# Functional a Posteriori Error Estimates for the Nonlinear Poisson-Boltzmann Equation

## Dr. Svetoslav Nakov

Nov. 21, 2017, 3:30 p.m. S2 416-1

In this talk, we show how to derive reliable functional a posteriori error estimates for the nonlinear Poisson-Boltzmann equation (PBE) which give not only an error indicator but also a tight bound on the error. The full solution $\phi$ is split in $\phi=G+u^h+u$ or in $\phi=G+u$, where $G$ is analytically known and $u^h$ is harmonic in the molecule domain and equal to $-G$ outside and is easy to approximate numerically. Our goal is to derive a posteriori error estimates for the component $u$ which satisfies a nonlinear elliptic equation with nonhomogeneous Dirichlet boundary condition and nonhomogeneous interface jump condition on the normal component of the flux. The idea is to do one more splitting $u=u^l+u^n$ (divide and conquer) where $u^l$ solves a linear nonhomogeneous interface elliptic problem and $u^n$ solves a nonlinear homogeneous elliptic problem which depends on $u^l$. This splitting has two advantages. Firstly, it allows to prove that $u\in L^\infty$. Secondly, it makes the actual derivation and practical application of the a posteriori error estimate easier since we can apply already available error estimates for linear interface problems. Finally, we analyze how the error in the computed approximation of $u^l$ affects the accuracy of the approximation to $u^n$ and thus the overall accuracy of $u$.\\
\indent We will finish the talk with numerical examples and demonstration of the effectiveness of the derived error estimates.