zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Optimal convergence analysis of mixed finite element methods for fourth-order elliptic and parabolic problems. (English) Zbl 1106.65081
Mixed finite element methods (FEMs) are considered on quasi-uniform rectangular decompositions of a spatial domain. The author applies a specific interpolation operator introduced by {\it V. Girault} and {\it P. A. Raviart} [Finite element methods for Navier-Stokes equations, Berlin: Springer (1986; Zbl 0585.65077)] to analyse the order of convergence in FEMs. First, a linear scalar elliptic equation of fourth order is given including generalised Neumannn boundary conditions. To employ a mixed FEM, a corresponding weak formulation consisting of two equations is constructed. A Galerkin approach yields a linear system for the numerical approximation. A corresponding error estimate is proved using the specific interpolation operator in a Sobolev space. Second, a linear scalar parabolic equation of fourth order, where generalised Neumann conditions in space and initial values in time arise, is discussed in a similar manner. Thereby, the derivative in time is discretised via the backward Euler scheme with equidistant step sizes. Consequently, the problem implies a sequence of weak formulations, which is solved by FEMs. A corresponding error bound is shown using the interpolation operator again. The author remarks that standard interpolation techniques can prove only lower orders of convergence. Finally, a brief numerical example of an elliptic problem is presented, which confirms the predicted order of convergence via refining the mesh widths.

65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65N15Error bounds (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J40Higher order elliptic equations, boundary value problems
35K30Higher order parabolic equations, initial value problems
65N12Stability and convergence of numerical methods (BVP of PDE)
Full Text: DOI