Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Andreev, V. B.

doi:10.1134/s0012266109070052

Cited by 8 publications

(10 citation statements)

References 6 publications

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“…A decomposition of the solution u into a regular component and singular components (more precisely two boundary layer functions and a corner layer function) is proven and precise pointwise bounds on these components are obtained. In [3] Andreev uses the same techniques to develop a solution decomposition without requiring that f (0, 0), f (0, 1) or f (1, 0) vanish, i.e. compatibility conditions are only posed at the inflow corner.…”

Section: Remark 12mentioning

confidence: 97%

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

Roos

Schopf

2014

Bit Numer Math

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The Il'in scheme is the most famous exponentially fitted finite difference scheme for singularly perturbed boundary value problems. In 1D, Kellogg and Tsan presented a precise error estimate for the scheme from which uniform first order convergence in the discrete maximum norm can be concluded. This estimate is optimal in the sense that one can supply easy examples with smooth data such that the order of uniform convergence of the Il'in scheme is one. In 2D the problem of proving an optimal error estimate remained open. Emel'janov conducted an error analysis giving uniform convergence orders close to one-half. Within the community of singularly perturbed problems it is a legend that this error estimate is sharp. Correcting this mistaken belief it is proven in the present paper that under some conditions the optimal uniform first order convergence of the Il'in scheme in 1D carries over to the two dimensional case. This new result is corroborated by numerical experiments which also shed light on the question in which cases the convergence rate deteriorates.

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Section: Remark 12mentioning

confidence: 97%

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

Roos

Schopf

2014

Bit Numer Math

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“…Lemma 2.1 [17] Assume (2). If w ∈ C 2 ( ) ∩ C 0 (¯ ) such that L ε,μ w| ≤ 0 and w| ∂ ≥ 0, then w|¯ ≥ 0.…”

Section: Preliminary a Priori Bounds On The Solutionmentioning

confidence: 97%

“…In the special case of μ = 0, parameter-explicit bounds on a suitable decomposition of the solution can be obtained assuming no compatibility conditions [1]. In the other special case of μ = 1, some compatibility conditions (but not as stringent as (1e), (1f)) are required to obtain suitable bounds on the derivatives of the components in a decomposition of the solution [2,8,11,16]. In the general case of arbitrary (ε, μ), the compatibility conditions (1e), (1f) suffice to construct our solution decomposition.…”

Section: Introductionmentioning

confidence: 99%

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

O’Riordan

Pickett

2010

Adv Comput Math

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In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.

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“…This leads to a restricted smoothness of derivatives [5] in neighborhoods of corner points, which results in additional difficulties of the analysis of the accuracy of the numerical solution and requires a separate investigation (see [9] and the bibliography therein). We approximate problem (1)-(3) by an implicit four-point difference scheme.…”

Section: Introductionmentioning

confidence: 99%

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

Zhemukhov

2014

Diff Equat

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We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ + N −2 ln 2 N ) ln(j + 1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.

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Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Cited by 8 publications

References 6 publications

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

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