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Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

Josef Dalík, Václav Valenta (2013)

Open Mathematics

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite...

Concepts—An object-oriented software package for partial differential equations

Philipp Frauenfelder, Christian Lage (2002)

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

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such...

Concepts—An Object-Oriented Software Package for Partial Differential Equations

Philipp Frauenfelder, Christian Lage (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into ...

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...

Coupling Darcy and Stokes equations for porous media with cracks

Christine Bernardi, Frédéric Hecht, Olivier Pironneau (2005)

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

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

Coupling Darcy and Stokes equations for porous media with cracks

Christine Bernardi, Frédéric Hecht, Olivier Pironneau (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

Discrete Sobolev inequalities and L p error estimates for finite volume solutions of convection diffusion equations

Yves Coudière, Thierry Gallouët, Raphaèle Herbin (2001)

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

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce L p error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

Domain optimization in 3 D -axisymmetric elliptic problems by dual finite element method

Ivan Hlaváček (1990)

Aplikace matematiky

An axisymmetric second order elliptic problem with mixed boundary conditions is considered. The shape of the domain has to be found so as to minimize a cost functional, which is given in terms of the cogradient of the solution. A new dual finite element method is used for approximate solutions. The existence of an optimal domain is proven and a convergence analysis presented.

Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd

M. Aurada, M. Feischl, J. Kemetmüller, M. Page, D. Praetorius (2013)

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

We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p = 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which...

Edge-based a Posteriori Error Estimators for Generating Quasi-optimal Simplicial Meshes

A. Agouzal, K. Lipnikov, Yu. Vassilevsk (2010)

Mathematical Modelling of Natural Phenomena

We present a new method for generating a d-dimensional simplicial mesh that minimizes the Lp-norm, p > 0, of the interpolation error or its gradient. The method uses edge-based error estimates to build a tensor metric. We describe and analyze the basic steps of our method

Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods

Ronald H.W. Hoppe, Barbara Wohlmuth (1995)

Applications of Mathematics

We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...

Embedding and a priori wavelet-adaptivity for Dirichlet problems

Andreas Rieder (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar,...

Enrichissement des interpolations d’éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2002)

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

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d’une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h ), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement p ). Néanmoins, le coût du calcul des fonctions d’interpolation est très...

Enrichissement des interpolations d'éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h), soit en augmentant l'ordre de consistance (analogue à la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctions d'interpolation est...

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