Stability of finite-difference schemes for elliptic equations with respect to Dirichlet boundary conditions

“…Note that another proof for the case where A 0 corresponds to the Neumann problem was given in [1]. Thus, let i/2 be determined from (8) where B is defined in (10) and u\ be found by the formula Set il/1 = u"-Ui, i= 1,2, ψ η = η η -w, where u = (u[,u*)\ is the exact solution to system (4).…”

Section: ^ 12) ~" X N-l -~·mentioning

confidence: 99%

On using the bordering method for solving systems of mesh equations

Matsokin¹,

Nepomnyaschikh²

1989

Russian Journal of Numerical Analysis and Mathematical Modelling

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The paper treats a version of the block bordering method for solving elliptic equations with piecewise-constant coefficients in domains composed of rectangles. An iterative process whose convergence is independent of the mesh step size is constructed, and the number of arithmetic operations required for realization of the method with a given accuracy is estimated.For the sake of simplicity, let us consider the following problem: ueW 2 (fl) Ln(u,9) where /eL 2 (Q), and Ω is a rectangle composed of three squares Ω ί? Ω = Uf el £ :Ω={(χ,3θ| 00 in Ω,, i=l,2,3.Problem (1) is replaced by an approximate problem using variational difference schemes with summatory approximation [7]. Using the results obtained in [6] an iterative method is constructed whose convergence rale is independent of a coefficient ρ and the mesh step size. The results of this paper are extended to the case of more general problems. Exploiting the algorithm with the capacitance matrix similar results for the L-shaped domains were obtained in [2].Let us construct in a domain Ω a regular triangulation Ω Λ = (J? = Χ Ω^ where h = l/n is the step size of the square mesh.Denote by H h the set of real-valued continuous functions linear on the triangles of the triangulation and vanishing on 3Ω, and by H h (Sli) denote the set of real-valued continuous piecewise-linear functions defined on Ω?.Replace problem (1) with an approximate problem u h eH H : L h *(u\ φ") = f p(Vu h · V

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“…2 where Γ is a symmetric positive semi-definite matrix and ker S = ker R = ker Γ. Henceforth, we need one more statement which follows from [2] (see also [4,9,23] for all veH 1^^ satisfying the condition J rh t;ds = 0, and there exists a constant c 3 such that…”

Section: Model Problemsmentioning

confidence: 99%

On some versions of the domain decomposition method

Agapov¹,

Kuznetsov²

1988

Russian Journal of Numerical Analysis and Mathematical Modelling

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The paper suggests a new approach to construction of preconditioned and convergence rate estimates for iterative methods to solve simultaneous linear algebraic equations arising in finite element approximations of elliptic problems. The approach is based on the decomposition of the original mesh domain into superelements. It is proved that for many important practical problems the convergence rate estimates of the methods considered do not depend on the mesh, and also on coefficients and boundary conditions of the original differential problem. Jfl[_iJ=l OXiOXj J J Flwhere Γ χ = 3Ω/Γ 0 , and the coefficients a ij9 b and σ are piece wise-smooth bounded functions and, moreover, b and σ are nonnegative. Consider for a given function /eL 2 = ί. 2 (Ω) the following variational problem: find ueV such that φ, !>) = (/, t?) VueK (0.2)Here, (·,·) stands for the standard scalar product in L 2 . In what follows, we consider independently two cases. First, it is the Neumann problem for Γ 0 = 0, b = 0 in Ω, σ = 0 on Γ! and the function / satisfies the condition | Ω /αΩ = 0. In this case, the solution to problem (0.2) does exist and is uniquely defined, for example, under the assumption Brought to you by | University of Arizona Authenticated Download Date | 7/8/15 10:09 AM

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Stability of finite-difference schemes for elliptic equations with respect to Dirichlet boundary conditions

Cited by 8 publications

References 0 publications

A capacitance matrix method for Dirichlet problem on polygon region

A capacitance matrix method for Dirichlet problem on polygon region

On using the bordering method for solving systems of mesh equations

On some versions of the domain decomposition method

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