Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 6th Japan--China joint seminar on numerical mathematics, university of Tsukuba, Japan, 5-9 August 2002
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International Journal of Computing Science and Mathematics
International Journal of Computing Science and Mathematics
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Computers & Mathematics with Applications
Optimal adaptive grids of least-squares finite element methods in two spatial dimensions
Journal of Computational and Applied Mathematics
Feedback guided dynamic integral partition: a new convergence case
ICCOMP'06 Proceedings of the 10th WSEAS international conference on Computers
Advances in Computational Mathematics
An adaptive algorithm for the Thomas---Fermi equation
Numerical Algorithms
Robust adaptive computation of a one-dimensional Q-tensor model of nematic liquid crystals
Computers & Mathematics with Applications
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A quasi-linear conservative convection-diffusion two-point boundary value problem is considered. To solve it numerically, an upwind finite difference scheme is applied. The mesh used has a fixed number (N+1) of nodes and is initially uniform, but its nodes are moved adaptively using a simple algorithm of de Boor based on equidistribution of the arc-length of the current computed piecewise linear solution. It is proved for the first time that a mesh exists that equidistributes the arc-length along the polygonal solution curve and that the corresponding computed solution is first-order accurate, uniformly in $\varepsilon$, where $\varepsilon$ is the diffusion coefficient. In the case when the boundary value problem is linear, if N is sufficiently large independently of $\varepsilon$, it is shown that after $O({\rm ln}(1/\varepsilon)/{\rm ln} N)$ iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves first-order accuracy in the $L^\infty[0,1]$ norm uniformly in $\varepsilon$. Numerical experiments are presented that support our theoretical results.