Uniform Quadratic Convergence of Monotone Iterates for Semilinear Singularly Perturbed Elliptic Problems

Boglaev, Igor

doi:10.1007/978-3-642-19665-2_5

Cited by 8 publications

(18 citation statements)

References 4 publications

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“…Similar to (15), using the mean-value theorem, from (9), we get the difference problem for W (1,2) :…”

Section: C75mentioning

confidence: 99%

“…A fruitful method for solving the nonlinear difference scheme is the method of upper and lower solutions and its associated C70 monotone iterations. Boglaev [2] and Pao [9] gave some accelerated monotone iterative methods with a quadratic rate for solving discrete elliptic boundary value problems. However, solving a monotone linear system at each iterative step can be expensive if the number of unknowns is large [2,9].…”

Section: Introductionmentioning

confidence: 99%

“…Boglaev [2] and Pao [9] gave some accelerated monotone iterative methods with a quadratic rate for solving discrete elliptic boundary value problems. However, solving a monotone linear system at each iterative step can be expensive if the number of unknowns is large [2,9]. It seems reasonable to solve the monotone linear system at each iterative step (outer iteration) only approximately with a fixed number of iterative steps (inner iterations).…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this article is to construct inexact monotone methods for solving semilinear elliptic problems, where the outer iterations are based on the monotone method discussed by Boglaev [2] and the inner iterations are based on the Jacobi and Gauss-Seidel iterative methods. The numerical results show that the inexact monotone methods converge with a fast linear rate of convergence (superlinear rate).…”

Section: Introductionmentioning

confidence: 99%

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Inexact monotone methods for solving nonlinear elliptic problems

Boglaev

2015

ANZIAMJ

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We numerically solving semilinear elliptic problems with the method of upper and lower solutions. Inexact monotone iterative methods are constructed, where monotone linear systems are solved by the Jacobi or Gauss-Seidel methods only approximately. The inexact monotone methods combine the quadratic monotone iterative method at outer iterations and the Jacobi or Gauss-Seidel methods at inner iterations, and possess global monotone convergence. Results of numerical experiments are presented.

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“…Similar to (15), using the mean-value theorem, from (9), we get the difference problem for W (1,2) :…”

Section: C75mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Inexact monotone methods for solving nonlinear elliptic problems

Boglaev

2015

ANZIAMJ

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“…Most of the publications on this topic involve monotone iterative schemes whose rate of convergence is of linear rate (cf. [1][2][3][4][5]). Some accelerated monotone iterative schemes for solving discrete elliptic boundary value problems are given in [6,7].…”

Section: Introductionmentioning

confidence: 98%

Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

Boglaev

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tThis paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.

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Iterative analytic approximation to one‐dimensional nonlinear reaction–diffusion equations

Kaushik

Sharma²,

Gupta

et al. 2020

Math Methods in App Sciences

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The paper is concerned with a class of nonlinear reaction-diffusion equations with a dissipating parameter. The problem is singularly perturbed from a mathematical perspective. Solutions of these problems are known to exhibit multiscale character. There are narrow regions in which the solution has a steep gradient.To approximate the multiscale solution, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. We obtain closed-form analytic approximation to nonlinear boundary value problems through iteration. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville-Green transforms. The idea of the paper is to overcome the well-known difficulties associated with numerical methods. Two test examples are taken into account, and rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.

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Uniform Quadratic Convergence of Monotone Iterates for Semilinear Singularly Perturbed Elliptic Problems

Cited by 8 publications

References 4 publications

Inexact monotone methods for solving nonlinear elliptic problems

Inexact monotone methods for solving nonlinear elliptic problems

Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

Iterative analytic approximation to one‐dimensional nonlinear reaction–diffusion equations

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