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A Reduced Basis Enrichment for the eXtended Finite Element Method

E. Chahine, P. Laborde, Y. Renard (2009)

Mathematical Modelling of Natural Phenomena

This paper is devoted to the introduction of a new variant of the extended finite element method (Xfem) for the approximation of elastostatic fracture problems. This variant consists in a reduced basis strategy for the definition of the crack tip enrichment. It is particularly adapted when the asymptotic crack-tip displacement is complex or even unknown. We give a mathematical result of quasi-optimal a priori error estimate and some computational tests including a comparison with some other strategies....

A reduced model for Darcy’s problem in networks of fractures

Luca Formaggia, Alessio Fumagalli, Anna Scotti, Paolo Ruffo (2014)

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

Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures...

A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

Tomás P. Barrios, Gabriel N. Gatica, María González, Norbert Heuer (2006)

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

In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable and efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities...

A residual based A POSTERIORI error estimator for an augmented mixed finite element method in linear elasticity

Tomás P. Barrios, Gabriel N. Gatica, María González, Norbert Heuer (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable and efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities...

A review of two different approaches for superconvergence analysis

Qiding Zhu (1998)

Applications of Mathematics

In 1995, Wahbin presented a method for superconvergence analysis called “Interior symmetric method,” and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different.

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Min Yang (2006)

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

In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in H 1 -norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Min Yang (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in H1-norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

Robert Renka (2013)

Open Mathematics

The velocity-vorticity-pressure formulation of the steady-state incompressible Navier-Stokes equations in two dimensions is cast as a nonlinear least squares problem in which the functional is a weighted sum of squared residuals. A finite element discretization of the functional is minimized by a trust-region method in which the trustregion radius is defined by a Sobolev norm and the trust-region subproblems are solved by a dogleg method. Numerical test results show the method to be effective.

A special finite element method based on component mode synthesis

Ulrich L. Hetmaniuk, Richard B. Lehoucq (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.

A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes

Malte Braack (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instance streamline upwind Petrov-Galerkin (SUPG), are optimal on isotropic meshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. We describe the new method and prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior...

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