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)
A robust finite element method for Darcy-Stokes flow. (English) Zbl 1037.65120
The authors study the uniform convergence of finite element methods (FEM) for a singularly perturbed saddle point problem in two dimensions. If the perturbation parameter is positive, the saddle point problem has the form of the Stokes equations with the operator $I-\varepsilon^2\Delta$ applied to the velocity instead of $-\Delta$. First, the existence and uniqueness of weak solutions is discussed. The results are based on a Babuška-Brezzi (BB) condition and the coercivity in an appropriate energy norm. The critical point is that the appropriate spaces and the energy norm are different in the cases $\varepsilon >0$ and $\varepsilon =0$. The uniform stability of a FEM relies on discrete versions of the BB condition and the coercivity. Numerical tests show that some commonly used FEM which are suited for $\varepsilon > 0$ fail as $\varepsilon \to 0$ and vice versa, FEM which work for $\varepsilon = 0$ diverge if $\varepsilon > 0$. Then, a new non-conforming finite element for triangular grids is constructed for which stability uniformly in $\varepsilon$ is proved. This finite element has nine degrees of freedom in each mesh cell which can be defined by integrals on the edges. Error estimates which show at least a linear rate of convergence independent of $\varepsilon$ are proved for smooth solutions which do not depend on $\varepsilon$. For $\varepsilon$-dependent solutions with boundary layers, it is shown that the rate of convergence in the energy norm is 0.5 uniformly in $\varepsilon$. Both analytical results are supported with numerical examples. Finally, it is shown by a numerical experiment that the new non-conforming FEM works also for a generalized singularly perturbed problem which possesses two parameters. The uniform convergence in the energy norm is proved for smooth solutions which do not depend on these parameters.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
35B25Singular perturbations (PDE)
35Q30Stokes and Navier-Stokes equations
35J25Second order elliptic equations, boundary value problems
76D07Stokes and related (Oseen, etc.) flows
76M10Finite element methods (fluid mechanics)
Full Text: DOI