Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method

2010 ◽  
pp. 375-382
Author(s):  
Roland Glowinski ◽  
Qiaolin He
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Least Squares ◽  
Elliptic Problems ◽  
Fictitious Domain ◽  
Robin Boundary Conditions ◽  
Fictitious Domain Method ◽  
Domain Method ◽  
Robin Boundary

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

2011 ◽  
Vol 9 (3) ◽  
pp. 587-606 ◽  
Author(s):  
Roland Glowinski ◽  
Qiaolin He
Keyword(s):  
Boundary Conditions ◽  
Least Squares ◽  
Gradient Algorithm ◽  
Fictitious Domain ◽  
Optimal Order ◽  
Robin Boundary Conditions ◽  
Fictitious Domain Method ◽  
Optimal Order Of Convergence ◽  
Domain Method ◽  
Robin Boundary

AbstractIn this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

scholarly journals Handling Neumann and Robin boundary conditions in a fictitious domain volume penalization framework

2022 ◽  
Vol 448 ◽  
pp. 110726
Author(s):  
Ramakrishnan Thirumalaisamy ◽  
Neelesh A. Patankar ◽  
Amneet Pal Singh Bhalla
Keyword(s):  
Boundary Conditions ◽  
Fictitious Domain ◽  
Robin Boundary Conditions ◽  
Robin Boundary

APPLICATION OF THE FICTITIOUS DOMAIN METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 74 (2) ◽  
pp. 5-12
Author(s):  
S. Kasenov ◽  
◽  
A.N. Temirbekov ◽  
A.ZH. Satybaev ◽  
L.N. Temirbekova ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Original Problem ◽  
Auxiliary Problem ◽  
Nonlinear Ordinary Differential Equation ◽  
Generalized Solutions ◽  
Fictitious Domain ◽  
Fictitious Domain Method ◽  
Domain Method

The article shows the ways of applying the method of fictitious domains in solving problems for ordinary differential equations. In the introduction, a small review of the literature on this method, as well as methods for the numerical solution of these problems, is made. The problem statement for the method of fictitious domains for ordinary differential equations is considered. Further, the inequality of estimates was shown. The solution of the auxiliary problem approximates the solution of the original problem with a certain accuracy. The inequality of estimates is obtained in the class of generalized solutions. For the purpose of visual application of the fictitious domain method in problems, a boundary value problem for a one-dimensional nonlinear ordinary differential equation is considered. The problem was written in the form of a difference scheme and led to a solution using the sweep method. In the numerical solution of the problem, numerical calculations were carried out for various values of the parameter included in the auxiliary problem, based on the method of fictitious domains. The numbers of iterations, execution time, and graphs of these calculations are presented and analyzed.

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