptive in space and time for nonlinear reaction-diffusion systems

Abstract 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 behavior of the proposed approaches is demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics, and combustion theory.

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