Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation (Q1985901)

The purpose is to investigate the well-posedness of a boundary problem related to the Poisson-Boltzmann equation expressed as: \(-\nabla.(\epsilon\nabla u)+k^2\sinh(u+w)=l\), in \(\Omega _1\cup\Omega_2\), with \([u]_\Gamma=0\), \([\frac{\partial u}{\partial n}]_{\Gamma}=0\), \(\Gamma\) being the boundary of \(\Omega _1\) which is an interior domain of \(\Omega\), \(\Omega\subset R^d\), \(d=2,3\), \(\Omega _2:=\Omega\backslash(\Omega_1\cup\Gamma)\) and \(u=0\) on \(\partial\Omega\). The bounded domain \(\Omega\) is assumed to have a Lipschitz boundary as well as its subdomain \(\Omega_1\), the coefficients \(\epsilon,k\in L^{\infty}(\Omega)\), \(\epsilon_{\max}\geq\epsilon\geq\epsilon_{\min}>0\), \(w\) is a measurable function and \(l\in L^2(\Omega)\). The paper considers the cases: \(k(x)\equiv 0\) in \(\Omega _1\), \(k_{\max}\geq k(x)\geq k_{\min}>0\) in \(\Omega_2\), \(w\in L^{\infty}(\Omega _2)\) only, that seem to be the most interesting in practice. The variational form of the problem is considered. One defines the weak solution, the corresponding functional and formulates the variational problem. The existence of a unique minimizer to this variational problem is proved. The next section starts with a recall of some results from the duality theory, Fenchel conjugates, error measures, obtains a special error identity and estimations and proves an approximation result. The rest of the paper is devoted to some numerical results where the performances are extensively discussed. Error estimations for variational problems and error identities are to be found mostly in earlier papers of the third author, S. I. Repin.