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

*(English)*Zbl 1451.65197Summary: The paper is concerned with the reliable numerical solution of a class of nonlinear interface problems governed by the Poisson-Boltzmann equation. Arising in electrostatic biomolecular models these problems typically contain measure-type source terms and their solution often exposes drastically different behaviour in different subdomains. The interface conditions reflect the requirement that the potential and its normal derivative must be continuous. In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this question is not trivial and requires additional analysis, which is based on a special splitting of the problem into simpler subproblems whose weak solutions can be defined in standard Sobolev spaces. This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully computable majorant of the difference between the exact solution of the original problem and any energy-type approximation of it. The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life biophysical systems.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35R06 | PDEs with measure |

35J20 | Variational methods for second-order elliptic equations |

35J61 | Semilinear elliptic equations |

49M29 | Numerical methods involving duality |

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

##### Keywords:

electrostatic biomolecular models; nonlinear elliptic interface problems; measure right-hand side; Poisson-Boltzmann equation; reliable computer simulation; adaptive methods; a posteriori error estimates
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\textit{J. Kraus} et al., Comput. Methods Appl. Math. 20, No. 4, 643--676 (2020; Zbl 1451.65197)

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