Displaying similar documents to “Computable error bounds with improved applicability conditions for collocation methods.”

Complementarity - the way towards guaranteed error estimates

Vejchodský, Tomáš

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This paper presents a review of the complementary technique with the emphasis on computable and guaranteed upper bounds of the approximation error. For simplicity, the approach is described on a numerical solution of the Poisson problem. We derive the complementary error bounds, prove their fundamental properties, present the method of hypercircle, mention possible generalizations and show a couple of numerical examples.

Two-sided bounds of the discretization error for finite elements

Michal Křížek, Hans-Goerg Roos, Wei Chen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

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We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis. ...

A unified approach to singular problems arising in the membrane theory

Irena Rachůnková, Gernot Pulverer, Ewa B. Weinmüller (2010)

Applications of Mathematics

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We consider the singular boundary value problem ( t n u ' ( t ) ) ' + t n f ( t , u ( t ) ) = 0 , lim t → 0 + t n u ' ( t ) = 0 , a 0 u ( 1 ) + a 1 u ' ( 1 - ) = A , where f ( t , x ) is a given continuous function defined on the set ( 0 , 1 ] × ( 0 , ∞ ) which can have a time singularity at t = 0 and a space singularity at x = 0 . Moreover, n ∈ ℕ , n ≥ 2 , and a 0 , a 1 , A are real constants such that a 0 ∈ ( 0 , ∞ ) , whereas a 1 , A ∈ [ 0 , ∞ ) . The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested...

Two-sided bounds of the discretization error for finite elements

Michal Křížek, Hans-Goerg Roos, Wei Chen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Similarity:

We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical...