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 new discontinuous Galerkin method for Kirchhoff plates. (English) Zbl 1231.74416
Summary: A general framework of constructing C 0 discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas in P. Castillo et al. [SIAM J. Numer. Anal. 38, No. 5, 1676–1706 (2000; Zbl 0987.65111)] and B. Cockburn [ZAMM, Z. Angew. Math. Mech. 83, No. 11, 731–754 (2003; Zbl 1036.65079)]. The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDG method, called the LCDG method, is particularly interesting in our study. It can be viewed as an extension to fourth-order problems of the LDG method studied in P. Castillo (2000; Zbl 0987.65111) and C. Cockburn (2003; Zbl 1036.65079). For this method, optimal order error estimates in certain broken energy norm and H 1 -norm are established. Some numerical results are reported, confirming the theoretical convergence orders.
MSC:
74S05Finite element methods in solid mechanics
74K20Plates (solid mechanics)
References:
[1]A. Adini, R.W. Clough, Analysis of Plate Bending by the Finite Element Method, NSF Report G. 7337, 1961.
[2]Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. anal. 39, 1749-1779 (2002) · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[3]Babus&breve, I.; Ka; Zlámal, M.: Nonconforming elements in the finite element method with penalty, SIAM J. Numer. anal. 10, 863-875 (1973)
[4]Baker, G. A.: Finite element methods for elliptic equations using nonconforming elements, Math. comput. 31, 44-59 (1977) · Zbl 0364.65085 · doi:10.2307/2005779
[5]G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz, Triangular elements in bending — conforming and non-conforming solutions, in: Proceedings of the Conference Matrix Methods in Structural Mechanics, Air Force Inst. Tech., Wright-Patterson AF Base, OH, 1965.
[6]Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods, (2008)
[7]Brenner, S. C.; Sung, L.: C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. sci. Comput. 22/23, 83-118 (2005) · Zbl 1071.65151 · doi:10.1007/s10915-004-4135-7
[8]Brezzi, F.; Cockburn, B.; Marini, L. D.; Süli, E.: Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput. methods appl. Mech. engrg. 195, 3293-3310 (2006) · Zbl 1125.65102 · doi:10.1016/j.cma.2005.06.015
[9]Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods, (1991) · Zbl 0788.73002
[10]Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. anal. 18, 1676-1706 (2000) · Zbl 0987.65111 · doi:10.1137/S0036142900371003
[11]Ciarlet, P. G.: The finite element method for elliptic problems, (1978)
[12]Cockburn, B.: Discontinuous Galerkin methods, ZAMM Z. Angew. math. Mech. 83, 731-754 (2003) · Zbl 1036.65079 · doi:10.1002/zamm.200310088
[13], Lecture notes in computational science in engineering 11 (2000)
[14]Cockburn, B.; Shu, C. -W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. anal. 35, 2440-2463 (1998) · Zbl 0927.65118 · doi:10.1137/S0036142997316712
[15]Dauge, M.: Elliptic boundary value problems on corner domains, Lecture notes in mathematics 1341 (1988) · Zbl 0668.35001
[16]Engel, G.; Garikipati, K.; Hughes, T. J. R.; Larson, M.; Mazzei, L.; Taylor, R.: 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. engrg. 191, 3669-3750 (2002) · Zbl 1086.74038 · doi:10.1016/S0045-7825(02)00286-4
[17]Ern, A.; Guermond, J. -L.: Discontinuous Galerkin methods for Friedrichs systems. Part I. General theory, SIAM J. Numer. anal. 44, 753-778 (2006) · Zbl 1122.65111 · doi:10.1137/050624133
[18]Ern, A.; Guermond, J. -L.: Discontinuous Galerkin methods for Friedrichs systems. Part II. Second-order elliptic pdes, SIAM J. Numer. anal. 44, 2363-2388 (2006) · Zbl 1133.65098 · doi:10.1137/05063831X
[19]Feng, K.; Shi, Z.: Mathematical theory of elastic structures, (1995)
[20]De Veubeke, B. Fraeins: Variational principles and the patch test, Int. J. Numer. methods engrg. 8, 783-801 (1974)
[21]Grisvard, P. G.: Singularities in boundary value problems, (1992) · Zbl 0766.35001
[22]Hughes, T. J. R.: The finite element method: linear static and dynamic finite element analysis, (2000)
[23]Morley, L. S. D.: The triangular equilibrium element in the solution of plate bending problems, Aero. quart. 19, 149-169 (1968)
[24]Mozolevski, I.; Bösing, P. R.: Sharp expressions for the stabilization parameters in symmetric interior-penalty discontinuous Galerkin finite element approximations of fourth-order elliptic problems, Comput. methods appl. Math. 7, 365-375 (2007) · Zbl 1136.65098 · doi:http://www.cmam.info/issues/?Vol=7&Num=4&ItID=181
[25]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, 596-607 (2003) · Zbl 1048.65100 · doi:http://www.cmam.info/issues/?Vol=3&Num=4&ItID=85&PHPSESSID=c89f68f4751ec0fa5c33d902db73486a
[26]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, 465-491 (2007) · Zbl 1116.65117 · doi:10.1007/s10915-006-9100-1
[27]Reddy, J. N.: Theory and analysis of elastic plates and shells, (2007)
[28]Süli, E.; Mozolevski, I.: Hp-version interior penalty dgfems for the biharmonic equation, Comput. methods appl. Mech. engrg. 196, 1851-1863 (2007) · Zbl 1173.65360 · doi:10.1016/j.cma.2006.06.014
[29]Wells, G. N.; Dung, N. T.: A C0 discontinuous Galerkin formulation for Kirchhoff plates, Comput. methods appl. Mech. engrg. 196, 3370-3380 (2007) · Zbl 1173.74447 · doi:10.1016/j.cma.2007.03.008