# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. (English) Zbl 1203.65249
Summary: We introduce and analyze a new discontinuous Galerkin method for solving the biharmonic problem ${{\Delta }}^{2}u=f$. The method has two main, distinctive features, namely, it is amenable to an efficient implementation, and it displays new superconvergence properties. Indeed, although the method uses as separate unknowns $u,\nabla u,{\Delta }u$ and $\nabla {\Delta }u$, the only globally coupled degrees of freedom are those of the approximations to $u$ and ${\Delta }u$ on the faces of the elements. This is why we say it can be efficiently implemented. We also prove that, when polynomials of degree at most $k\ge 1$ are used on all the variables, approximations of optimal convergence rates are obtained for both $u$ and $\nabla u$; the approximations to ${\Delta }u$ and $\nabla {\Delta }u$ converge with order $k+1/2$ and $k-1/2$, respectively. Moreover, both the approximation of $u$ as well as its numerical trace superconverge in ${L}^{2}$-like norms, to suitably chosen projections of $u$ with order $k+2$ for $k\ge 2$. This allows the element-by-element construction of another approximation to $u$ converging with order $k+2$ for $k\ge 2$. For $k=0$, we show that the approximation to $u$ converges with order one, up to a logarithmic factor. Numerical experiments are provided which confirm the sharpness of our theoretical estimates.
##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N12 Stability and convergence of numerical methods (BVP of PDE)
COMODI
##### References:
 [1] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985) [2] Babuska, I.: Finite element method for domains with corners. Computing 6, 264–273 (1970) · Zbl 0224.65031 · doi:10.1007/BF02238811 [3] Babuška, I., Osborn, J., Pitkaranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput. 35(152), 1039–1062 (1980) [4] Bacuta, C., Nistor, V., Zikatanov, L.: Improving the rate of convergence of ’high order finite elements’ on polygons and domains with cusps. Numer. Math. 100(2), 165–184 (2005) · Zbl 1116.65119 · doi:10.1007/s00211-005-0588-3 [5] Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977) · doi:10.1090/S0025-5718-1977-0431742-5 [6] Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980) · Zbl 0445.35023 · doi:10.1002/mma.1670020416 [7] Brenner, S.C., Sung, L.-Y.: C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005). MR MR2142191 (2005m:65258) · Zbl 1071.65151 · doi:10.1007/s10915-004-4135-7 [8] Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Armsterdam (1978) [9] Ciarlet, P.G., Raviart, P.-A.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, 1974, pp. 125–145 (1974) [10] Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008) · Zbl 1198.65193 · doi:10.1090/S0025-5718-08-02123-6 [11] Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. (to appear) [12] Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009) · Zbl 1198.65194 · doi:10.1090/S0025-5718-08-02146-7 [13] Comodi, M.I.: The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing. Math. Comput. 52, 17–29 (1989) · doi:10.1090/S0025-5718-1989-0946601-7 [14] Engel, G., Garikipati, K., Hughes, J.T.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4 [15] Falk, R.S.: Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal. 15(3), 556–567 (1978). MR MR0478665 (57 #18142) · Zbl 0383.65059 · doi:10.1137/0715036 [16] From, S.: Potential space estimates for green potentials in convex domains. Proc. Am. Math. Soc. 119, 225–233 (1993) · doi:10.1090/S0002-9939-1993-1156467-3 [17] Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO Modél. Math. Anal. Numér. 23, 103–128 (1989) [18] Glowinski, R., Pironneau, O.: Numerical methods for the first biharmonic equation and the two-dimensional Stokes problem. SIAM Rev. 21(2), 167–212 (1979). MR MR524511 (80e:65101) · Zbl 0427.65073 · doi:10.1137/1021028 [19] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985) [20] Mozolevski, I., Süli, E.: A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3(4), 596–607 (2003) (electronic). MR MR2048235 (2005c:65106) [21] Mozolevski, I., Süli, E., Bösing, P.R.: hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30(3), 465–491 (2007). MR MR2295480 (2008c:65341) · Zbl 1116.65117 · doi:10.1007/s10915-006-9100-1 [22] Raugel, G.: Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C R. Acad. Sci. Paris Sér. A-B 286(18), A791–A794 (1978) (French) [23] Scholz, R.: Approximation von sattelpunkten mit finiten elementen. In: Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975), no. 89, pp. 53–66 (1976) [24] Scholz, R.: A mixed method for 4th order problems using linear finite elements. RAIRO Modél. Math. Anal. Numér. 12(1), 85–90 (1978) [25] Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25, 151–167 (1991) [26] Süli, E., Mozolevski, I.: hp-version interior penalty DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Eng. 196(13–16), 1851–1863 (2007). MR MR2298696 (2008c:65350) · Zbl 1173.65360 · doi:10.1016/j.cma.2006.06.014