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Inner products in covolume and mimetic methods

Kathryn A. Trapp (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This...

Internal finite element approximation in the dual variational method for the biharmonic problem

Ivan Hlaváček, Michal Křížek (1985)

Aplikace matematiky

A conformal finite element method is investigated for a dual variational formulation of the biharmonic problem with mixed boundary conditions on domains with piecewise smooth curved boundary. Thus in the problem of elastic plate the bending moments are calculated directly. For the construction of finite elements a vector potential is used together with C 0 -elements. The convergence of the method is proved and an algorithm described.

Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries

Ivan Hlaváček, Michal Křížek (1984)

Aplikace matematiky

Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an internal approximation of the dual problem for second order elliptic equations is defined. A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation...

Interpolation of non-smooth functions on anisotropic finite element meshes

Thomas Apel (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, several modifications of the quasi-interpolation operator of Scott and Zhang [30] are discussed. The modified operators are defined for non-smooth functions and are suited for application on anisotropic meshes. The anisotropy of the elements is reflected in the local stability and approximation error estimates. As an application, an example is considered where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges.

Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle

Jean Chanzy (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Ce travail a pour objet l’étude d’une méthode de « discrétisation » du Laplacien dans le problème de Poisson à deux dimensions sur un rectangle, avec des conditions aux limites de Dirichlet. Nous approchons l’opérateur Laplacien par une matrice de Toeplitz à blocs, eux-mêmes de Toeplitz, et nous établissons une formule donnant les blocs de l’inverse de cette matrice. Nous donnons ensuite un développement asymptotique de la trace de la matrice inverse, et du déterminant de la matrice de Toeplitz....

Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and in vivo stress incorporation

Joris Bols, Joris Degroote, Bram Trachet, Benedict Verhegghe, Patrick Segers, Jan Vierendeels (2013)

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

In vivo visualization of cardiovascular structures is possible using medical images. However, one has to realize that the resulting 3D geometries correspond to in vivo conditions. This entails an internal stress state to be present in the in vivo measured geometry of e.g. a blood vessel due to the presence of the blood pressure. In order to correct for this in vivo stress, this paper presents an inverse method to restore the original zero-pressure geometry of a structure, and to recover the in vivo...

Inverse source problem in a space fractional diffusion equation from the final overdetermination

Amir Hossein Salehi Shayegan, Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie (2019)

Applications of Mathematics

We consider the problem of determining the unknown source term f = f ( x , t ) in a space fractional diffusion equation from the measured data at the final time u ( x , T ) = ψ ( x ) . In this way, a methodology involving minimization of the cost functional J ( f ) = 0 l ( u ( x , t ; f ) | t = T - ψ ( x ) ) 2 d x is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence...

Inverted finite elements : a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2005)

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

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Inverted finite elements: a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Isogeometric analysis for fluid flow problems

Bastl, Bohumír, Brandner, Marek, Egermaier, Jiří, Michálková, Kristýna, Turnerová, Eva (2015)

Programs and Algorithms of Numerical Mathematics

The article is devoted to the simulation of viscous incompressible fluid flow based on solving the Navier-Stokes equations. As a numerical model we chose isogeometrical approach. Primary goal of using isogemetric analysis is to be always geometrically exact, independently of the discretization, and to avoid a time-consuming generation of meshes of computational domains. For higher Reynolds numbers, we use stabilization techniques SUPG and PSPG. All methods mentioned in the paper are demonstrated...

Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

Pulin Kumar Bhattacharyya, Neela Nataraj (2002)

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

Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field Ψ = ( ψ i j ) 1 i , j 2 and displacement...

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