Search results for " 65N15"

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Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation

2020

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 computa…

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 equationdifferentiaaliyhtälöt
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Functional a posteriori error estimates for boundary element methods

2019

Functional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results.

osittaisdifferentiaaliyhtälötDiscretizationApplied MathematicsComputationNumerical analysisNumerical Analysis (math.NA)adaptive mesh-refinementFinite element methodMathematics::Numerical Analysisboundary element methodComputational MathematicsComputer Science::Computational Engineering Finance and ScienceCollocation methodMathematikFOS: MathematicsApplied mathematicsA priori and a posterioriMathematics - Numerical Analysisnumeerinen analyysivirheanalyysiGalerkin methodBoundary element methodfunctional a posteriori error estimate65N38 65N15 65N50MathematicsNumerische Mathematik
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