An adaptive grid with directional control
Journal of Computational Physics
Multigrid Methods for Anisotropic Edge Refinement
SIAM Journal on Numerical Analysis
Anisotropic adaptive refinement algorithms for finite element methods
Anisotropic adaptive refinement algorithms for finite element methods
Efficient image representation by anisotropic refinement in matching pursuit
ICASSP '01 Proceedings of the Acoustics, Speech, and Signal Processing, 2001. on IEEE International Conference - Volume 03
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In this paper we construct an algorithm that generates a sequence of continuous functions that approximate a given real valued function f of two variables that have jump discontinuities along a closed curve. The algorithm generates a sequence of triangulations of the domain of f. The triangulations include triangles with high aspect ratio along the curve where f has jumps. The sequence of functions generated by the algorithm are obtained by interpolating f on the triangulations using continuous piecewise polynomial functions. The approximation error of this algorithm is O(1/N^2) when the triangulation contains N triangles and when the error is measured in the L^1 norm. Algorithms that adaptively generate triangulations by local regular refinement produce approximation errors of size O(1/N), even if higher-order polynomial interpolation is used.