### Abstract

Let k be a sequence of triangulations of a polyhedron n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m . Let u k S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet-Neumann boundary conditions on a polygon.

Original language | English (US) |
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Pages (from-to) | 613-639 |

Number of pages | 27 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 26 |

Issue number | 6 |

DOIs | |

State | Published - Sep 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

*Numerical Functional Analysis and Optimization*,

*26*(6), 613-639. https://doi.org/10.1080/01630560500377295

}

*Numerical Functional Analysis and Optimization*, vol. 26, no. 6, pp. 613-639. https://doi.org/10.1080/01630560500377295

**Improving the rate of convergence of high-order finite elements on polyhedra I : A priori estimates.** / Bacuta, Constantin; Nistor, Victor; Zikatanov, Ludmil.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Improving the rate of convergence of high-order finite elements on polyhedra I

T2 - A priori estimates

AU - Bacuta, Constantin

AU - Nistor, Victor

AU - Zikatanov, Ludmil

PY - 2005/9/1

Y1 - 2005/9/1

N2 - Let k be a sequence of triangulations of a polyhedron n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m . Let u k S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet-Neumann boundary conditions on a polygon.

AB - Let k be a sequence of triangulations of a polyhedron n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m . Let u k S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet-Neumann boundary conditions on a polygon.

UR - http://www.scopus.com/inward/record.url?scp=29044437906&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29044437906&partnerID=8YFLogxK

U2 - 10.1080/01630560500377295

DO - 10.1080/01630560500377295

M3 - Article

AN - SCOPUS:29044437906

VL - 26

SP - 613

EP - 639

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 6

ER -