A mixed method for 4th order problems using linear finite elements

Scholz, Reinhard

doi:10.1051/m2an/1978120100851

Cited by 95 publications

(94 citation statements)

References 6 publications

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“…The use of mixed and hybrid methods for the approximate solution of source problems has received considerable attention. We mention the works of Herrmann [20], [21], Glowinski [19], Miyoshi [29], Oden [33], Johnson [25], Mercier [27], Ciarlet-Raviart [10], Brezzi [6], [7], Scholz [42], [43], BrezziRaviart [8], Oden-Reddy [35], Raviart-Thomas [40], [41], Falk [14], Falk-Osborn [15], Rannacher [39], and Babuska-Osborn-Pitkäranta [5].…”

mentioning

confidence: 99%

Eigenvalue Approximation by Mixed and Hybrid Methods

Mercier¹,

Osborn²,

Rappaz³

et al. 1981

Mathematics of Computation

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Abstract. Rate of convergence estimates are derived for the approximation of eigenvalues and eigenvectors by mixed and hybrid methods. Several closely related abstract results on spectral approximation are proved. These results are then applied to a variety of finite element methods of mixed and hybrid type: a mixed method for 2nd order problems, mixed methods for 4th order problems, a hybrid method for 2nd order problems, and two mixed methods for the Stokes eigenvalue problem.

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mentioning

confidence: 99%

Eigenvalue Approximation by Mixed and Hybrid Methods

Mercier¹,

Osborn²,

Rappaz³

et al. 1981

Mathematics of Computation

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“…Therefore, piece-wise continuous approximations are used for both variables. The error analysis of the C-R method can be found in [25,4,39,38]. Optimal convergence rates for the approximation to u are obtained however sub-optimal convergence rates are proved for the approximation to u.…”

Section: Introductionmentioning

confidence: 99%

“…Optimal convergence rates for the approximation to u are obtained however sub-optimal convergence rates are proved for the approximation to u. More precisely, assuming certain L ∞ smoothness assumptions on the derivatives of the solutions, the approximation to u convergences with rate h k−1/2 if piecewise polynomials of degree k are used; see [39]. It is well known that this result is in fact sharp.…”

Section: Introductionmentioning

confidence: 99%

A Mixed Method for the Biharmonic Problem Based On a System of First-Order Equations

Behrens¹,

Guzmán²

2011

SIAM J. Numer. Anal.

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Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is re-written as a system of four first-order equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation. For k ≥ 1 a projection of the primal variable error superconverges with order k + 3 while the error itself converges with order k + 1 only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k + 3. We provide numerical experiments that validate our theoretical results.

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“…For example, (1.2) is needed in order to establish the numerically observed L 2 error estimate for bi-harmonic problems (cf. [37]). By taking χ = 0 in (1.2), we obtain the following stability result…”

Section: Introductionmentioning

confidence: 99%

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

et al. 2009

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Abstract. A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green's function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in W 1 ∞ . In contrast to previously known results, W 2 p regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green's function estimates allow us to obtain localized error estimates at a point.

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A mixed method for 4th order problems using linear finite elements

Cited by 95 publications

References 6 publications

Eigenvalue Approximation by Mixed and Hybrid Methods

Eigenvalue Approximation by Mixed and Hybrid Methods

A Mixed Method for the Biharmonic Problem Based On a System of First-Order Equations

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

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