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Schur Complement Domain Decomposition in Conjunction with Algebraic Multigrid Methods Based on Generic Approximate Inverses

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Schur Complement Domain Decomposition in Conjunction with Algebraic Multigrid Methods Based on Generic Approximate Inverses Proceedings of the 2013 Federated Conference on Computer Science and Information Systems pp. 487–493 Schur Complement Domain Decomposition in conjunction with Algebraic Multigrid methods based on Generic Approximate Inverses P.I. Matskanidis G.A. Gravvanis Department of Electrical and Computer Engineering, Department of Electrical and Computer Engineering, School of Engineering, Democritus University of School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, Thrace, University Campus, Kimmeria, GR 67100 Xanthi, Greece GR 67100 Xanthi, Greece Email: [email protected] Email: [email protected] In non-overlapping DD methods, also referred to as itera- Abstract—For decades, Domain Decomposition (DD) tech- tive substructuring methods, the subdomains intersect only niques have been used for the numerical solution of boundary on their interface. Non-overlapping methods can further- value problems. In recent years, the Algebraic Multigrid more be distinguished in primal and dual methods. Primal (AMG) method has also seen significant rise in popularity as well as rapid evolution. In this article, a Domain Decomposition methods, such as BDDC [7], enforce the continuity of the method is presented, based on the Schur complement system solution across the subdomain interface by representing the and an AMG solver, using generic approximate banded in- value of the solution on all neighbouring subdomains by the verses based on incomplete LU factorization. Finally, the appli- same unknown. In dual methods, such as FETI [10], the cability and effectiveness of the proposed method on character- continuity is further enforced by the use of Lagrange multi- istic two dimensional boundary value problems is demonstrated pliers. Hybrid methods, such as FETI-DP [8],[9],[16], have and numerical results on the convergence behavior are given. also been introduced. In the past decades, the development of multigrid meth- I. INTRODUCTION ods has also been critical for the numerical solution of OMAIN decomposition includes a significant range of PDEs. An essential component of the multigrid method is Dcomputing techniques for the numerical solution of the relaxation scheme, which efficiently reduces high fre- Partial Differential Equations (PDEs). Domain decomposi- quency components of the error, however is inefficient at re- tion techniques are based on splitting the computational do- ducing the lower frequency ones [2]. Transferring the prob- main into smaller subdomains, with or without overlap. The lem to a coarser grid, those low frequency errors become problems in the subdomains are independent, thus rendering more oscillatory and can be effectively damped by a station- domain decomposition methods suitable for parallelization. ary iterative method. Recursive application of this process Domain decomposition techniques can themselves be used produces the multigrid methods [18]. as stationary iterative schemes, as well as preconditioners in The algebraic multigrid algorithm (AMG) was first intro- order to accelerate the convergence of other iterative meth- duced over twenty years ago [1],[20]. Unlike geometric ods, specifically Krylov subspace methods [14]. multigrid, the algebraic multigrid method does not require Domain decomposition methods can be split into two cat- knowledge of the geometry of the problem to define its com- egories: overlapping and non-overlapping methods. In over- ponents. This is the reason AMG is perfectly suited for un- lapping DD methods, often referred to as Schwarz methods structured grids, both in two and three dimensions, and com- due to Schwarz’s work in 1870 [22], the subdomains overlap plicated domains. Specifically, by considering a linear sys- by more than the interface. The overlapping methods have a tem Au=f, the AMG method requires only the coefficient simple algorithmic structure, since there is no need to solve matrix A and the right-hand side vector f. As a result, AMG special interface problems between neighbouring subdo- solvers can easily be integrated into existing problem solv- mains. This feature differentiates overlapping from ing environments as standard solvers or precondition- non-overlapping DD methods [5],[23]. Overlapping meth- ers [18]. ods operate by an iterative procedure, where the PDE is re- Consider a linear system Au=f, where A= a ,i,j∈[ 1,n] peatedly solved within every subdomain. For each subdo- ( i,j) main, the artificial internal boundary condition is provided is an (n×n) coefficient matrix. A “grid” is a set of indices of by its neighbouring subdomains. The convergence of the so- the variables, thus the first grid is Ω={1,2, ... ,n } . Since lution on these internal boundaries ensures the convergence AMG is independent of the geometry of the problem, the of the solution in the entire solution domain. coarser grids, where the successive corrections to the solu- 978-1-4673-4471-5/$25.00 c 2013, IEEE 487 488 PROCEEDINGS OF THE FEDCSIS. KRAKOW,´ 2013 tion will be obtained, have to be constructed by the coarsen- Au=f (2) ing process, which is an essential component of the AMG al- with a symmetric, positive definite matrix A. Partitioning the gorithm. degrees of freedom to those internal to Ω1, Ω2 and those in- The components needed for AMG, where superscripts in- terior of Γ, the matrix A and vectors u, f can be expressed as: dicate the level with 1 being the finest level [4],[27], are the following: (1) ( 1) (1 ) ( 1) AII 0 AIΓ u f 1 2 Ν I I • Grids Ω ⊃Ω ⊃...⊃Ω (Ω1=Ω) containing the fol- (2 ) ( 2) A= 0 A A ,u= (2) ,f = ( 2 ) (3) lowing two disjoint subsets: II IΓ uI f I k A(1) A(2 ) A u f Coarse points set (C-points): C ,k=1,...,N−1 . [ ΓI ΓI ΓΓ ] [ Γ ] [ Γ ] k Fine points set (f-points): F ,k=1,...,N−1 . The first step of many iterative domain decompositions • Grid operators: A1 ,A2 ,... ,AN , where A1=A . methods eliminates the unknowns in the interior of the sub- (i) • Interpolation and restriction operators: domains uI . This leads to a block factorization of the matrix k k+1 A (3) [25]: I k+1 ,k=1,...,N−1 ; I k ,k=1,...,N−1 . • A smoother (relaxation scheme) for each level. AMG consists of two main phases: the setup phase, where (1 ) (1) I 0 0 A 0 A the above components are created, and the solution phase II IΓ 0 I 0 ( 2 ) (2) that utilizes the components in the recursively defined multi- A=LR= 0 A A −1 −1 II IΓ grid cycle. A( 1) A(1) A( 2) A(2) I [ ΓI II ΓI II ][ 0 0 S ] In this article, a domain decomposition method is pre- sented, based on the Schur complement system. An Alge- (4) braic Multigrid method is used to solve the resulting linear and the resulting linear system: systems for each domain, based on the use of generic ap- (1) (1 ) (1 ) proximate banded inverses, derived from the ILU(0) factor- AII 0 AIΓ f I ization [13],[18]. In section II, the Schur complement ( 2 ) ( 2) u= (2 ) (5) method is showcased, while in section III, the AMG method 0 AII AIΓ f I is presented. 0 0 S g [ ] [ Γ ] Finally in section IV, the applicability of the new pro- posed scheme on two dimensional boundary value problems where I is the identity matrix and is demonstrated and numerical results on the convergence (1 ) (1)−1 (1) ( 2) (2)−1 (2) is the behavior and performance are given. S= AΓΓ− AΓI AII AΙΓ −AΓI AII AΙΓ Schur complement matrix relative to the unknowns on Γ. II.THE SCHUR COMPLEMENT METHOD Defining the local Schur complements by 2 In this section, the Schur complement domain decomposi- (i ) (i) (i ) (i)−1 ( i) tion method is presented. S := AΓΓ−∑ AΓI AII AΙΓ (6) The Schur complement method is the earliest version of i=1 non-overlapping DD methods. Methods such as Dirich- let-Neumann and Neumann-Neumann are essentially the Schur complement method with the use of particular precon- ditioners. Let us consider the Poisson equation on a region Ω, with zero Dirichlet data given on ∂Ω , the boundary of Ω. Let us also suppose that Ω is partitioned into two non-overlapping subdomains Ωi: Ω=Ω1∪Ω2 ,Ω1∩Ω2=∅,Γ =∂Ω1∩∂Ω2 as shown in Fig. 1 [25]. Assuming the boundaries of the subdomains are Lipschitz continuous, we consider the problem: −Δu(x,y )= f ( x,y)∈Ω (1) u(x,y)=0 ( x,y)∈∂Ω (1.a) Fig. 1 Ω partitioned into two non-overlapping subdomains. Considering a triangulation of the domain Ω and a finite we find the Schur complement system for uΓ to be [25]: element approximation of the problem (1), assuming that subdomains consist of unions of elements, leads to a linear SuΓ = g Γ (7) system with P. I. MATSKANIDIS, G. A. GRAVVANIS: SCHUR COMPLEMENT DOMAIN DECOMPOSITION 489 1 2 H1 S=S( )+S( ) (8) • : For each point j that strongly influences a fine-grid point i, j is either a coarse-grid point or strongly −1 depends on a coarse-grid point that also strongly g :=g(1)+g( 2)=( f (1 )− A( 1) A(1 ) f ( 1)) Γ Γ Γ Γ ΓI II I (9) influences i. −1 (2) ( 2) (2) (2) • H2: C is a maximal set with the property that no +( f Γ − AΓI AII f I ) C-point influences another C-point. Once uΓ is found by solving (7), the internal components Condition H1 ensures the quality of the interpolation and can be found by using (5): condition H2 restricts the size of coarser grids. The coarsening schemes of early AMG methods are based ( i) (i )−1 (1) (i ) u I =AII ( f I −AIΓ uΓ ) (10) on the Ruge-Stüben (RS) coarsening method [20]. The Ruge-Stüben coarsening is the classical coarse-grid selection Equations (8) and (9) can be extended to a generic case, algorithm, based on enforcing heuristic H1, while implicitly where there are more than two domains.
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