Search results for "error estimate"

On a posteriori error bounds for approximations of the generalized Stokes problem generated by the Uzawa algorithm

2012

In this paper, we derive computable a posteriori error bounds for approximations computed by the Uzawa algorithm for the generalized Stokes problem. We show that for each Uzawa iteration both the velocity error and the pressure error are bounded from above by a constant multiplied by the L2-norm of the divergence of the velocity. The derivation of the estimates essentially uses a posteriori estimates of the functional type for the Stokes problem. peerReviewed

Analysis of errors caused by incomplete knowledge of material data in mathematical models of elastic media

2011

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…

Functional a posteriori error equalities for conforming mixed approximations of elliptic problems

2014

Guaranteed error bounds for linear algebra problems and a class of Picard-Lindelöf iteration methods

2012

This study focuses on iteration methods based on the Banach fixed point theorem and a posteriori error estimates of Ostrowski. Their application for systems of linear simultaneous equations, bounded linear operators, as well as integral and differential equations is considered. The study presents a new version of the Picard–Lindelöf method for ordinary differential equations (ODEs) supplied with guaranteed and explicitly computable upper bounds of the approximation error. The estimates derived in the thesis take into account interpolation and integration errors and, therefore, provide objective information on the accuracy of computed approximations.

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.

A posteriori error control for Maxwell and elliptic type problems

2014

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…

Functional Type Error Control for Stabilised Space-Time IgA Approximations to Parabolic Problems

2018

The paper is concerned with reliable space-time IgA schemes for parabolic initial-boundary value problems. We deduce a posteriori error estimates and investigate their applicability to space-time IgA approximations. Since the derivation is based on purely functional arguments, the estimates do not contain mesh dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they imply estimates for discrete norms associated with stabilised space-time IgA approximations. Finally, we illustrate the reliability and efficiency of presented error estimates for the approximate solutions recovered with IgA techniques on a model example. peerReviewed

A posteriori error estimates for variational problems in the theory of viscous fluids

2016

The papers included in the thesis are focused on functional type a posteriori error estimates for the Stokes problem, the Stokes problem with friction type boundary conditions, the Oseen problem, and the anti-plane Bingham problem. In the summary of the thesis we consider only the Oseen problem. The papers present and justify special forms of these estimates which are suitable for the approximations generated by the Uzawa algorithm. The estimates are of two main types. Estimates of the first type use exact solutions obtained on the steps of the Uzawa algorithm. They show how errors encompassed in Uzawa approximations behave and have mainly theoretical meaning. Estimates of the second type o…