Numerical methods for reaction-diffusion problems with non-differentiable kinetics
A class of steady-state semilinear reaction-diffusion problems with non- differential kinetics is considered. It is described by the scalar equation \(\Delta u=\lambda f(u(x)),\) \(0<\lambda =const\), \(x\in \Omega\) (u\(\geq 0\) in \(\Omega)\), \(u=1\), \(x\in \partial \Omega\), with \(\Omega\) \(\subset R\) n, \(n=1,2,3\), \(f(t)=t\) p if \(t>0\), \(0<p<1\), \(f(t)=0\), if \(t\leq 0.\) For a finite element scheme with linear triangular elements the estimates \(\| v-v\) \(h\| \leq ch\), if \(0.5<p<1\), and \(\| v-v\) \(h\| \leq ch^{2p}\), if \(0<p<0.5\), are proved in which v is the exact solution of the problem, v h is the finite element solution and \(\| \cdot \|\) is the norm in the Sobolev space H 1(\(\Omega)\). The authors note that their numerical results show that for all \(p\in (0,1)\) the rate of convergence is O(h).
steady-state semilinear reaction-diffusion problems
scientific article; zbMATH DE number 4070246
scientific article