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The G method for heterogeneous anisotropic diffusion on general meshes

Léo Agélas, Daniele A. Di Pietro, Jérôme Droniou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method. A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method. Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity,...

The method of fictitious right-hand sides

Milan Práger (1984)

Aplikace matematiky

The paper deals with the application of a fast algorithm for the solution of finite-difference systems for boundary-value problems on a standard domain (e.g. on a rectangle) to the solution of a boundary-value problem on a domain of general shape contained in the standard domain. A simple iterative procedure is suggested for the determination of fictitious right-hand sides for the system on the standard domain so that its solution is the desired one. Under the assumptions that are usual for matrices...

The output least squares identifiability of the diffusion coefficient from an H 1 –observation in a 2–D elliptic equation

Guy Chavent, Karl Kunisch (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation

Guy Chavent, Karl Kunisch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

The role of the patch test in 2D atomistic-to-continuum coupling methods∗

Christoph Ortner (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction,...

The role of the patch test in 2D atomistic-to-continuum coupling methods∗

Christoph Ortner (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction,...

The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

Miroslaw Lustyk, Julian Janus, Marzenna Pytel-Kudela, Anatoliy Prykarpatsky (2009)

Open Mathematics

The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated...

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Ivana Pultarová (2005)

Applications of Mathematics

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.

Total overlapping Schwarz' preconditioners for elliptic problems

Faker Ben Belgacem, Nabil Gmati, Faten Jelassi (2011)

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

A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math. 336 (2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains. That same method turns to be an efficient tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the behavior of the solution...

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