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A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

Gabriel R. Barrenechea, Volker John, Petr Knobloch (2013)

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

An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived...

A localized orthogonal decomposition method for semi-linear elliptic problems

Patrick Henning, Axel Målqvist, Daniel Peterseim (2014)

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

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions...

A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

Carlo Lovadina, David Mora, Rodolfo Rodríguez (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to develop a finite element method which allows computing the buckling coefficients and modes of a non-homogeneous Timoshenko beam. Studying the spectral properties of a non-compact operator, we show that the relevant buckling coefficients correspond to isolated eigenvalues of finite multiplicity. Optimal order error estimates are proved for the eigenfunctions as well as a double order of convergence for the eigenvalues using classical abstract spectral approximation theory...

A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

Carlo Lovadina, David Mora, Rodolfo Rodríguez (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to develop a finite element method which allows computing the buckling coefficients and modes of a non-homogeneous Timoshenko beam. Studying the spectral properties of a non-compact operator, we show that the relevant buckling coefficients correspond to isolated eigenvalues of finite multiplicity. Optimal order error estimates are proved for the eigenfunctions as well as a double order of convergence for the eigenvalues using classical abstract spectral approximation theory...

A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients

Chokri Chniti (2011)

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

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci. 342 (2006) 883–886; CALCOLO 45 (2008) 111–147; J. Sci. Comput. 38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...

A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients

Chokri Chniti (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...

A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations

Christophe Prud'homme, Dimitrios V. Rovas, Karen Veroy, Anthony T. Patera (2002)

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

We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space W N spanned...

A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

Christophe Prud'homme, Dimitrios V. Rovas, Karen Veroy, Anthony T. Patera (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space WN spanned...

A matrix constructive method for the analytic-numerical solution of coupled partial differential systems

Lucas Jódar, Enrique A. Navarro, M. V. Ferrer (1995)

Applications of Mathematics

In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation u t - A u x x - B u = 0 , where B is an arbitrary square complex matrix and A ia s matrix such that the real part of the eigenvalues of the matrix 1 2 ( A + A H ) is positive. Given an admissible error ε and a finite domain G , and analytic-numerical solution whose error is uniformly upper bounded by ε in G , is constructed.

A mechanochemical model of angiogenesis and vasculogenesis

Daphne Manoussaki (2003)

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

Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming...

A mechanochemical model of angiogenesis and vasculogenesis

Daphne Manoussaki (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming...

A mimetic discretization method for linear elasticity

Lourenco Beirão Da Veiga (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A Mimetic Discretization method for the linear elasticity problem in mixed weakly symmetric form is developed. The scheme is shown to converge linearly in the mesh size, independently of the incompressibility parameter λ, provided the discrete scalar product satisfies two given conditions. Finally, a family of algebraic scalar products which respect the above conditions is detailed.

A mixed finite element method for plate bending with a unilateral inner obstacle

Ivan Hlaváček (1994)

Applications of Mathematics

A unilateral problem of an elastic plate above a rigid interior obstacle is solved on the basis of a mixed variational inequality formulation. Using the saddle point theory and the Herrmann-Johnson scheme for a simultaneous computation of deflections and moments, an iterative procedure is proposed, each step of which consists in a linear plate problem. The existence, uniqueness and some convergence analysis is presented.

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