Symmetric marching technique for the poisson equation. I. Dirichlet boundary conditions

1984 ◽  
Vol 15 (2) ◽  
pp. 137-149
Author(s):  
Kamal K. Bharadwaj ◽  
Mohan K. Kadalbajoo ◽  
R. Sankar
Keyword(s):  
Boundary Conditions ◽  
Poisson Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
The Poisson Equation

scholarly journals Homogenization of the Poisson equation in a non-periodically perforated domain

2021 ◽  
pp. 1-27
Author(s):  
Xavier Blanc ◽  
Sylvain Wolf
Keyword(s):  
Poisson Equation ◽  
Dirichlet Boundary ◽  
Rocky Mountain ◽  
Dirichlet Boundary Conditions ◽  
Perforated Domain ◽  
Periodic Case ◽  
Localized Deformation ◽  
The Poisson Equation ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Periodic Setting

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by ε > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit ε → 0) is well understood (see (Rocky Mountain J. Math. 10 (1980) 125–140)). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem, and two-scale expansion.

Estimates for the energy of the solutions to elliptic Dirichlet problems on convex domains

2004 ◽  
Vol 134 (1) ◽  
pp. 89-107 ◽  
Author(s):  
Graziano Crasta
Keyword(s):  
Boundary Conditions ◽  
Isoperimetric Inequality ◽  
Variational Formulation ◽  
Dirichlet Boundary ◽  
Main Tool ◽  
Dirichlet Boundary Conditions ◽  
Convex Domains ◽  
Dirichlet Problems ◽  
The Poisson Equation ◽  
Minkowski Theorem

We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain Ω ⊂ Rn. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of Ω. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.

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