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An adaptive Rothe method for nonlinear reaction-diffusion systems. (English) Zbl 0789.65075
The authors study a reaction-diffusion system of the form (1) $$u_ t - \nabla \cdot (a(x)\nabla u) = f(u,t)$$ for $$x\in \Omega \subset \mathbb{R}^ 1$$, $$t\in [0,T]$$, $$u: \Omega \times [0,T] \to \mathbb{R}^ k$$, where $$u$$ satisfies some boundary conditions and takes an initial value $$u_ 0$$ in $$L_ 2(\Omega)$$. The problem (1) is reformulated as an abstract Cauchy problem as follows:(2) $$\dot u(t) + Au(t) = f(u(t),t)$$, $$t\in [0,T]$$, $$u(0) = u_ 0$$, where $$u: [0,T] \to L_ 2(\Omega)$$ and $$A$$ denotes the weak representation of the diffusion operator in the original problem (1).
After adding on both sides of (2) a bounded linear operator representing (with the sign minus) an approximation of the Jacobian $$f_ u(u_ 0)$$ one can write out explicitly, using the theory of semigroups, a formal solution for this modified problem (2). The numerical analogue of the abstract solution formula is obtained via an adaptive two-stage Runge- Kutta method. Using suitable stability functions one gets strongly $$A$$- stable or $$L$$-stable Euler methods. A way to achieve a prescribed tolerance of the local error is given.
Elliptic problems arising during the realization of one time step of the above adaptive Runge-Kutta method are reformulated in variational forms and then solved approximately by the finite element method applying quadratic trial functions. Both local and global error estimates of the finite element method are given. For the whole solution process some criteria for global error estimation are proposed and investigated.
Next, the full adaptive algorithm is written out in a pseudocode. Finally, the authors use their method for the numerical solution of two interesting nonlinear reaction-diffusion systems, one from population ecology and the second one from flame propagation.
Reviewer: S.Burys (Kraków)

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 35K57 Reaction-diffusion equations
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