# 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)
Galerkin domain decomposition procedures for parabolic equations on rectangular domain. (English) Zbl 1187.65109
Solving parabolic equation using an implicit method has an advantage from a stability point of view, but on the other hand in practical engineering problems a large global system of equations must be solved in each time step. To overcome this problem the domain decomposition leads to solve instead the large original problem several smaller ones. This technique enables also to use parallel computations. This paper is a continuation of {\it C. N. Dawson} and {\it T. F. Dupont}’s explicit/implicit non-overlapping domain decomposition procedures [Math. Comput. 58, No. 197, 21--34 (1992; Zbl 0746.65072)] and presents two kinds of new Galerkin non-overlapping domain decomposition methods which differs in a prediction of the fluxes on the boundary $\Gamma$ between subdomains. In the first method (IM-PGDD) the explicit flux calculation is just the mean integral value of the directional derivative of the solution on the boundary $\Gamma$. The second method (EIM-PGDD) is based on an extrapolation of the flux calculation in order to improve the high-order accuracy with respect to the parameter $H$ which represents the size of the boundary region $\Gamma$. Both methods are presented on a rectangular computational domain by decomposing it into two subdomains for solving a parabolic equation with a homogeneous Neumann boundary condition but the extension for other boundary conditions is included too. Then for the IM-PGDD method an error estimation of the type $$\max_{1\leq n \leq N} \| u^n -U^n \|_{L_2(\Omega)} \leq C ( \Delta t + h^{r+1} + H^{\frac52})$$ is derived, where $r$ represents polynomial degree of finite elements. For the scheme (EIM-PGDD) the final result is $$\max_{1\leq n \leq N} \|u^n -U^n \|_{L_2(\Omega)} \leq C ( \Delta t + h^{r+1} + H^{\frac92} ),$$ both under some relation on $\Delta t$ and $H$. Finally, numerical experiments that confirm the accuracy and efficiency of both presented schemes are presented.

##### MSC:
 65M55 Multigrid methods; domain decomposition (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 35K20 Second order parabolic equations, initial boundary value problems
Full Text: