Difference Methods for Singular Perturbation Problems

Shishkin, G. I.; Shishkina, Lidia P.

doi:10.1201/9780203492413

Cited by 82 publications

(94 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Согласно теории (см., например, [2]), решение дифференциальной модельной задачи (1)-(3) может быть представлено в виде суммы двух функций…”

Section: декомпозиция решения задачи (1)-(3)unclassified

“…Задача Неймана для начально-краевой задачи для сингулярно возмущенного пара-болического уравнения диффузии исследуется подобно задаче для параболического уравнения реакции-диффузии и конвекции-диффузии (см., например, [2]). Справедлива следующая Теорема об априорных оценках.…”

Section: априорные оценкиunclassified

“…возмущенного параболического уравнения, мы изучаем природу ошибки в прибли-женном решении и его регулярной и сингулярной компонентах для стандартной (классической) разностной схемы на равномерных сетках и для специальной раз-ностной схемы на сетке Шишкина кусочно-равномерной по x сетке, сгущающейся в окрестностях пограничных слоев, и равномерной по t (см., например, [2]). …”

unclassified

See 2 more Smart Citations

Numerical Study of an Initial-Boundary Value Neumann Problem for a Singularly Perturbed Parabolic Equation

Shishkina¹

2016

Model. anal. inf. sist.

View full text Add to dashboard Cite

Section: декомпозиция решения задачи (1)-(3)unclassified

Section: априорные оценкиunclassified

unclassified

See 1 more Smart Citation

Numerical Study of an Initial-Boundary Value Neumann Problem for a Singularly Perturbed Parabolic Equation

Shishkina¹

2016

Model. anal. inf. sist.

View full text Add to dashboard Cite

“…Here we define our standard and stabilised finite difference methods, and a Shishkin mesh [13,17,18] that is tailored to (1.1).…”

Section: Standard and Stabilised Numerical Methods Shishkin Meshmentioning

confidence: 99%

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Kopteva

Stynes

2011

Numer. Math.

View full text Add to dashboard Cite

Abstract.A semilinear reaction-diffusion two-point boundary value problem, whose secondorder derivative is multiplied by a small positive parameter ε 2 , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult.To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub-and super-solutions. Convergence results are deduced that depend on the relative sizes of ε and N , where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton's method to compute a discrete solution are discussed.

show abstract

“…(For a discussion of Shishkin meshes see [10,11].) Our finite element method on this mesh will use globally continuous piecewise bilinears.…”

Section: The Shishkin Meshmentioning

confidence: 99%

Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities

2011

View full text Add to dashboard Cite

Abstract. An error analysis of Galerkin and streamline diffusion finite element methods for the numerical solution of a singularly perturbed convectiondiffusion problem is given. The problem domain is the unit square. The solution contains boundary layers and corner singularities. A tensor product Shishkin mesh is used, with piecewise bilinear trial functions. The error bounds are uniform in the singular perturbation parameter. Numerical results supporting the theory are given.

show abstract

Difference Methods for Singular Perturbation Problems

Cited by 82 publications

References 0 publications

Numerical Study of an Initial-Boundary Value Neumann Problem for a Singularly Perturbed Parabolic Equation

Numerical Study of an Initial-Boundary Value Neumann Problem for a Singularly Perturbed Parabolic Equation

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities

Contact Info

Product

Resources

About