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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)
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