### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Applied Numerical Mathematics*,

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

**An efficient interpolation algorithm on anisotropic grids for functions with jump discontinuities in 2-D.** / Aguilar, Juan C.; Goodman, Jonathan.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 55, no. 2, pp. 137-153. https://doi.org/10.1016/j.apnum.2005.02.001

}

TY - JOUR

T1 - An efficient interpolation algorithm on anisotropic grids for functions with jump discontinuities in 2-D

AU - Aguilar, Juan C.

AU - Goodman, Jonathan

PY - 2005/10

Y1 - 2005/10

N2 - 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.

AB - 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.

KW - Adaptive mesh refinement

KW - Anisotropic triangulations

KW - Jump discontinuities

KW - Local error estimate

KW - Polynomial interpolation

UR - http://www.scopus.com/inward/record.url?scp=23144444753&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23144444753&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2005.02.001

DO - 10.1016/j.apnum.2005.02.001

M3 - Article

AN - SCOPUS:23144444753

VL - 55

SP - 137

EP - 153

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 2

ER -