# The Nonlinear Poisson-Boltzmann Equation: well-posedness analysis and a posteriori error estimate

## Dr. Svetoslav Nakov

June 30, 2015, 3:30 p.m. S2 059

In this talk, we show a short derivation of the Poisson-Boltzmann equation and then discuss the well-posedness of the problem as it appears in the state of the art. The main difficulties related to PBE are the strong nonlinearity of exponential type and the delta distributions on the right hand side of the equation. A series of splittings of the solution are made first to eliminate the singular delta distributions on the right and then for further analysis. A priori $L_\infty$ estimates are shown for the respective components of the solution. Using these a-priori $L_\infty$ estimates it is possible to show existence and uniqueness. \\
\indent Next, the focus goes on deriving a functional a posteriori error estimate for the PBE. The advantage of the functional a posteriori error estimates based on the duality theory is that only the structure of the equation alone is exploited and therefore no global or local constants enter in the estimate. This is in contrast to other methods, e.g a residual based one, which depend on the particular triangulation. Therefore functional type a posteriori error estimates give not only an error indicator, but also a guaranteed tight bound on the error. We discuss two cases for the coefficient in the nonlinear term. First, we show the error estimates for a uniformly positive coefficient. Second, we discuss an approach to obtain such error estimates when this coefficient is identically zero in parts of the domain with positive measure.