### Abstract

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/N2) when the triangulation contains N triangles and when the error is measured in the L1 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.

Original language | English (US) |
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Pages (from-to) | 137-153 |

Number of pages | 17 |

Journal | Applied Numerical Mathematics |

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2005 |

### Keywords

- Adaptive mesh refinement
- Anisotropic triangulations
- Jump discontinuities
- Local error estimate
- Polynomial interpolation

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Applied Numerical Mathematics*,

*55*(2), 137-153. https://doi.org/10.1016/j.apnum.2005.02.001