Uniform Pointwise Convergence of Finite Difference Schemes Using Grid Equidistribution

Linß, Torsten

doi:10.1007/s006070170037

Cited by 32 publications

(16 citation statements)

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“…Monitor functions are used by many authors (see, e.g., [4][5][6][7]9,12,18,20]) to drive adaptive algorithms that produce layer-resolving meshes in solving differential equations. We first give a general description of this methodology, then use the theoretical results of section 3 to choose monitor functions that are appropriate for (1.1).…”

Section: Monitor Functionsmentioning

confidence: 99%

“…Grid equidistribution has already been applied to convection-diffusion problems by several researchers (e.g., [3,9,12,14]), where the arclength of the computed solution and other monitor functions are considered. While equidistribution of the arc-length is intuitively a reasonable way of controlling an adaptive algorithm, nevertheless for reaction-diffusion problems such as (1.1) it may yield a solution that is much less accurate than that generated by the equidistribution of an alternative measure of the computed solution; compare tables 1 and 2 below.…”

Section: Introductionmentioning

confidence: 99%

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Grid equidistribution for reaction–diffusion problems in one dimension

2005

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The numerical solution of a linear singularly-perturbed reaction-diffusion two-point boundary value problem is considered. The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic argument based on truncation error analysis leads to several suitable monitor functions, but also shows that the standard arc-length monitor function is unsuitable for this problem. Numerical results are provided to demonstrate the effectiveness of our preferred monitor function.

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Section: Monitor Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Grid equidistribution for reaction–diffusion problems in one dimension

2005

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“…A similar theorem was derived in [3, Theorem 3] for linear problem (2.1). This analysis was extended in [17] to the quasi-linear problem (1.1). From the proof of Theorem 1 and Remark 2 in [17] one can quickly get (3.2a).…”

Section: First-order Error Estimatesmentioning

confidence: 99%

“…This will enable us to estimate the maximum norm of the error in terms of difference derivatives of the numerical solution (Theorems 3.2 and 4.2), which is new to our knowledge. Such estimates can be considered as a theoretical framework for using grid equidistribution (which implies that meshes are generated to equidistribute some positive monitor function based on a posteriori information) [6,17,19,20,23,24] and they can also serve as maximum-norm error estimators in adaptive mesh refinement.…”

Section: Introductionmentioning

confidence: 99%

Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem

Kopteva¹

2001

SIAM J. Numer. Anal.

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A singularly perturbed quasi-linear two-point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first-and second-order difference schemes, including upwind schemes. We give first-and second-order maximum norm a posteriori error estimates that are based on difference derivatives of the numerical solution and hold true uniformly in the small parameter. Numerical experiments support the theoretical results.

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“…The first one is the fitted operator method, which reflects the qualitative behavior of the solution; such fitted methods can be found in [3 − 5, 22, 26, 28] and references therein. The second one is the fitted mesh method, which contains finite difference operators on specially designed mesh in the boundary-layer regions, such as Shishkin mesh [5,22,27] and grid equidistribution [18,23,25].…”

Section: Introductionmentioning

confidence: 99%

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

Salama¹,

Alamery²

2016

Turk J Math

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In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. To obtain parameter-uniform convergence, the present method is constructed on a piecewise-uniform Shishkin mesh. The error estimate is discussed and it is shown that the method is uniformly convergent with respect to the singular perturbation parameter. Moreover, a bound of the global error is also derived. The effect of small shifts on the solution behavior is shown by numerical computations. Several numerical examples are presented to support the theoretical results, and to demonstrate the efficiency and the high-order accuracy of the proposed method.

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Uniform Pointwise Convergence of Finite Difference Schemes Using Grid Equidistribution

Cited by 32 publications

References 0 publications

Grid equidistribution for reaction–diffusion problems in one dimension

Grid equidistribution for reaction–diffusion problems in one dimension

Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

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