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Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces

Sergey I. Repin, Tatiana S. Samrowski, Stéfan A. Sauter (2012)

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

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...

Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces

Sergey I. Repin, Tatiana S. Samrowski, Stéfan A. Sauter (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...

Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces

Sergey I. Repin, Tatiana S. Samrowski, Stéfan A. Sauter (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...

Complementarity - the way towards guaranteed error estimates

Vejchodský, Tomáš (2010)

Programs and Algorithms of Numerical Mathematics

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.

Complexity of the method of averaging

Dalík, Josef (2010)

Programs and Algorithms of Numerical Mathematics

The general method of averaging for the superapproximation of an arbitrary partial derivative of a smooth function in a vertex a of a simplicial triangulation 𝒯 of a bounded polytopic domain in ℜ d for any d ≥ 2 is described and its complexity is analysed.

Contact between elastic bodies. II. Finite element analysis

Jaroslav Haslinger, Ivan Hlaváček (1981)

Aplikace matematiky

The paper deals with the approximation of contact problems of two elastic bodies by finite element method. Using piecewise linear finite elements, some error estimates are derived, assuming that the exact solution is sufficiently smooth. If the solution is not regular, the convergence itself is proven. This analysis is given for two types of contact problems: with a bounded contact zone and with enlarging contact zone.

Contact between elastic bodies. III. Dual finite element analysis

Jaroslav Haslinger, Ivan Hlaváček (1981)

Aplikace matematiky

The problem of a unilateral contact between elastic bodies with an apriori bounded contact zone is formulated in terms of stresses via the principle of complementary energy. Approximations are defined by means of self-equilibriated triangular block-elements and an L 2 -error estimate is proven provided the exact solution is regular enough.

Contact problem of two elastic bodies. II

Vladimír Janovský, Petr Procházka (1980)

Aplikace matematiky

The goal of the paper is the study of the contact problem of two elastic bodies which is applicable to the solution of displacements and stresses of the earth continuum and the tunnel wall. In this first part the variational formulation of the continuous and discrete model is stated. The second part covers the proof of convergence of finite element method to the solution of continuous problem while in the third part some practical applications are illustrated.

Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Robert Eymard, Raphaèle Herbin, Jean-Claude Latché, Bruno Piar (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup...

Convergence analysis of the lowest order weakly penalized adaptive discontinuous Galerkin methods

Thirupathi Gudi, Johnny Guzmán (2014)

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

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides...

Convergence and quasi-optimal complexity of a simple adaptive finite element method

Roland Becker, Shipeng Mao (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We prove convergence and quasi-optimal complexity of an adaptive finite element algorithm on triangular meshes with standard mesh refinement. Our algorithm is based on an adaptive marking strategy. In each iteration, a simple edge estimator is compared to an oscillation term and the marking of cells for refinement is done according to the dominant contribution only. In addition, we introduce an adaptive stopping criterion for iterative solution which compares an estimator for the iteration error...

Convergence and regularization results for optimal control problems with sparsity functional

Gerd Wachsmuth, Daniel Wachsmuth (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical...

Convergence and regularization results for optimal control problems with sparsity functional

Gerd Wachsmuth, Daniel Wachsmuth (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical...

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