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Convergence and stability of higher-order finite element solution of reaction-diffusion equation with Turing instability. (English) Zbl 1363.65147

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, November 18–21, 2015. In honor of the birthday anniversaries of Ivo Babuška (90), Milan Práger (85), and Emil Vitásek (85). Prague: Czech Academy of Sciences, Institute of Mathematics (ISBN 978-80-85823-65-3). 140-147 (2015).
The author employs numerical tests to investigate the convergence of higher-order finite element method when applied to a 2D nonlinear reaction-diffusion problem exhibiting the Turing instability. There are not enough mathematical results to carry out the investigation theoretically.
The second order Crank-Nicolson method is used for the time discretization and various versions of the finite element method for the space discretization that follows (the Rothe method). The computation continues in time until the steady-state of the solution \(u\), \(v\) is reached.
Special attention is paid to the dependence of the solution on the scaling parameter \(\delta\) at the \(\Delta u\) and \(\Delta v\) terms. Several figures and a table show that the convergence of the finite element method can differ from usual patterns.
For the entire collection see [Zbl 1329.00187].

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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