Numerical Algorithms manuscript No. (will be inserted by the editor)

A note on the H 1 -convergence of the overlapping Schwarz waveform relaxation method for the heat equation† Hui Zhang · Yao-Lin Jiang

Received: date / Accepted: date

Abstract The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time. Keywords Schwarz waveform relaxation · Convergence · Energy norm

1 Introduction The Schwarz waveform relaxation (SWR) methods are a class of parallel methods for solving time-dependent PDEs, inspired by the WR methods for ODEs (see e.g. [1, 2, 3, 4]). At each iteration of SWR, space-time subdomain problems across a time interval are solved. Compared to the time stepping methods with space–uniform time steps (see e.g. [5]), the SWR methods allow local spacetime adaptivity. The SWR methods also have less communication overhead because no communication is needed until every subdomain has completed †The work was supported by the Natural Science Foundation of China (11071192) and the International Science and Technology Cooperation Program of China (2010DFA14700). H. Zhang School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China Department of Mathematics, University of Geneva, Geneve CH-1211, Switzerland E-mail: [email protected] Y.-L. Jiang (B) School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China E-mail: [email protected]

2

Hui Zhang, Yao-Lin Jiang

the time integration over the whole time interval. This is in contrast to the time stepping methods that demand communication at each time step. The memory requirement is small because only the data along the interior spacetime interfaces need to be stored in the iterative process. We refer to [6, 7, 8, 9] for the early development of the overlapping SWR method, [10, 11, 12, 13] for convergence analyses of the method, [14, 15, 16] for extension to nonlinear problems, [17, 18,19] for local space-time adaptivity, [20] for parallel experiments, [15, 18, 21, 22, 23, 24, 25] for using Robin-like transmission conditions and optimization of parameters. It is also worthwhile to note that SWR methods can be used to solve problems with non-local coupling in time, see e.g. [26] for time-periodic problems and [27, 28] for time-delay problems. The convergence of the overlapping SWR method for parabolic equations has been analyzed at the continuous level [10] (from which we will cite a result, see Thm. 1 in the next section) and at the discrete level [11], both by maximum principles and under L∞ -norm. It is natural to compare the results with the convergence of the Schwarz method for elliptic equations. Note that in the latter case the Schwarz method converges also under H 1 norm, which is important for many applications where the flux or the gradient of the solution is concerned. However, it was unknown before this work whether SWR converges under H 1 -norm. Of course, at the discrete level any norms are equivalent, but the constant involved in the equivalence goes to infinity when the mesh size tends to zero. Hence, an analysis at the continuous level is useful. Inspired by [29] which proves the Schwarz method converges under H 1 -norm given it converges under L∞ -norm, we prove in this paper a similar result holds for SWR for the heat equation, from which the H 1 -convergence of the overlapping SWR is obtained for the first time.

2 Previous results We consider the time-dependent heat equation in a bounded domain Ω ⊂ Rd : ut − 4u = f (x, t), (x, t) ∈ Ω × (0, T ), u(x, t) = g(x, t), (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u0 (x), x ∈ Ω.

(1)

To apply the multiplicative SWR method, we decompose Ω into overlapping subdomains Ω1 , Ω2 such that Ω1 ∪ Ω2 = Ω. Given an initial guess u0 (x, t) satisfying u0 (x, t) = g(x, t) for (x, t) ∈ ∂Ω × (0, T ), we iterate

and

u2n−1 − 4u2n−1 = f (x, t), (x, t) ∈ Ω1 × (0, T ), t u2n−1 (x, t) = u2n−2 (x, t), (x, t) ∈ (Ω − Ω1 ) × (0, T ), u2n−1 (x, 0) = u0 (x), x ∈ Ω1 ,

(2)

2n u2n = f (x, t), (x, t) ∈ Ω2 × (0, T ), t − 4u 2n u (x, t) = u2n−1 (x, t), (x, t) ∈ (Ω − Ω2 ) × (0, T ), u2n (x, 0) = u0 (x), x ∈ Ω2 ,

(3)

H 1 -convergence of the overlapping Schwarz waveform relaxation

3

for n = 1, 2, · · · . The well-posedness results for the problems (1), (2) and (3) can be found in [30, p. 32] and [31]. We recall the following convergence result. Theorem 1 ([10, Theorem 3.2, Theorem 4.3]) Let the errors of the SWR method be n := u − un . Denote by k · kT the norm in L∞ (Ω × (0, T )). For any T > 0, we have kn+k kT ≤ γ(n) kk kT , ∀n ≥ 1, k ≥ 0, where 0 < γ(n) < 1, γ(n) → 0 as n → ∞ (see [10] for detailed estimates of γ(n)).

3 Convergence under H 1 -norm Let (·, ·), k · k be the inner product and norm of L2 (Ω), respectively. Denote by ∇x u the gradient of u w.r.t. space variables and by W 1,∞ (Ω) := {u ∈ L∞ (Ω) | ∇x u ∈ (L∞ (Ω))d }. We need the following assumption about the existence of partition of unity functions that are bounded in W 1,∞ (Ω). Assumption 1 For the overlapping partition Ω = Ω1 ∪Ω2 , there exist ξ1 , ξ2 ∈ W 1,∞ (Ω) such that 0 ≤ ξ1 , ξ2 ≤ 1, on Ω, supp ξi ⊂ Ωi ,

i = 1, 2,

ξ1 + ξ2 = 1,

on

Ω.

Theorem 2 Suppose the errors generated by the SWR method satisfy n ∈ L∞ (Ω × [0, T ]) ∩ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; L2 (Ω)) for all n ≥ 0. Let Assumption 1 hold. Then for all n ≥ 1 we have Z

T


k∇x 2n k2 dt ≤ 2|Ω1 | + 80 T k∇x ξ1 k2 k2n−1 k2T + 2 T k∇x ξ1 k2 k2n k2T .

0

(4) Proof We proceed in three steps labeled by (i), (ii) and (iii) below. (i) From (2) the error 2n−1 (n ≥ 1) satisfies (2n−1 , v) + (∇x 2n−1 , ∇x v) = 0, ∀ v ∈ H01 (Ω1 ). t

(5)

Let v = ξ12 2n−1 where ξ1 is the function given in Assumption 1. Since v ∈ H01 (Ω1 ), substituting it into (5) we get (2n−1 , ξ12 2n−1 ) + (ξ1 ∇x 2n−1 , ξ1 ∇x 2n−1 ) = −2(ξ1 ∇x 2n−1 , 2n−1 ∇x ξ1 ). t

4

Hui Zhang, Yao-Lin Jiang

Integration over t ∈ [0, T ] and applying the Cauchy-Schwarz inequality to the right side yields 1 2n−1 (·, T ), ξ1 2n−1 (·, T )) 2 (ξ1 

+

RT 0

(ξ1 ∇x 2n−1 , ξ1 ∇x 2n−1 ) dt

RT RT ≤ 2 { 0 (ξ1 ∇x 2n−1 , ξ1 ∇x 2n−1 ) dt}1/2 · { 0 (2n−1 ∇x ξ1 , 2n−1 ∇x ξ1 ) dt}1/2 Hence, T

Z

kξ1 ∇x 2n−1 k2 dt ≤ 4

Z

0

T

k2n−1 ∇x ξ1 k2 dt ≤ (4 T k∇x ξ1 k2 ) k2n−1 k2T .

0

(6) (ii) In (5) taking the test function v = ξ12 w ∈ H01 (Ω1 ) with an arbitrary w ∈ H01 (Ω) ∩ L∞ (Ω), we obtain (∂t 2n−1 , ξ12 w) + (∇x 2n−1 , ∇x (ξ12 w)) = 0. From the above equality and the fact that the second term equals (∇x 2n−1 , ∇x (ξ12 w)) = (ξ1 ∇x 2n−1 , ξ1 ∇x w) + 2(ξ1 ∇x 2n−1 , w∇x ξ1 ), we apply the Cauchy-Schwarz inequality and use 0 ≤ ξ1 ≤ 1 (Assumption 1) to get |(∂t 2n−1 , ξ12 w)| ≤ kξ1 ∇x 2n−1 k · {k∇x wk + 2 k∇x ξ1 k kwkT }.

(7)

(iii) From (3) we have 2n 1 (2n t , v) + (∇x  , ∇x v) = 0, ∀ v ∈ H0 (Ω2 ).

(8)

From (3) and Assumption 1 we also know that v = 2n − ξ12 2n−1 belongs to H01 (Ω2 ). Substitution of this choice of v into (8) and integration over t ∈ [0, T ] gives us Z

T

(∂t 2n , 2n )+(∇x 2n , ∇x 2n ) dt =

T

Z

0

(∂t 2n , ξ12 2n−1 )+(∇x 2n , ∇x (ξ12 2n−1 )) dt.

0

Integration by parts for the first terms of the left and the right hand sides gives 1 2n k (·, T )k2 + 2

Z

T

k∇x 2n k2 dt

0

=(2n (·, T ), ξ12 2n−1 (·, T )) −

Z

T

(ξ12 2n , ∂t 2n−1 ) dt

0

Z +

T

(∇x 2n , ξ12 ∇x 2n−1 ) dt + 2

0

:=I1 + I2 + I3 + I4 .

Z 0

T

(∇x 2n , 2n−1 ξ1 ∇x ξ1 ) dt

(9)

H 1 -convergence of the overlapping Schwarz waveform relaxation

5

We now estimate I1 , I2 , I3 , I4 separately. First, by the Cauchy-Schwarz inequality and Assumption 1, |I1 | ≤

1 2n |Ω1 | 2n−1 2 k (·, T )k2 + k kT . 2 2

(10)

Let σ1 , σ2 , σ3 , η > 0 be any positive numbers. Using (7) we have Z T kξ1 ∇x 2n−1 k · {k∇x 2n k + 2 k∇x ξ1 k k2n kT } dt |I2 | ≤ 0

≤

1 2σ1

Z 0

T

k∇x 2n k2 dt +

T σ1 k∇x ξ1 k2 k2n k2T + ( + η) η 2

Z

T

kξ1 ∇x 2n−1 k2 dt,

0

(11) and by substituting (6) into the last term of (11),   Z T 1 1 2n 2 k∇x 2n k2 dt + (T k∇x ξ1 k2 ) k kT + (2σ1 + 4η)k2n−1 k2T . |I2 | ≤ 2σ1 0 η (12) By the Cauchy-Schwarz inequality, Assumption 1 and (6), we obtain Z T 1 k∇x 2n k2 dt + 2σ2 (T k∇x ξ1 k2 ) k2n−1 k2T . (13) |I3 | ≤ 2σ2 0 Again, using the Cauchy-Schwarz inequality and Assumption 1, we have Z T 1 |I4 | ≤ k∇x 2n k2 dt + σ3 (T k∇x ξ1 k2 ) k2n−1 k2T . (14) σ3 0 Combining (9),(10),(12),(13) and (14), we get Z T 1 1 1 (1 − − − ) k∇x 2n k2 dt 2σ1 2σ2 σ3 0   |Ω1 | 1 2 + (2σ1 + 4η + 2σ2 + σ3 )(T k∇x ξ1 k ) k2n−1 k2T + (T k∇x ξ1 k2 ) k2n k2T . ≤ 2 η Now taking σ1 = σ2 = η = 2, σ3 = 4, we arrive at the conclusion (4). Remark 1 A similar result holds for the errors on Ω1 , that is Z T
k∇x 2n+1 k2 dt ≤ 2|Ω2 | + 80 T k∇x ξ2 k2 k2n k2T + 2 T k∇x ξ2 k2 k2n+1 k2T . 0

Remark 2 Convergence of SWR in L2 (0, T ; H 1 (Ω))∩C(0, T ; L2 (Ω)) is a direct consequence of Theorem 1 and Theorem 2. Remark 3 The regularity assumed for n will become true after sufficiently many iterations even if the initial error 0 is less smooth. This is because of the interior regularity of the heat equation. We say that the SWR method has a smoothing effect.

6

Hui Zhang, Yao-Lin Jiang

Note that Assumption 1 holds in common cases. In particular, from [32, Lemma 3.4], when Ω1 , Ω2 are shape regular of diameter H we have k∇x ξi k < C/δ where δ > 0 is the overlap size. Note also that k2n kT ≤ k2n−1 kT by Theorem 1. So we have Corollary 1 Under the assumptions of [32, Lemma 3.4] and of Theorem 2, the errors of SWR satisfy Z T T Hd k∇x n k2 dt ≤ C max{1, 2 } kn−1 k2T . δ 0 4 Numerical experiments To illustrate our theoretical results, we numerically solve (1) by the SWR method. For simplicity, we take Ω = (0, 1)2 , f = 0, g = 0, u0 = 0, overlap size 1/8 and a random initial guess u0 . We use P1 elements to discretize space with mesh size h = 1/16, 1/32 and implicit Euler to discretize time with the timestep size ∆t = 1/10, 1/20. The implementation is done with FreeFem++[33]. We can see from Fig.1 that both the infinity norm and the H 1 semi-norm of the errors of iterates converge to zero with linked behaviors: on the long time interval both converge linearly and on the short interval both converge superlinearly, which confirms our result in Theorem 2. It is interesting to note that the curve for H 1 -semi-norm becomes lower than the curve for infinity norm when the time interval is shortened, which agrees with our estimate in Cor. 1. We can also compare the last two plots in Fig. 1 to see that the convergence under H 1 -semi-norm does not deteriorate when the space-time mesh is refined and the overlap fixed, which again confirms our estimate. Acknowledgements This paper was partially written when the first author stays as a postdoc in University of Geneva. He would like to thank the university and his postdoctoral advisor Martin J. Gander for their support. We also thank the reviewers for their helpful comments that improved our presentation dramatically.

References 1. Lelarasmee E., Ruehli A.E., Sangiovanni-Vincentelli A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1(3), 131–145 (1982) 2. Jiang Y.L.: Waveform Relaxation Methods (in Chinese). Science Press, Beijing (2009) 3. Jiang Y.L.: On time-domain simulation of lossless transmission lines with nonlinear terminations, SIAM J. Numer. Anal., 42(3), 1018–1031 (2004) 4. Jiang Y.L.: A general approach to waveform relaxation solutions of differential-algebraic equations: the continuous-time and discrete-time cases, IEEE Transactions on Circuits and Systems - Part I, 51(9), 1770–1780 (2004) 5. Cai X.C.: Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math., 60, 41–62 (1990) 6. Gander M.J.: Overlapping Schwarz for linear and nonlinear parabolic problems, Proceedings of the 9th International Conference on Domain Decomposition, 97–104, ddm.org (1996)

H 1 -convergence of the overlapping Schwarz waveform relaxation

7

1

log10 of infinity norm log10 of H1 semi-norm

0

log10 of iterative error

-1

-2

-3

-4

-5

-6

-7

-8

0

5

10

15

20

25

30

35

40

iteration

1 log10 of infinity norm log10 of H1 semi−norm

0

log10 of iterative error

−1 −2 −3 −4 −5 −6 −7 −8 0

5

10

15

20 iteration

25

30

35

40

1 log10 of infinity norm log10 of H1 semi−norm

0

log10 of iterative error

−1 −2 −3 −4 −5 −6 −7 −8 0

5

10

15

20 iteration

25

30

35

40

RT R 2n 2 1/2 against the even Fig. 1 Logarithm of kuh − u2n h kT and ( 0 Ω |∇x (uh − uh )| dx dt) iteration number 2n. The overlap is always 1/8. From top to bottom, T = 1, 1/5, 1/5, h = 1/16, 1/16, 1/32 and ∆t = 1/10, 1/10, 1/20, respectively.

8

Hui Zhang, Yao-Lin Jiang

7. Gander M.J.: A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numer. Linear Algebra Appl. 6, 125–145 (1998) 8. Gander M.J., Stuart A.M.: Space-time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comput. 19, 2014–2031 (1998) 9. Giladi E., Keller H.: Space time domain decomposition for parabolic problems, Numer. Math. 93, 279–313 (2002) 10. Gander M.J., Zhao H.: Overlapping Schwarz waveform relaxation for the heat equation in N dimensions, BIT 42, 779–795 (2002) 11. Mathew T.P., Russo G.: Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids, Math. Comp. 72, 619–656 (2003) 12. Jiang Y.L., Zhang H.: Schwarz waveform relaxation methods for parabolic equations in space-frequency domain, Comput. Math. Appl., 55, 2924–2939 (2008) 13. Daoud D.S., Gander M.J.: Overlapping Schwarz waveform relaxation for advection reaction diffusion problems, Bol. Soc. Esp. Mat. Apl., 46, 75–90 (2009) 14. Gander M.J., Rohde C.: Overlapping Schwarz waveform relaxation for convection dominated dissipative conservation laws, SIAM J. Sci. Comput. 27, 415–439 (2005) 15. Caetano F., Gander M.J., Halpern L., Szeftel J.: Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Networks and Heterogeneous Media, 5(3), 487–505 (2010) 16. Tran M.B.: Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain, C. R. Acad. Sci. Paris, Ser. I 348, 795–799 (2010) 17. Haynes R.D., Russell R.D.: A Schwarz waveform moving mesh method, SIAM J. Sci. Comput. 29, 656–673 (2007) 18. Halpern L., Japhet C., Szeftel Z.: Optimized Schwarz waveform relaxation and discontinuous Galerkin time Stepping for heterogeneous problem, SIAM J. Numer. Anal., 50(5), 2588–2611 (2012) 19. Zhang H., Song B., Jiang Y.L. A new domain decomposition waveform relaxation algorithm with local time-stepping (in Chinese), Sci. Sin. Math., 42(5): 501514 (2012) 20. Ltaief H., Garbey M.: A parallel Aitken-additive Schwarz waveform relaxation suitable for the grid, Parallel Computing 35, 416–428 (2009) 21. Halpern L.: Absorbing boundary conditions and optimized Schwarz waveform relaxation, BIT, 46, 21–34 (2006) 22. Gander M.J., Halpern L.: Optimized Schwarz waveform relaxation for advection reaction diffusion problems, SIAM J. Numer. Anal. 45, 666–697 (2007) 23. Bennequin D., Gander M.J., Halpern L.: A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp., 78, 185–223 (2009) 24. Courvoisier Y., Gander M.J.: Optimization of Schwarz waveform relaxation over short time windows, Numerical Algorithms, doi: 10.1007/s11075-012-9662-y (2012) 25. Wu S.L.: Convergence analysis for discrete Schwarz waveform relaxation algorithm of Robin type (in Chinese), Sci. Sin. Math., 43: 211–234 (2013) 26. Zhang H., Jiang Y.L.: Schwarz waveform relaxation methods of parabolic time-periodic problems (in Chinese), Sci. Sin. Math., 40(5), 497–516 (2010) 27. Gander M.J., Vandewalle S.: Optimized overlapping Schwarz methods for parabolic PDEs with time-delay. In: Kornhuber R., et al (eds.) Domain Decomposition Methods in Science and Engineering, 291–298, Springer, Berlin (2005) 28. Wu S.L., Huang C.M., Huang T.Z.: Convergence analysis of the overlapping Schwarz waveform relaxation algorithm for reaction-diffusion equations with time delay, IMA Journal of Numerical Analysis, 32(2), 632–671 (2012) 29. Lions P.L.: On the Schwarz alternating method II: Stochastic interpretation and order properties. In: Chan T., et al (eds.) Domain Decomposition Methods, 47–70, SIAM, Philadelphia (1989) 30. Lions J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications II. Springer-Verlag, New York-Heidelberg (1972) 31. Mikhailov V.P. (originator): Mixed and boundary value problems for parabolic equations and systems, Encyclopedia of Mathematics (2013) 32. Toselli A., Widlund O.B.: Domain Decomposition Methods–Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005) 33. Hecht F., Pironneau O., Morice J., Ohtsuka K.: FreeFem++. Universite Pierre et Marie Curie, Paris, http://www.freefem.org/ff++/ftp/freefem++doc.pdf (2012)