Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation
This work provides rigorous error estimation for numerical solutions of the Poisson-Boltzmann equation, which is important for biophysics simulations, but the approach is an extension of existing methods to a specific class of problems.
The authors prove mathematical correctness and derive guaranteed, fully computable error bounds for a class of nonlinear elliptic problems arising from the Poisson-Boltzmann equation in biophysics. Numerical tests on 2D and 3D domains confirm the theoretical results.
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 [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] 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.