Higher-order finite element methods for elliptic problems with interfaces

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.Comment: 26 pages, 6 figures. An earlier version of this paper appeared on November 13, 2014 in http://www.brown.edu/research/projects/scientific-computing/reports/201

Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation

In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear Poisson-Boltzmann equation and its regularizations. The algorithm we study consists of the standard SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element algorithms, but where the SOLVE step involves only a full solve on the coarsest level, and the remaining levels involve only single Newton updates to the previous approximate solution. We summarize a recently developed AFEM convergence theory for inexact solvers, and present a sequence of numerical experiments that give evidence that the theory does in fact predict the contraction properties of AFEM with inexact solvers. The various routines used are all designed to maintain a linear-time computational complexity.Comment: Submitted to DD20 Proceeding

Variational approach for electrolyte solutions: from dielectric interfaces to charged nanopores

A variational theory is developed to study electrolyte solutions, composed of interacting point-like ions in a solvent, in the presence of dielectric discontinuities and charges at the boundaries. Three important and non-linear electrostatic effects induced by these interfaces are taken into account: surface charge induced electrostatic field, solvation energies due to the ionic cloud, and image charge repulsion. Our variational equations thus go beyond the mean-field theory. The influence of salt concentration, ion valency, dielectric jumps, and surface charge is studied in two geometries. i) A single neutral air-water interface with an asymmetric electrolyte. A charge separation and thus an electrostatic field gets established due to the different image charge repulsions for coions and counterions. Both charge distributions and surface tension are computed and compared to previous approximate calculations. For symmetric electrolyte solutions close to a charged surface, two zones are characterized. In the first one, with size proportional to the logarithm of the coupling parameter, strong image forces impose a total ion exclusion, while in the second zone the mean-field approach applies. ii) A symmetric electrolyte confined between two dielectric interfaces as a simple model of ion rejection from nanopores. The competition between image charge repulsion and attraction of counterions by the membrane charge is studied. For small surface charge, the counterion partition coefficient decreases with increasing pore size up to a critical pore size, contrary to neutral membranes. For larger pore sizes, the whole system behaves like a neutral pore. The prediction of the variational method is also compared with MC simulations and a good agreement is observed.Comment: This version is accepted for publication in Phys. Rev. E