Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains

1997 ◽  
Vol 127 (1) ◽  
pp. 77-125 ◽  
Author(s):  
Benqi Guo ◽  
Ivo Babuška
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Elliptic Problems ◽  
Regularity Theory ◽  
Weighted Spaces ◽  
Regularity Of Solutions ◽  
The Finite Element Method ◽  
Normed Spaces ◽  
Nonsmooth Domains

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3. Part II: Regularity in neighbourhoods of edges

1997 ◽  
Vol 127 (3) ◽  
pp. 517-545 ◽  
Author(s):  
Benqi Guo ◽  
Ivo Babuška
Keyword(s):  
Finite Elements ◽  
Weighted Sobolev Spaces ◽  
Elliptic Problems ◽  
Regularity Of Solutions ◽  
Normed Spaces ◽  
Qualitative And Quantitative ◽  
The Poisson Equation ◽  
Nonsmooth Domains ◽  
Polyhedral Domain ◽  
Derivatives Of

This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.

The Convergence of the H-P Version of the Finite Element Method with Quasi-Uniform Meshes for Three Dimensional Poisson Problems with Edge Singularity

2014 ◽  
Vol 644-650 ◽  
pp. 1551-1555
Author(s):  
Jian Ming Zhang ◽  
Yong He
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Smooth Functions ◽  
Edge Singularity ◽  
Theoretical Predictions ◽  
Theoretical Results ◽  
Element Method

This paper is concerned with the convergence of the h-p version of the finite element method for three dimensional Poisson problems with edge singularity on quasi-uniform meshes. First, we present the theoretical results for the convergence of the h-p version of the finite element method with quasi-uniform meshes for elliptic problems on polyhedral domains on smooth functions in the framework of Jacobi-weighted Sobolev spaces. Second, we investigate and analyze numerical results for three dimensional Poission problems with edge singularity. Finally, we verified the theoretical predictions by the numerical computation.

Embeddings of weighted Sobolev spaces and degenerate elliptic problems

2017 ◽  
Vol 60 (8) ◽  
pp. 1399-1418 ◽  
Author(s):  
ZongMing Guo ◽  
LinFeng Mei ◽  
FangShu Wan ◽  
XiaoHong Guan
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Elliptic Problems ◽  
Degenerate Elliptic

ON THE OSEEN PROBLEM IN THREE-DIMENSIONAL EXTERIOR DOMAINS

2006 ◽  
Vol 04 (02) ◽  
pp. 133-162 ◽  
Author(s):  
CHÉRIF AMROUCHE ◽  
ULRICH RAZAFISON
Keyword(s):  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Exterior Domains ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Oseen Problem

In this paper, we prove existence and uniqueness results for the Oseen problem in exterior domains of ℝ3. To prescribe the growth or decay of functions at infinity, we set the problem in weighted Sobolev spaces. The analysis relies on a Lp-theory for any real p such that 1 < p < ∞.

The Wente Problem Associated to the Modified Helmholtz Operator on Weighted Sobolev Spaces

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
Ines Ben Omrane ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Sobolev Spaces ◽  
Positive Constant ◽  
Weighted Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Weighted Version ◽  
Helmholtz Operator

AbstractIn this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator -Δ + αI, where α is a positive constant.

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

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