scholarly journals INTERIOR ERROR ESTIMATE FOR PERIODIC HOMOGENIZATION

2006 ◽  
Vol 04 (01) ◽  
pp. 61-79 ◽  
Author(s):  
GEORGES GRISO
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Problem ◽  
Global Error ◽  
Periodic Homogenization ◽  
Open Set ◽  
Interior Error Estimate

In a previous paper about homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary, we proved that the global error is of order ε1/2. Now, for an open set Ω with sufficiently smooth boundary [Formula: see text] and homogeneous Dirichlet or Neumann limit conditions, we show that in any open set strongly included in Ω the error is of order ε. If the open set Ω ⊂ ℝn is of polygonal (n = 2) or polyhedral (n = 3) boundary, we also give the global and interior error estimates.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

scholarly journals A Posteriori Error Estimates for the Generalized Overlapping Domain Decomposition Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hakima Benlarbi ◽  
Ahmed-Salah Chibi
Keyword(s):  
Error Estimate ◽  
Domain Decomposition ◽  
Error Estimates ◽  
Domain Decomposition Method ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Decomposition Methods ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Overlapping Domain Decomposition

A posteriori error estimates for the generalized overlapping domain decomposition method (GODDM) (i.e., with Robin boundary conditions on the interfaces), for second order boundary value problems, are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residuala posteriorierror analysis to get ana posteriorierror estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

UNIFORM INF–SUP CONDITIONS FOR THE SPECTRAL DISCRETIZATION OF THE STOKES PROBLEM

1999 ◽  
Vol 09 (03) ◽  
pp. 395-414 ◽  
Author(s):  
C. BERNARDI ◽  
Y. MADAY
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Best Approximation ◽  
Stokes Problem ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Discretization Parameter ◽  
The Best Approximation ◽  
Discrete Spaces ◽  
Spectral Discretization

In standard spectral discretizations of the Stokes problem, error estimates on the pressure are slightly less accurate than the best approximation estimates, since the constant of the Babuška–Brezzi inf–sup condition is not bounded independently of the discretization parameter. In this paper, we propose two possible discrete spaces for the pressure: for each of them, we prove a uniform inf–sup condition, which leads in particular to an optimal error estimate on the pressure.

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

scholarly journals Error Estimate of Eigenvalues of Perturbed Higher-Order Discrete Vector Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Haiyan Lv ◽  
Yuming Shi ◽  
Guojing Ren
Keyword(s):  
Error Estimate ◽  
Boundary Value Problems ◽  
Function Space ◽  
Error Estimates ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Admissible Function ◽  
Variational Formula ◽  
Direct Consequence ◽  
Higher Order

This paper is concerned with the eigenvalues of perturbed higher-order discrete vector boundary value problems. A suitable admissible function space is first introduced, a new variational formula of eigenvalues is then established under certain nonsingularity conditions, and error estimates of eigenvalues of problems with small perturbation are finally given by using the variational formula. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the nonsingularity conditions. In addition, two special perturbed cases are discussed.

Sign in / Sign up

Export Citation Format

Share Document