Theoretical and numerical analysis of a class of nonlinear elliptic equations (Q2504859)

This paper considers the existence and numerical computation of an ordinary quasilinear second order first-kind boundary value problem possessing the following difficulties: the nonlinearity is in \(u'\) convex but may grow arbitrarily fast, moreover, the right-hand side may contain a finite measure with respect to \(x\), e.g. a delta-function. It is shown that the problem has a weak solution in a Sobolev space if it has a weak super-solution, Numerically, the latter is approximated first to start an iteration using Yosida approximation and Newton on the nonlinearity, and combining overlapping domain decomposition and a finite element method. Convergence of the whole procedure is not considered but the domain decomposition is shown to converge. Several computations on problems with delta-functions and with algebraic growth in \(u'\) and \(u\) illustrate the approach.