When Bingham meets Bratu: mathematical and computational investigations

Abstract

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.

Adjoint-Based Estimation of Sensitivity of Clinical Measures to Boundary Conditions for Arteries

“…Sensitivity to Changes in Inflow Position. Consider next the variation of the Lagrangian L given in (7) with respect to the normal n. We recall that after simplifications, we have…”

“…Another possibility is via adjoints [16,23,13,12,2]. We also refer to a series of works by Glowinski and collaborators on the role of adjoints in optimization [8,11,21,1,7,9]. See also [22,14,10].…”

Nonlinear iterative approximation of steady incompressible chemically reacting flows

“…Glowinski's papers [3,4] with Americo Marrocco were some of the earliest contributions to the finite element approximation of p-Laplace type nonlinear elliptic equations and associated convex energy-minimization problems for functionals with p-growth of the kind that appear in models of steady incompressible quasi-Newtonian fluids. Glowinski's subsequent work over the past five decades on the Bingham model [5][6][7][8][9][10][11] involved a range of new ideas, including domain decomposition and operator splitting methods, the analysis of qualitative properties of Bingham flows, particularly large-time stabilization and, most recently in [12], the numerical solution of the Bingham-Bratu-Gelfand problem, a non-smooth nonlinear eigenvalue problem associated with the total variation integral that includes an additional exponential nonlinearity.…”