Antonietti, Paola F.

Houston, Paul

Nottingham eTheses

Antonietti, Paola F. and Houston, Paul (2011) Preconditioning high-order discontinuous Galerkin discretizations of elliptic problems. In: 20th International Conference on Domain Decomposition Methods, 7-11 Feb 2011, UC San Diego, California, USA. (Submitted) Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/1474/1/AntoniettiHouston_dd20_v4.pdfCopyright and reuse: The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions.This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence.  For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdfA note on versions: The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.ukPreconditioning high–orderDiscontinuous Galerkin discretizationsof elliptic problemsPaola F. Antonietti1 and Paul Houston21 IntroductionIn recent years, attention has been devoted to the development of efficient it-erative solvers for the solution of the linear system of equations arising fromthe discontinuous Galerkin (DG) discretization of a range of model prob-lems. In the framework of two level preconditioners, scalable non-overlappingSchwarz methods have been proposed and analyzed for the h–version of theDG method in the articles [1, 2, 6, 7, 9]. Recently, in [3] it has been provedthat the non-overlapping Schwarz preconditioners can also be successfully em-ployed to reduce the condition number of the stiffness matrices arising froma wide class of high–order DG discretizations of elliptic problems. In thisarticle we aim to validate the theoretical results derived in [3] for the multi-plicative Schwarz preconditioner and for its symmetrized variant by testingtheir numerical performance. This article is organized as follows. In Section 2we introduce the model problem and its DG approximation. In Section 3 weconstruct the Schwarz preconditioners, and recall the main theoretical resultsshown in [3]. Finally, in Section 4 we present some numerical results obtainedwith the multiplicative Schwarz preconditioner and its symmetrized variant.2 Model problem and DG discretizationIn this section we introduce the model problem under consideration and itsDG approximation, working, for the sake of simplicity, with the SIPG formu-1MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci32, 20133 Milano ITALY. paola.antonietti@polimi.it ·2 School of MathematicalSciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.Paul.Houston@nottingham.ac.uk12 Paola F. Antonietti and Paul Houstonlation proposed in [4].We consider, for simplicity, the weak formulation of the Poisson problemwith homogeneous Dirichlet boundary conditions: find U ∈ H10 (Ω) such that(∇U ,∇v)Ω = (f, v)Ω ∀ v ∈ H10 (Ω), (1)where Ω is a bounded polygonal domain in Rd, d = 2, 3, f ∈ L2(Ω) is a givensource term and (·, ·)Ω is the standard inner product in [L2(Ω)]d.Let Th be a shape-regular, not necessarily matching partition of Ω intodisjoint open elements K (with diameter hK), where each K is the affine imageof a fixed master element K̂, i.e., K = FK(K̂), where K̂ is either the open unitd-simplex or the d-hypercube in Rd, d = 2, 3. We define the mesh-size hby h := maxK∈Th hK, and assume that Th satisfies a bounded local variationproperty: for any pair of neighboring elements K1,K2 ∈ Th, hK1 ≈ hK2 .For a given approximation order p ≥ 1, we define the DG spaceVh,p := {v ∈ L2(Ω) : v|K ◦ FK ∈Mp(K̂) ∀K ∈ Th},where Mp(K̂) is either the space of polynomials of degree at most p on K̂, ifK̂ is the reference d-simplex, or the space of polynomials of degree at most pin each variable on K̂, if K̂ is the reference d-hypercube.Next, for any internal face F = ∂K+ ∩ ∂K− shared by two adjacent ele-ments K±, with outward unit normal vectors n±, respectively, we define[[τ ]] := τ+ · n+ + τ− · n−, [[v]] := v+n+ + v−n−,{{τ}} := (τ+ + τ−)/2, {{v}} := (v+ + v−)/2,where τ± and v± denote the traces on ∂K± taken from the interior of K± ofthe (sufficiently regular) functions τ and v, respectively (cf. [5]). On a bound-ary face F = ∂K∩∂Ω, we set [[τ ]] := τ ·n, [[v]] := v n, {{τ}} := τ , and {{v}} := v.We collect all interior (respectively, boundary) faces in the set FIh (respec-tively, FBh ), define Fh := FIh ∪FBh , and introduce on Vh,p×Vh,p the followingthe bilinear formA(u, v) :=∑K∈Th∫K∇u · ∇v dx+∑K∈Th∫K∇u · R([[v]]) dx+∑K∈Th∫KR([[u]]) · ∇v dx+∑F∈Fh∫Fαp2|F |[[u]] · [[v]] ds,where α > 0 is a parameter at our disposal. The lifting operator R(·) isdefined as: R(τ ) :=∑F∈FhrF (τ ), where rF : [L2(F )]d → [Vh,p]d is given byPreconditioning high–order DG discretizations of elliptic problems 3∫ΩrF (τ ) · η dx := −∫Fτ · {{η}} ds ∀η ∈ [Vh,p]d ∀F ∈ Fh.The DG discretization of problem (1) reads:Find u ∈ Vh,p such that A(u, v) =∫Ωfv dx ∀v ∈ Vh,p. (2)Let ϕj , j = 1, . . . , Nph := dim(Vh,p), be a set of basis functions that spanVh,p, then (2) can be written in the following equivalent form:Find u ∈ RNph such that Au = f, (3)where here (and in the following) we use the bold notation to denote thespaces of degrees of freedom (vectors) and discrete linear operators (matrices).The following result provides an estimate for the spectral condition numberof A; we refer to [3] for the proof.Proposition 1 ([3]). For a set of basis functions which are orthonormal onthe reference element K̂ ⊂ Rd, d = 2, 3, the condition number κ(A) of thestiffness matrix A can be bounded byκ(A) . αp4h2.Remark 1. We are working, for the sake of simplicity, with the SIPG formula-tion proposed in [4], but the results shown in Proposition 1 and in Theorem 1below also hold for a wide class of DG methods; we refer to [3] for details.3 Two level non-overlapping Schwarz preconditionersIn this section we introduce the non-overlapping Schwarz preconditioners.Subdomain partition.We decompose the domainΩ intoN non-overlappingsubdomains Ωi, i.e., Ω = ∪Ni=1Ωi. Next, we consider two levels of nested par-titions of the domain Ω: i) a coarse partition TH (with mesh-size H); ii) afine partition Th (with mesh-size h). We will suppose that the subdomainpartition does not cut any element of TH (and therefore of Th).Local solvers. For i = 1, . . . , N , we define the local DG spaces asV ih,p := {v ∈ L2(Ωi) : v|K ◦ FK ∈Mp(K̂) ∀ K ∈ Th,K ⊂ Ωi}.Denoting by RTi : Vih,p −→ Vh,p the classical injection operator from Vih,p toVh,p, the local solvers Ai : Vih,p × Vih,p −→ R are defined as4 Paola F. Antonietti and Paul HoustonAi(ui, vi) := A(RTi ui, RTi vi) ∀ui, vi ∈ Vih,p, i = 1, . . . , N.Coarse solver. For an integer 0 ≤ q ≤ p, we define the coarse space V 0H,q asV 0H,q := {v ∈ L2(Ω) : v|D ◦ FD ∈ MqD (K̂) ∀ D ∈ TH},and the coarse solver A0 : V0H,q × V0H,q −→ R asA0(u0, v0) := A(RT0 u0, RT0 v0) ∀u0, v0 ∈ V0H,q, (4)where RT0 : V0H,q −→ Vh,p is the classical injection operator from V0H,q to Vh,p.Let the local projection operators be defined asP˜i : Vh,p → Vih,p : Ai(P˜iu,RTi vi) := A(u,RTi vi) ∀vi ∈ Vih,p, i = 1, . . . , N,P˜0 : Vh,p → V0H,q : A0(P˜0u,RT0 v0) := A(u,RT0 v0) ∀v0 ∈ V0H,q,and define the projection operators as Pi := RTi P˜i : Vh,p −→ Vh,p, i =0, 1, . . . , N . The multiplicative Schwarz operator and its symmetrized variantare then defined asPmu := I − (I − PN )(I − PN−1) · · · (I − P0), (5)P Smu := I − (I − P0)T · · · (I − PN )T (I − PN ) · · · (I − P0), (6)respectively (cf. [10]). The Schwarz method consists in solving, by a suitableKrylov-subspace type method, one of the following linear systems of equationsPmuu = gmu, PSmuu = gSmu,for suitable right hand sides gmu and gSmu, respectively. It can be shown thatthe operator defined in (6) is symmetric and positive definite; we thereforeconsider the conjugate gradient (CG) algorithm for the solution of P Smuu =gSmu. An estimate of the condition number of PSmu isκ(P Smu) :=λmax(PSmu)λmin(P Smu),where λmax(PSmu) and λmin(PSmu) are the extremal eigenvalues of the operatorP Smu. On the other hand, the multiplicative operator Pmu is non-symmetric;we therefore consider a Richardson iteration applied to Pmuu = gmu, andshow that the norm of the error propagation operator Emu := (I − PN )(I −PN−1) · · · (I − P0) is strictly less than one, i.e.,‖Emu‖2A := supv∈Vh,pv 6=0A(Emuv, Emuv)A(v, v)< 1,Preconditioning high–order DG discretizations of elliptic problems 5and therefore a Richardson iteration applied to the preconditioned systemconverges. The following result provides a bound for the norm of the errorpropagation operator of the multiplicative Schwarz operator, and for thecondition number of the symmetrized Schwarz operator (we refer to [3] forthe proof).Theorem 1 ([3]). There exists constants C1, C2 ≥ 1, independent of themesh-size and the polynomial degree, such that‖Emu‖2A ≤ 1−hC1α p2H, κ(P Smu) ≤ C2αp2Hh.Theorem 1 also guarantees that the multiplicative Schwarz method can beaccelerated with the GMRES iterative solver. Indeed, according to [8], theGMRES method applied to the preconditioned system Pmuu = gmu doesnot stagnate (i.e., the iterative method makes some progress in reducing theresidual at each iteration step) provided that: i) ‖Pmu‖A is bounded; ii) thesymmetric part of Pmu is positive definite, i.e., there exists cp > 0 such thatA(v, Pmuv) > cpA(v, v) for all v ∈ Vh,p. Condition i) follows directly from thedefinition of Pmu and Theorem 1: ‖Pmu‖A = ‖I−Emu‖A ≤ 1+‖Emu‖A < 2.To prove condition ii), it can be shown thatA(Pmuv, v) = A(v, v) −A(Emuv, v) ≥ (1− ‖Emu‖A) A(v, v).Therefore, condition ii) holds true with cp = 1 − ‖Emu‖A which is positivedue to Theorem 1.4 Numerical resultsIn this section we present some numerical experiments to highlight the prac-tical performance of the multiplicative and symmetrized non-overlappingSchwarz preconditioners. Throughout this section we have set the penaltyparameter α := 10 (see (2)). We consider a subdomain partition consisting ofN = 16 squares, and consider the initial Cartesian and unstructured triangu-lar partitions shown in Figure 1, and denote by H0 and h0 the correspondinginitial coarse and fine mesh-sizes, respectively. We consider n successive globaluniform refinements of these initial grids so that the resulting mesh-sizes areHn = H0/2n and hn = h0/2n, with n = 0, 1, 2, 3, respectively. The (relative)tolerance is set equal to 10−9 (respectively, 10−6) for the CG (respectively,GMRES) iterative solver.We first address the performance of the multiplicative Schwarz precondi-tioner by keeping the mesh fixed, and varying the polynomial approximationdegree p. In Table 1 we compare the GMRES iteration counts for both thepreconditioned and non-preconditioned (in parenthesis) systems, for differentpolynomial approximation degrees and different mesh configurations. These6 Paola F. Antonietti and Paul Houston(a) Initial coarse grids (mesh-size H0) (b) Initial fine grids (mesh-size h0)Fig. 1 Initial Cartesian and triangular coarse and fine grids on a 16 subdomain partition.Table 1 GMRES iteration counts. Multiplicative Schwarz preconditioner with a piecewiseconstant coarse solver (q = 0). Unstructured triangular grids.h = h0/2 h = h0/4 h = h0/4H = H0 H = H0 H = H0/2p = 1 23 (94) 33 (199) 25 (199)p = 2 45 (259) 64 (540) 49 (540)p = 3 66 (470) 93 (996) 74 (996)p = 4 85 (713) 124 (1546) 97 (1546)p = 5 105 (1004) 153 (2187) 123 (2187)p = 6 124 (1342) 183 (2924) 144 (2924)p = 7 143 (1727) 209 (3742) 167 (3742)p = 8 162 (2148) 235 (4673) 189 (4673)p− rate 0.93 (1.63) 0.88 (1.66) 0.93 (1.66)results have been obtained on unstructured triangular grids (cf. Figure 1).Comparing the iteration counts of the preconditioned systems with the un-preconditioned ones for a fixed p, it is clear that the proposed preconditioneris very efficient. Indeed, we observe a reduction in the number of iterationsneeded to achieve convergence of around one order of magnitude when theproposed preconditioner is employed. The last row of Table 1 shows the com-puted growth rate in the number of iterations: we observe that the numberof iterations needed to obtain convergence increases linearly as a function ofp for the preconditioned system of equations, whereas this quantity growsalmost quadratically for the non-preconditioned problem.In Figure 2 we report the condition number estimates of the symmetrizedSchwarz operator and the corresponding iteration counts versus the poly-nomial degree p. The solid lines refer to the mesh configuration h = h0/2,H = H0, whereas the dashed lines refer to the mesh configuration h = h0/4,H = H0/2. This set of numerical experiments has been obtained on Cartesianmeshes, employing a piecewise linear coarse solver. As predicted by the the-oretical estimates, the condition number of the preconditioned system growsquadratically as a function of p. Moreover, we clearly observe that, for fixedp, by refining both the fine and the coarse grid, but keeping the ratio of thePreconditioning high–order DG discretizations of elliptic problems 71 2 3 4 5 6 7 8101102p21κ(PSmu)κ(PSmu)it(PSmu)it(PSmu)Fig. 2 Condition number estimates of the symmetrized Schwarz operator and correspond-ing iteration counts versus the polynomial degree p on Cartesian grids for different dis-cretization steps (solid line: h = h0/2, H = H0; dashed line h = h0/4, H = H0/2).Piecewise linear coarse solver.Table 2 Condition number estimates for the symmetrized Schwarz operator with p = 2,q = 0 (left) and p = 3, q = 1 (right). Cartesian grids.h ↓ H → H0 H0/2 H0/4 H0/8 H0 H0/2 H0/4 H0/8h0 5.32e2 1.12e3 4.01e3 7.08e3 4.81e1 9.5925e1 1.9168e2 3.9068e2h0/2 2.74e2 4.71e2 2.80e3 5.59e3 2.14e1 4.35e1 8.70e1 1.75e2h0/4 – 2.60e2 1.18e3 3.42e3 – 2.09e1 4.24e1 8.44e1h0/8 – – 3.45e2 1.75e3 – – 2.05e1 4.26e1κ(A) 2.88e5 1.18e6 4.89e6 1.99e7 7.44e5 2.81e6 1.11e7 4.55e7fine and coarse mesh-sizes constant, the condition number (and therefore thenumber of iterations needed to obtain convergence) remains constant.Next, we consider the performance of the symmetrized Schwarz precondi-tioner when varying the coarse and fine mesh-size, and keeping the polyno-mial approximation degree p fixed. In Table 2 (left) we report the conditionnumber estimates for the symmetrized Schwarz operator employing piecewisebiquadratic elements (p = 2) and a piecewise constant coarse solver (q = 0);whereas, in Table 2 (right) the analogous results obtained with piecewisebicubic elements (p = 3) and a piecewise linear coarse solver (q = 1) areshown. We clearly observe that the condition number grows as O(Hh−1),as predicted by Theorem 1. Moreover, we clearly observe that employing apiecewise linear coarse solver (q = 1) rather than a piecewise constant coarsesolver (q = 0) significantly improves the performance of the preconditioner.Indeed, comparing the condition number estimates of the preconditioned sys-tem with the analogous ones obtained for the non-preconditioned problem(last row of Table 2) we clearly observe that the condition number of the8 Paola F. Antonietti and Paul Houstonnon-preconditioned system is reduced with respect to the condition numberof the preconditioned system by approximately 5 orders of magnitude forq = 1 and 4 orders of magnitude for q = 0.AcknowledgementPH acknowledges the financial support of the EPSRC under the grantEP/H005498.References[1] P. F. Antonietti and B. Ayuso. Schwarz domain decomposition precon-ditioners for discontinuous Galerkin approximations of elliptic problems:non-overlapping case. M2AN Math. Model. Numer. Anal., 41(1):21–54,2007.[2] P. F. Antonietti and B. Ayuso. Multiplicative Schwarz methods for dis-continuous Galerkin approximations of elliptic problems. M2AN Math.Model. Numer. Anal., 42(3):443–469, 2008.[3] P. F. Antonietti and P. Houston. A class of domain decomposition pre-conditioners for hp-discontinuous Galerkin finite element methods. J.Sci. Comp., 46(1):124–149, 2011.[4] D. N. Arnold. An interior penalty finite element method with discontin-uous elements. SIAM J. Numer. Anal., 19(4):742–760, 1982.[5] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified anal-ysis of discontinuous Galerkin methods for elliptic problems. SIAM J.Numer. Anal., 39(5):1749–1779 (electronic), 2001/02.[6] A. T. Barker, S. C. Brenner, P. Eun-Hee, and L.-Y. Sung. Two-leveladditive Schwarz preconditioners for a weakly over-penalized symmetricinterior penalty method. J. Sci. Comp., 47:27–49, 2011.[7] S. C. Brenner and K. Wang. Two-level additive Schwarz preconditionersfor C0 interior penalty methods. Numer. Math., 102(2):231–255, 2005.[8] S. C. Eisenstat, H. C. Elman, and M. H. Schultz. Variational iterativemethods for nonsymmetric systems of linear equations. SIAM J. Numer.Anal., 20(2):345–357, 1983.[9] X. Feng and O. A. Karakashian. Two-level additive Schwarz methodsfor a discontinuous Galerkin approximation of second order elliptic prob-lems. SIAM J. Numer. Anal., 39(4):1343–1365 (electronic), 2001.[10] A. Toselli and O. Widlund. Domain decomposition methods—algorithmsand theory, volume 34 of Springer Series in Computational Mathematics.Springer-Verlag, Berlin, 2005.

Preconditioning high-order discontinuous Galerkin discretizations of elliptic problems

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Preconditioning high-order discontinuous Galerkin discretizations of elliptic problems

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