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)
Mixed finite elements for second order elliptic problems in three variables. (English) Zbl 0631.65107
The authors introduce two families of spaces of mixed finite elements to approximate the solution of Dirichlet problems of the form -div(agradu)=f in G 3 , u=g on G. By substituting q=-agradu the authors use a weak formulation of the equivalent first order system q+agradu=0, divq=f, for q and u to approximate the solution. The first family of spaces are spaces over simplices with flat faces in G; boundary simplices have one curved face lying in the boundary of G. The second family are spaces over cubes (i.e. rectangular parallelepipeds) in G and simplicial boundary elements with one curved face as in the first family. The elements are based on polynomials of total degree j for the vector variable q and total degree j-1 for the scalar variable u. Error estimates in L 2 and H -s are derived. In addition it is shown that the solution of the resulting algebraic equations may be simplified by introducing a Lagrange multiplier to enforce the continuity of the normal components of the approximation of q across interelement boundaries; this method allows a post-processing of the approximation of u which improves the convergence from O(h j ) to O(h j+2 ) for j>1. Finally, an Arrow-Hurwitz-type alternating-direction technique for the solution of the algebraic equations is described briefly.
Reviewer: J. Weisel

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65F10Iterative methods for linear systems
65N15Error bounds (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
References:
[1]Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing, and error estimates. M2AN,19, 7-32 (1985)
[2]Arnold, D.N., Brezzi, F., Douglas, J., Jr.: Peers: a new mixed finite element for plane elasticity. Japan J. Appl. Math.1, 347-367 (1984) · Zbl 0633.73074 · doi:10.1007/BF03167064
[3]Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.47, 217-235 (1985) · Zbl 0599.65072 · doi:10.1007/BF01389710
[4]Brezzi, F., Douglas, J., Jr., Marini, L.D.: Variable degree mixed methods for secod order elliptic problems. Mat. Apl. Comput.4, 19-34 (1985)
[5]Brown, D.C.: Alternating-direction iterative schemes for mixed finite element methods for second order elliptic problems. Thesis, University of Chicago 1982
[6]Douglas, J., Jr.: Alternating direction methods for three space variables. Numer. Math.4, 41-63 (1962) · Zbl 0104.35001 · doi:10.1007/BF01386295
[7]Douglas, J., Jr., Durán, R., Pietra, P.: Formulation of alternating-direction iterative methods for mixed methods in three space. Proceedings of the Simposium Internacional de Analisis Numérico, Madrid, September 1985
[8]Douglas, J., Jr., Durán, R., Pietra, P.: Alternating-direction iteration for mixed finite element methods. Proceedings of the Seventh International Conference on Computing Methods in Applied Sciences and Engineering, Versailles, December 1985
[9]Douglas, J., Jr., Pietra, P.: A description of some alternating-direction iterative techniques for mixed finite element methods. Proceedings. SIAM/SEG/SPE conference, Houston, January 1985
[10]Douglas, J., Jr., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput.44, 39-52 (1985) · doi:10.1090/S0025-5718-1985-0771029-9
[11]Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev space. Math. Comput.34, 441-463 (1980) · doi:10.1090/S0025-5718-1980-0559195-7
[12]Fraeijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. In: Stress analysis (O.C. Zienkiewicz, G. Holister, eds.). New York: John Wiley 1965
[13]Fraeijs de Veubeke, B.X.: Stress function approach. World Congress on the Finite Element Method in Structural Mechanics. Bournemouth, 1965
[14]Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equation. Lecture Notes in Mathematics, Vol. 749. Berlin, Heidelberg, New York: Springer 1979
[15]Nedelec, J.C.: Mixed finite elements inR 3. Numer. Math.35, 315-341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[16]Nedelec, J.C.: A new family of mixed finite elements inR 3 Numer. Math.50, 57-82 (1986) · Zbl 0625.65107 · doi:10.1007/BF01389668
[17]Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36, 9-15 (1970/1971) · Zbl 0229.65079 · doi:10.1007/BF02995904
[18]Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of the finite element method. Lecture Notes in Mathematics, Vol. 606. Berlin, Heidelberg, New York: Springer 1977