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

Abstract

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 [25] 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.Keywords: Poisson-Boltzmann equation, semilinear partial differential equations, existence and uniqueness of solutions, convergence of finite element approximations, a priori error estimates, guaranteed and efficient a posteriori error bounds, error indicators, adaptive mesh refinement, qualified and unqualified convergence.

Reliable computer simulation methods for electrostatic biomolecular models based on the Poisson-Boltzmann equation

“…Then, for the minimizer u, using Lebesgue Dominated Convergence Theorem, we can prove that it is indeed a solution to (2.28). The uniqueness of the solution u of (2.28) is proven in a similar way to the approach in [21]. However, an easier approach is to take advantage of the fact that we have already shown existence and uniqueness of a solution to problems (2.23) and (2.26).…”

“…The present paper is a continuation of a recent work by the authors ( [21]) and is devoted to adaptive modeling of electrostatic interactions of biomolecules. We use two test systems on which the theoretical findings are demonstrated.…”

“…To see that (2.28) has a solution, we can define the energy functional J over H 1 g (Ω), like in [21]…”

“…Now, we turn our attention to the problem (2.26) which falls in the class of problems that we have considered in [21]. Since in practice, we only have an approximation ũL Ωs) .…”

Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

“…The fact that the regular component of a solution obtained by such a decomposition satisfies a weak formulation involving H 1 spaces means that this component can be numerically approximated by means of well studied methods, such as standard conforming finite elements. Besides, it also means that the duality approach for error estimation is applicable to obtain both a priori near-best approximation results and to compute guaranteed a posteriori error bounds, as done in [40,41]. Having a C 1 interface also has practical implications, since in this case it is easier to represent exactly with curved elements or isogeometric analysis.…”