Search results for "guaranteed error bounds"

showing 3 items of 3 documents

Guaranteed error bounds and local indicators for adaptive solvers using stabilised space-time IgA approximations to parabolic problems

2019

The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and investigate their efficiency. The derivation of error estimates is based on the analysis of the corresponding integral identity and exploits purely functional arguments in the maximal parabolic regularity setting. The estimates are valid for any approximation from the admissible (energy) class and do not contain mesh-dependent constants. They provide computable and fully guaranteed error bounds for the norms arising in stabilised space–time approximations. Furthermore, a p…

osittaisdifferentiaaliyhtälötominaisarvotfunctional error estimatesguaranteed error boundsadaptive space–time schemesnumeerinen analyysivirheanalyysistabilised space–time IgA schemesparabolic initial-value boundary problems
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Rank structured approximation method for quasi--periodic elliptic problems

2016

We consider an iteration method for solving an elliptic type boundary value problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter $\epsilon$) . The method is based on using a simpler operator $\mathcal{A}_0$ (inversion of $\mathcal{A}_0$ is much simpler than inversion of $\mathcal{A}$), which can be viewed as a preconditioner for $\mathcal{A}$. We prove contraction of the iteration method and establish explicit estimates of the contraction factor $q$. Certainly the value of $q$ depends on the difference between $\mathcal{A}$ and $\mathcal…

Discrete mathematicsNumerical AnalysisRank (linear algebra)PreconditionerApplied Mathematicsprecondition methodsguaranteed error boundsOrder (ring theory)65F30 65F50 65N35 65F10tensor type methods010103 numerical & computational mathematicsNumerical Analysis (math.NA)elliptic problems with periodic and quasi-periodic coefficients01 natural sciencesFinite element method010101 applied mathematicsComputational MathematicsOperator (computer programming)Simple (abstract algebra)FOS: MathematicsBoundary value problemTensorMathematics - Numerical Analysis0101 mathematicsMathematics
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Guaranteed error bounds for a class of Picard-Lindelöf iteration methods

2013

We present a new version of the Picard-Lindelof method for ordinary dif- ¨ ferential equations (ODEs) supplied with guaranteed and explicitly computable upper bounds of an approximation error. The upper bounds are based on the Ostrowski estimates and the Banach fixed point theorem for contractive operators. The estimates derived in the paper take into account interpolation and integration errors and, therefore, provide objective information on the accuracy of computed approximations. peerReviewed

Discrete mathematicsClass (set theory)Banach fixed-point theoremOdeguaranteed error boundsPicard-Lindelöf methodsinversio-ongelmatelliptic boundary value problemsPower iterationApproximation errorOrdinary differential equationComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATIONApplied mathematicsa posteriori estimatesObjective informationInterpolationMathematics
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