scholarly journals The Fictitious Domain Method with Sharp Interface for Elasticity Systems with General Jump Embedded Boundary Conditions

2021 ◽  
Vol 13 (1) ◽  
pp. 119-139
Author(s):  
global sci
Keyword(s):  
Boundary Conditions ◽  
Sharp Interface ◽  
Fictitious Domain ◽  
Fictitious Domain Method ◽  
Domain Method ◽  
Embedded 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 MATHEMATICAL MODELING OF THE BOUNDARY CONDITIONS OF THE OCEANALOGY WITH THE HELP PHOTO AREA METHOD

2020 ◽  
Vol 72 (4) ◽  
pp. 73-77
Author(s):  
L.M. Tukenova ◽  
Keyword(s):  
Boundary Conditions ◽  
Ocean Model ◽  
Stability And Convergence ◽  
Navier Stokes ◽  
Fictitious Domain ◽  
Fictitious Domain Method ◽  
Computational Mathematics ◽  
Domain Method ◽  
The Stability ◽  
Differential Relations

Mathematical models of oceanology are equations of the Navier-Stokes type, the construction of stable effective algorithms for their solution is associated with certain difficulties due to the well-known problems of setting boundary conditions, the presence of integro-differential relations, etc. In practice, when solving problems of oceanology, finitedifference methods are widely used, but there are no works in the literature devoted to theoretical studies of the stability and convergence of the algorithms used. In most cases, stability and convergence tests are established through computational experiments. Therefore, we believe that the development and mathematical substantiation of converging methods for solving the system of oceanology equations are urgent problems of computational mathematics. The paper studies variants of the fictitious domain method for a nonlinear ocean model. An existence theorem for the convergence of solutions to approximate models obtained using the fictitious domain method is investigated. An unimprovable estimate of the rate of convergence of the solution of the fictitious domain method is derived.

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