Finite volume schemes for fully non-linear elliptic equations in divergence form

Jérôme Droniou

Displaying similar documents to “Finite volume schemes for fully non-linear elliptic equations in divergence form”

Weak solutions for elliptic systems with variable growth in Clifford analysis

Yongqiang Fu, Binlin Zhang (2013)

Czechoslovak Mathematical Journal

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In this paper we consider the following Dirichlet problem for elliptic systems: $\begin{matrix} \bar{D A (x, u (x), D u (x))} = & B (x, u (x), D u (x)), x \in Ω, u (x) = & 0, x \in \partial Ω, \end{matrix}$ where $D$ is a Dirac operator in Euclidean space, $u (x)$ is defined in a bounded Lipschitz domain $Ω$ in $ℝ^{n}$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the...

External approximation of first order variational problems estimates

Cesare Davini, Roberto Paroni (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving $W^{- 1, p}$ norms obtained by Nečas and on the general framework of -convergence theory.

Regularity of weak solutions to certain degenerate elliptic equations

Albo Carlos Cavalheiro (2006)

Commentationes Mathematicae Universitatis Carolinae

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In this article we establish the existence of higher order weak derivatives of weak solutions of Dirichlet problem for a class of degenerate elliptic equations.

Equi-integrability results for 3D-2D dimension reduction problems

Marian Bocea, Irene Fonseca (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $(\nabla_{α} u_{ε} | \frac{1}{ε} \nabla_{3} u_{ε})$ bounded in $L^{p} (Ω; ℝ^{9}), 1 < p < + \infty .$ Here it is shown that, up to a subsequence, $u_{ε}$ may be decomposed as $w_{ε} + z_{ε},$ where $z_{ε}$ carries all the concentration effects, i.e. $\{{|(\nabla_{α} w_{ε} | \frac{1}{ε} \nabla_{3} w_{ε})|}^{p}\}$ is equi-integrable, and $w_{ε}$ captures the oscillatory behavior, i.e. $z_{ε} \to 0$ in measure. In addition, if ${u_{ε}}$ is a recovering sequence then $z_{ε} = z_{ε} (x_{α})$ nearby $\partial Ω .$

Spatial heterogeneity in 3D-2D dimensional reduction

Jean-François Babadjian, Gilles A. Francfort (2005)

ESAIM: Control, Optimisation and Calculus of Variations

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A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as...

-convergence of functionals on divergence-free fields

Nadia Ansini, Adriana Garroni (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of -convergence. We prove that the -limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the -limit is also stable under volume constraint and various type of boundary conditions.