Numerical Methods for Grid Equations

Samarskiî, A. A.; Nikolaev, Evgenii S.

doi:10.1007/978-3-0348-9142-4

Cited by 134 publications

(200 citation statements)

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“…Although A is not a diagonal dominant matrix, which is a classical requirement, see [16,17], for the correctness and stability of the Thomas method, we can prove the following statement.…”

Section: M+2mentioning

confidence: 85%

“…For linear FEM solution, the details are given in [13]. For quadratic FEM, using (4.3)-(4.6), just as in the classical theory, [16,17] and applying the full induction method, we can prove the inequalities…”

Section: M+2mentioning

confidence: 99%

“…The proof is achieved taking advantage of the classical approach, see [16,17], to analyze the Thomas method solution by means of the corresponding coefficients. We shall discuss in details the solution, obtained by quadratic FEM.…”

Section: Discrete Mp and Convergencementioning

confidence: 99%

See 2 more Smart Citations

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

Koleva¹

2008

Mathematical Modelling and Analysis

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Abstract. We consider stationary linear problems on non-connected layers with distinct material properties. Well posedness and the maximum principle (MP) for the differential problems are proved. A version of the finite element method (FEM) is used for discretization of the continuous problems. Also, the MP and convergence for the discrete solutions are established. An efficient algorithm for solution of the FEM algebraic equations is proposed. Numerical experiments for linear and nonlinear problems are discussed.

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“…Although A is not a diagonal dominant matrix, which is a classical requirement, see [16,17], for the correctness and stability of the Thomas method, we can prove the following statement.…”

Section: M+2mentioning

confidence: 85%

Section: M+2mentioning

confidence: 99%

See 1 more Smart Citation

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

Koleva¹

2008

Mathematical Modelling and Analysis

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“…In the case if the size of the system (2.1) is too large to solve it by direct methods one should use iterative methods (see [9]). Then the gain of the parametric extrapolation in computational amount is great.…”

Section: 12)mentioning

confidence: 99%

Parametric extrapolation method for degenerate system of linear algebraic equations

Á¹

2012

JCC

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Abstract. In this paper we propose an extrapolation method by a spectrum shift parameter for solving degenerate system of linear algebraic equations. An estimate of the computational work for achieving the normal solution with a given accuracy as well as the advantages of the method are shown theoretically and on examples.T6m t~t. Trong bai nay chung t5i de xuat phiro'ng ph ap ngoai suy theo tham s6 dich chuygn ph5 M giii h~phtro-ng trlnh dai s6 tuyen tinh suy bien, trrrc hro'ng kh6i hrong tinh toan c'an thiet M dat dtro'c nghiem chu.rn tltc v&i d9 chinh xac cho triro'c cling nhir tinh iru vi~t ctia phirong phap du'o'c chi ra bhg ly thuydt va b~ng cac vi du.

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“…After finding EVs, the EF corresponding to s+1 λ m is calculated by inverse iteration method [7,9,25]. Then the obtained EF is used for constructing a new matrix s+1 Λ on the following iterative step.…”

Section: Eigenfunction Problemmentioning

confidence: 99%

Parameter Control of Optical Soliton in One‐dimensional Photonic Crystal

Trofimov

Lysak

Matusevich

et al. 2010

Mathematical Modelling and Analysis

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Abstract. The paper deals with finding of soliton solution for Schrödinger equation with periodic linear and nonlinear properties of medium in 1D case. Such structure is named as photonic crystal. To find soliton solution the corresponding problem for finding of eigenfunctions and eigenvalues is formulated. Iterative process is proposed for solution of this problem. Using the technique of continuation on parameter we investigate a dependence of soliton location on its maximum intensity, on ratio between light frequency and frequency of structure, on ratio between dielectric permittivity of alternating linear and nonlinear layers and on position between centre of initial distribution of eigenfunction and center of considered photonic structure area. The results of this paper confirm the features of soliton self-formation investigated early in our papers [37,38,39,40,41,42], in which one considered a propagation of femtosecond laser pulse through nonlinear layered structure.

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Numerical Methods for Grid Equations

Cited by 134 publications

References 0 publications

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

Finite Element Solution of Boundary Value Problems With Nonlocal Jump Conditions

Parametric extrapolation method for degenerate system of linear algebraic equations

Parameter Control of Optical Soliton in One‐dimensional Photonic Crystal

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