Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation
| Autor: | Johannes Kraus, Sergey Repin, Svetoslav Nakov |
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| Rok vydání: | 2018 |
| Předmět: |
a priori error estimates
Class (set theory) Correctness 010103 numerical & computational mathematics 01 natural sciences Measure (mathematics) guaranteed and efficient a posteriori error bounds FOS: Mathematics Applied mathematics Polygon mesh Mathematics - Numerical Analysis 0101 mathematics error indicators and adaptive mesh refinement Mathematics Numerical Analysis Applied Mathematics Regular polygon Numerical Analysis (math.NA) convergence of finite element approximations Lipschitz continuity 010101 applied mathematics Computational Mathematics Nonlinear system existence and uniqueness of solutions semilinear partial differential equations 65J15 49M29 65N15 65N30 65N50 35J20 Mathematik A priori and a posteriori Poisson-Boltzmann equation differentiaaliyhtälöt |
| DOI: | 10.48550/arxiv.1803.05668 |
| Popis: | 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. Comment: 34 pages, 18 figures |
| Databáze: | OpenAIRE |
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