# 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)
Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems. (English) Zbl 1222.65106

The authors combine their procedures from the paper by K. Ma, T. Sun and D. Yang [Numer. Methods Partial Differ. Equations 25, No. 5, 1167–1194 (2009; Zbl 1173.65061)] with the finite difference streamline diffusion method (FDSD method) for time-dependent convection-diffusion problem on a general domain. The two approximation schemes established in Section 2 use an implicit FDSD method in the sub-domains and explicit flux calculations on the inter-domain boundary ${\Gamma }$. These two schemes are called integral mean parallel FDSD scheme (IM-PFDSD scheme) and extrapolation-integral mean parallel FDSD scheme (EIM-PFDSD scheme), respectively.

In Sections 3 and 4, artificial diffusion parameters $\delta$ are given and time step constraints ${\Delta }t\le C{H}^{2}$ are derived. By analysis, the optimal order error estimate is derived in a norm which is stronger than the ${L}^{2}$-norm for IM-PFDSD scheme and EIM-PFDSD scheme, respectively. This error estimate not only includes the optimal ${H}^{1}$-norm error estimate, but also includes the error estimate along the streamline direction $\parallel \beta ·\nabla \left(u-U\right)\parallel$, which cannot be achieved by standard finite element method. For these purposes, new nonstandard elliptic projections are defined and analyzed, which include some terms on ${\Gamma }$. The EIM-PFDSD scheme has higher order accuracy with respect to $H$ than the IM-PFDSD scheme, which means that the EIM-PFDSD scheme can use larger $H$ than the IM-PFDSD scheme so that the time step constraint is weaker than that of the IM-PFDSD scheme and larger time step size can be used. Section 5 presents results of some numerical experiments, which confirm the theoretical results.

The novel procedures of this paper not only hold the advantages of that of the above quoted paper, but also hold the advantages of the FDSD method: good stability properties, higher accuracy and error estimates along the streamline direction. These advantages are not shared by previous known finite element methods.

##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M06 Finite difference methods (IVP of PDE) 65M55 Multigrid methods; domain decomposition (IVP of PDE) 35K20 Second order parabolic equations, initial boundary value problems 65M15 Error bounds (IVP of PDE) 65Y05 Parallel computation (numerical methods)