When Bingham meets Bratu: Mathematical and computational investigations

In this article, we discuss the numerical solution of the Bingham-Bratu-Gelfand (BBG) problem, a non-smooth nonlinear eigenvalue problem associated with the total variation integral and an exponential nonlinearity. Using the fact that one can view the nonlinear eigenvalue as a possible Lagrange multiplier associated with a constrained minimization problem from Calculus of Variations, we associate with the BBG problem an initial value problem (dynamical flow), well suited to time-discretization by operator-splitting. Various mathematical results are proved, including the convergence of a finite element approximation of the BBG problem. The operator-splitting/finite element methodology discussed in this article is robust and easy to implement. We validate the implementation by first solving the classical Bratu-Gelfand problem, obtaining and reporting results consistent with those found in the literature. We then explore the full capability of the implementation by solving the viscoplastic BBG problem, obtaining and reporting results for several values of the plasticity yield. We conclude by exhibiting and discussing the bifurcation diagrams corresponding to these same values of the plasticity yield, and by reporting and examining some finer details of the solver discovered during the course of our investigation.