Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

Efficient solution and value function for non-convex variational problems

In this paper, we want to investigate a wide range of non-convex variational problems and obtain some sufficient and necessary conditions for existence of a feasible solution for these problems. Hence, we define optimal value function corresponding to these problems and obtain a relationship between subdifferential of the optimal value function and the set of Lagrange multipliers.