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A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Komla Domelevo, Pascal Omnes (2005)

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

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Komla Domelevo, Pascal Omnes (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...

A Fortin operator for two-dimensional Taylor-Hood elements

Richard S. Falk (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.

A free boundary value problem in potential theory

Guido Stampacchia, D. Kinderlehrer (1975)

Annales de l'institut Fourier

This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain Ω and a function u defined in Ω satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of Ω . The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists a unique solution...

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy (2007)

Annales Polonici Mathematici

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to L u = d i v f → in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the L q ( Ω , d μ ) norm of |∇u| is dominated by the L p ( Ω , d v ) norms of d i v f → and | f → | . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

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...

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