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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.

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
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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)
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