##
**Finite elements in H(curl) with applications to elasticity.**
*(English)*
Zbl 0712.73088

(From the author’s introduction.) In this paper we introduce a family of finite elements in the space H(curl, \(\Omega)=\{u\in L^ 2+L^ 2:\) curl \(u\in L^ 2\}\), where \(\Omega \subset {\mathbb{R}}^ 2\) is a convex polygonal domain. These elements are constructed following the ideas of F. Brezzi, J. Douglas jun., and L. D. Marini [e.g.: Numer. Math. 47, 217-235 (1985; Zbl 0599.65072), Mat. Apl. Comput. 4, 19- 34 (1985; Zbl 0592.65073)], who introduced new elements in H (div,\(\Omega\)) for second order elliptic problems. The elements are used to approximate the solution of the classical linear elasticity equations \(-\mu \Delta u+(\lambda +\mu)\nabla (div u)=f,\) when the normal component of the force acting on the boundary is known and a nonslipping condition on the boundary is assumed. They can also be used to approximate the solution of the Stokes equations describing the flow of a viscous fluid. The formulation used here was proposed by J.-C. Nedelec [e.g.: Numer. Math. 50, 57-81 (1986; Zbl 0625.65107)]. The error analysis is based on the abstract theory on the approximation of saddle point problems introduced by F. Brezzi [e.g.: Revue Franc. Automat. Inform. Rech. Operat. 8, R-2, 129-151 (1974; Zbl 0338.90047)] and developed further by M. Fortin and A. Fortin [e.g.: Commun. Appl. Numer. Methods 1, 205-208 (1985; Zbl 0592.76040)], and R. S. Falk and J. E. Osborn [e.g.: RAIRO Anal. Numer. 14, 249-277 (1980; Zbl 0467.65062)]. However, in this theory the error in the scalar p and the vector field u are bounded together with the error for curl u.

In this paper we improve the results of Nedelec in the two-dimensional case, obtaining error estimates in \(L^ 2\) for the approximation of u and p independent of the approximation to the term curl u. These results allow us to use polynomials of degree k to obtain convergence of order \(k+1\), that is optimal, because we do not need to approximate curl u with the same order.

In this paper we improve the results of Nedelec in the two-dimensional case, obtaining error estimates in \(L^ 2\) for the approximation of u and p independent of the approximation to the term curl u. These results allow us to use polynomials of degree k to obtain convergence of order \(k+1\), that is optimal, because we do not need to approximate curl u with the same order.

Reviewer: Michael Sever (Jerusalem)

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

### Citations:

Zbl 0599.65072; Zbl 0592.65073; Zbl 0625.65107; Zbl 0338.90047; Zbl 0592.76040; Zbl 0467.65062
PDFBibTeX
XMLCite

\textit{R. G. Durán}, Numer. Methods Partial Differ. Equations 6, No. 2, 167--175 (1990; Zbl 0712.73088)

Full Text:
DOI

### References:

[1] | Brezzi, R.A.I.R.O. Anal. Numer. 2 pp 129– (1974) |

[2] | The Finite Element Method for Elliptic Problems, North Holland, Amsterdam & New York, 1978. |

[3] | Brezzi, Numer. Math. 47 pp 217– (1985) |

[4] | Dupont, Math. Comp. 34 pp 441– (1980) |

[5] | Fortin, R.A.I.R.O. Anal. Numer. 11 pp 341– (1977) |

[6] | and , Error Estimates for Mixed Methods, Tech. Report, Math. Research Center, Univ. of Wisconsin, Madison, 1979. |

[7] | Nedelec, Numer. Math. 35 pp 315– (1980) |

[8] | Nedelec, Numer. Math. 50 pp 57– (1986) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.