The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities (Q1899115)

This paper is concerned with a coupled semilinear elliptic boundary value problem which arises in the modelling of irreversible, isothermal chemical reactions involving two chemicals. The unknown variables are the concentrations \(u\) and \(v\) of the chemicals, and the problem takes the form \[ \begin{alignedat}{2} -\nabla u + \lambda f(u)g(v) & = 0 \text{ in }\Omega,&&\quad u = a \text{ on }\partial \Omega,\\ - \nabla v + k \lambda f(u) g(v) & = 0 \text{ in }\Omega,&&\quad v= b \text{ on } \partial \Omega.\end{alignedat} \] Here \(\Omega\) is a bounded Lipschitz domain, \(a\) and \(b\) are nonnegative functions, \(k\) and \(\lambda\) are positive constants. The functions \(f\) and \(g\) are continuous on \(\mathbb{R}\), continuously differentiable on \((0,\infty)\), and monotonically increasing with \(f(t) = g(t) = 0\) for \(g \leq 0\). In addition they satisfy the growth and Hölder conditions \[ \begin{alignedat}{1} f(t), g(t) & \leq C(1 + t^\gamma)\;\text{ for } t \geq 0 \text{ and }\gamma \geq 0,\\ \text{ and }|f(t_1) - f(t_2)|& \leq C|t_1 - t_2 |^p,\;|g(t_1) - g(t_2)|\leq C|t_1 - t_2|^q \end{alignedat} \] for \(t_1,t_2 \in [0,\varepsilon_0]\) and \(p, q \in (0,1]\). The paper studies finite element approximations of this problem using piecewise linear Galerkin approximations, and numerical integration. First, the so-called catalyst problem is addressed; this is the case in which \(k = 0\) and \(a\) and \(b\) are constants, so that \(v\) is constant, and may be thought of as a catalyst. If \(p \in (0,1)\), so that \(f\) is not locally Lipschitz, there is the possibility of a `dead core' region in which \(u\) is zero. The non-differentiability of \(f\) is dealt with by introducing a regularization \(f_\varepsilon\). For this case the authors summarise results obtained previously by \textit{R. Nochetto} [Numer. Math. 54, No. 3, 243-255 (1988; Zbl 0663.65125)]. They then go on to introduce a numerical integration scheme for the term in the variational problem involving \(f\), and obtain an \(O(h^2 \ln h^{- 1})^{1/(2-p)}\) error estimate in the \(W^{1,\infty}\)-norm, for the regularized problem with numerical integration, for a specific choice of \(\varepsilon\). \(H^1\)-error estimates are also derived. The authors return to the coupled problem, and with the assumption \(p \leq q\), adapt the method used by Nochetto [loc. cit.] to obtain error bounds in the \(W^{0,\infty}\)- and \(H^1\)-norms, for the error due to regularization. The discrete problem is considered both with and without numerical integration, and the authors are able, by adapting the approach used by Nochetto [loc. cit.], to obtain error bounds similar to those obtained for the catalyst problem. [Reviewer's remark: The numbering of references in the text of the paper differs from that in the list of references, in that reference \(N\) is quoted in the text as reference \((N - 1)\)].