6533b7d6fe1ef96bd126678d
RESEARCH PRODUCT
Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation
Johannes KrausSergey RepinSvetoslav Nakovsubject
a priori error estimatesClass (set theory)Correctness010103 numerical & computational mathematics01 natural sciencesMeasure (mathematics)guaranteed and efficient a posteriori error boundsFOS: MathematicsApplied mathematicsPolygon meshMathematics - Numerical Analysis0101 mathematicserror indicators and adaptive mesh refinementMathematicsNumerical AnalysisApplied MathematicsRegular polygonNumerical Analysis (math.NA)convergence of finite element approximationsLipschitz continuity010101 applied mathematicsComputational MathematicsNonlinear systemexistence and uniqueness of solutionssemilinear partial differential equations65J15 49M29 65N15 65N30 65N50 35J20MathematikA priori and a posterioriPoisson-Boltzmann equationdifferentiaaliyhtälötdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2020-01-01 |