### Abstract

We study the approximation properties of some general finiteelement spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the L^{p}-version of these spaces. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, "skinny" triangles. Our results are then used to obtain L^{2}-error estimates and h^{m}-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.

Original language | English (US) |
---|---|

Pages (from-to) | 2191-2220 |

Number of pages | 30 |

Journal | Mathematics of Computation |

Volume | 84 |

Issue number | 295 |

DOIs | |

State | Published - Jan 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*84*(295), 2191-2220. https://doi.org/10.1090/S0025-5718-2015-02934-2

}

*Mathematics of Computation*, vol. 84, no. 295, pp. 2191-2220. https://doi.org/10.1090/S0025-5718-2015-02934-2

**Graded mesh approximation in weighted sobolev spaces and elliptic equations in 2D.** / Adler, James H.; Nistor, Victor.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Graded mesh approximation in weighted sobolev spaces and elliptic equations in 2D

AU - Adler, James H.

AU - Nistor, Victor

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We study the approximation properties of some general finiteelement spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the Lp-version of these spaces. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, "skinny" triangles. Our results are then used to obtain L2-error estimates and hm-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.

AB - We study the approximation properties of some general finiteelement spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the Lp-version of these spaces. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, "skinny" triangles. Our results are then used to obtain L2-error estimates and hm-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.

UR - http://www.scopus.com/inward/record.url?scp=84930856676&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930856676&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2015-02934-2

DO - 10.1090/S0025-5718-2015-02934-2

M3 - Article

VL - 84

SP - 2191

EP - 2220

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 295

ER -