A parallel partition of unity method for the time-dependent convection

A parallel partition of unity method for the time-dependent convection-diffusion equations Guangzhi Du and Yanren Hou School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China Abstract

A parallel partition of unity method 1947 Received 10 September 2014 Revised 7 November 2014 Accepted 13 November 2014

Purpose – The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations. Design/methodology/approach – This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method. Findings – This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time. Originality/value – In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem. Keywords Partition of unity, Time-dependent convection-diffusion equations Paper type Research paper

1. Introduction Parallel finite element computations have received great attention and become a very active research in the last decades, due to its significance in modern scientific and engineering computing. A number of local and parallel finite element algorithms based on two-grid discretizations have been proposed for a class of linear and nonlinear elliptic boundary value problems by Xu and Zhou (1999a, b, 2001). Then this method was further extended to the Stokes equations (He et al., 2008), Navier-Stokes equations (He et al., 2006; Ma et al., 2007) and time-dependent convection-diffusion equations (Liu and Hou, 2009). In view of the partition of unity method (Babuška and Melenk, 1997; Melenk and Babuška, 1996), Holst combined the partition of unity with the parallel adaptive finite element and proposed a parallel partition of unity method (PPUM) in Holst (2001, 2003) and Huang and Xu (2003) proposed a new finite element discretization for the elliptic boundary value problems, Bacuta et al. (2011) developed a partition of unity refinement method to improve the local approximations of ellipitc boundary value problems. Wang et al. (2008) presented a two-grid partition of unity method, where the partition of unity method was employed to glue all the local solutions together to get the global continuous solution. In Yu et al. (2014) and Zheng et al. (n.d.), some new local and parallel finite element methods based on partition of unity method were proposed for solving the steady elliptic boundary value problem and the incompressible Stokes problem. Moreover, in Larson and Målqvist (2007, 2009), Larson et al. proposed the adaptive variational multi-scale method based on the partition of unity method, the idea is that to decouple the fine scale equations by including a partition of unity in the right-hand side and then solve the resulting problems on pathes. A parallel least-square finite element algorithm used in Zhang and Subsidized by NSFC(Grant Nos 11571274 & 11171269) and the PhD Programs Foundation of Ministry of Education of China(20110201110027).

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 25 No. 8, 2015 pp. 1947-1956 © Emerald Group Publishing Limited 0961-5539 DOI 10.1108/HFF-09-2014-0277

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Yang (2011) and the localization technique applied in Song et al. (2013) are both associate to the partition of unity method. In this paper, following the idea of Larson and Målqvist (2007, 2009), we propose a parallel method based on the partition of unity method. The main idea is that, at each time step, a global problem can be rewritten into the sum of a suite of sub-problems by the partition of unity and superposition principle. Each sub-problem can be approximated by one local sub-problem defined on the relevant sub-domain ωi. Clearly, those local sub-problems are independent of each other and can be solved in parallel. The rest of the paper is organized as follows. In Section 2, some preliminary materials are provided. In Section 3, a PPUM is constructed. Finally, some numerical experiments are given. 2. Preliminaries Let Ω be a bound domain in Rd (d ¼ 2, 3) with a Lipschitz continuous boundary ∂Ω, and T be a finite time. We consider the following time-dependent convection-diffusion equation: 8 @u > < @t Du þ bUru ¼ f ; 8ðx; t ÞA O  ½0;T ; u¼ 0; 8ðx; t ÞA @O  ½0;T ; (1) > : uðx; 0Þ ¼ u ; 8x A O: 0 where u(x, t) is the unknown scalar function, b ∈ L∞(0, T; L∞(Ω)d) and f a forcing function. Throughout this paper, (⋅, ⋅) denotes the usual L2 – inner product on Ω and :U:0 the corresponding norm. Let us introduce the following Hilbert space: n o X ¼ H 10 ðOÞ ¼ vA H 1 ðOÞ : v9@O ¼ 0 ; which is equipped with the following inner product and associated norm: ððu; vÞÞ ¼ ðru; rvÞ

1

and:u:X ¼ ððu; vÞÞ2 ;

8u;

vAX:

Moreover, we define a bilinear form a(⋅, ⋅) on X × X by: aðu; vÞ ¼ ðru; rvÞþ ðbUru; vÞ;

8u;

vAX:

With the above notation, the weak formulation of problem (1) reads: find u(t) ∈ X for all t ∈ [0, T] such that for all v ∈ X: ðut ; vÞþ aðu; vÞ ¼ ðf ; vÞ; uð0Þ ¼ u0 : (2)  h h Let h be a real positive parameter, T ðOÞ ¼ tO be a regular triangulation of Ω,    where h ¼ max diam thO is the mesh parameter. Associated with the mesh h Th(Ω), set: thO A T ðOÞ n o (3) S h ðOÞ ¼ vh A C 0 ðOÞ : vh 9th A P rth ; 8thO AT h ðOÞ ; O

O

S h0 ðOÞ ¼ S h ðOÞ \ H 10 ðOÞ;

(4)

where r ⩾ 1 is a positive integer and P rth is the space of polynomials of degree O not greater than r defined on thO . For simplicity and without loss of generality, we only

consider the case that Sh is a space of piecewise linear polynomials, i.e., r ¼ 1. Let P h : X -S h0 denote the L2-orthogonal projection defined by: ðP h v; vh Þ ¼ ðv; vh Þ;

8vA X ;

vh A S h0 :

Let kW 0 be a time step length and set tn ¼ nk for 0 ⩽ n ⩽ N ¼ [T/k]. Then the fully discrete standard Galerkin scheme for solving (1) based on backward Euler time discretization strategy reads: let uh, 0 ¼ Phu0, for 1 ⩽ n ⩽ N, find uh;n A S h0 such that:  h; n        u ; v þ ka uh; n ; v ¼ uh; n1 ; v þ k f n ; v ; 8vA S h0 : (5) Assume that f ∈ L∞( [0, T], L2(Ω)), it is classical that:   :uðt n Þuh; n :0;O p c k þ h2 ; 1 p n p N :

(6)

3. PPUM In this section, we will present a PPUM for the time-dependent convection-diffusion problem. Let us denote: w^ n ¼ uh; n uh; n1 ;

n ¼ 1; 2; :::;N :

Then the following equation holds:  n        w^ ; v þ ka w^ n ; v ¼ k f n ; v ka uh;n1 ; v ; 8vA S h0 : (7)  I Assume that fi i¼1 is anpartition of unity on Ω o for aPgiven integer I ⩾ 1 such that: d I O ⊂ [i¼1 suppfi ⊂ Oe ¼ x A R : distðx; OÞo e and Ii¼1 fi  1 on Ω. Then (7) can be rewritten as: ! ! I I X X  n   n  n h; n1 w^ ; v þ ka w^ ; v ¼ k f ; fi v  ka u ; fi v ; 8vA S h0 : (8) i¼1

i¼1

By superposition principle, the above Equation (8) is equivalent to the following sub-problems: for i ¼ 1, 2, ..., I:  n        w^ i ; v þ ka w^ ni ; v ¼ k f n ; fi v  ka uh;n1 ; fi v ; 8v A S h0 ; (9) P and w^ n ¼ Ii¼1 w^ ni . Moreover, we observe that each sub-problem is a equation driven by right-hand side term of a very small compact support with homogeneous Dirichlet boundary condition defined in the entire domain Ω, whose solution decays fast away from the small support. This suggests us to localize the Equation (9) by restricting the equation in a suitable neighborhood ωi of supp ϕi and therefore to reduce the computational scale. We restrict the above sub-problem in such a local domain ωi, which contains supp ϕi and is chosen large enough to obtain a good of n approximation o h h ^wni . For each local sub-domain ωi, wenassume  that o T ðoi Þ ¼ toi is a regular triangulation. Here: h ¼ max

max

1 p i p N th A T h ðoi Þ o i

diam thoi

. We also have the spaces Sh(ωi)

A parallel partition of unity method 1949

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and S h0 ðoi Þ, which are defined by (3) and (4). For simplicity, we assume that T h ðoi Þ ¼ T h ðOÞ9oi throughout this paper. Since the functions in S h0 ðoi Þ can be extended to functions in S h0 ðOÞ with zero value outside ωi, we regard S h0 ðoi Þ as a sub-space of S h0 ðOÞ in the sense of such zero extension. By using the above idea, we propose the PPUM as follows. PPUM: Step 1: let u h;0 ¼ P h u0 . h Step 2: for 1 ⩽ n ⩽ N, find w^ h;n i A S 0 ðoi Þ such that for all i ¼ 1, 2, …, I, 

    n   h;n1  ^ h;n w^ h;n ; fi v ; i ; v þ ka wi ; v ¼ k f ; fi v ka u

8vA S h0 ðoi Þ:

It is clear that the above sub-problems are independent of each other and can be solved in parallel. Note that w^ h;n can be extended to the entire domain Ω with zero i n ^ h; outside ωi in H 10 ðOÞ, we still use w to denote such extension in the follows. i P I h; n Step 3: u h; n ¼ u h;n1 þ i¼1 w^ i : The crucial issue for the implementation  Iof the proposed PPUM Steps 1-3 is the construction of the partition of unity fi i¼1 on Ω and the determination of the associatedsub-domains ωi. For this purpose, we define another regular triangulation  T H ðOÞ ¼ tH O with the mesh parameter H W h, and its related piecewise linear finite element space SH(Ω). For simplicity, we assume that TH(Ω) aligns with Th(Ω). A simple choice of the partition of unity is the piecewise linear Lagrange basis functions in SH(Ω). In this case, I is the number of vertices of TH(Ω) (including boundary vertices). For each vertex xi of the coarse grid, let o0i ¼ supp fi . We call o0i the patch of layer 0 defined on the ith vertex. Then we can define the patch of layer Li (Li ⩾ 1, an integer) based on vertex i recursively. That is: oLi i ¼

[

L 1

xj A oi i

o0j ;

and the vertices on the boundary are included. For a proper chosen Li, we denote oi ¼ oLi i , see Figures 1 and 2 for example. 1.00

Y

Figure 1. The case o0i

0.500

0.000 0.00000

0.500

1.00

A parallel partition of unity method

1.00

1951 Y

0.500

Figure 2. The case o1i

0.000 0.00000

0.500

1.00

4. Numerical experiments In this section, we construct two 2D numerical examples defined on a unit square Ω ¼ (0, 1) × (0, 1) to confirm the effectiveness of the scheme we proposed. The difference of the two numerical experiments lies in the choice of the exact solutions. In the first experiment, the exact solution is chosen in the form of product of polynomials and trigonometric function and T ¼ 1. In the second experiment, the exact solution is in the form of product of trigonometric functions and T ¼ 1. H For given another uniform triangulation  h mesh T (Ω), see Figure 3,Hwe construct h T ðOÞ ¼ tO with h o H such that T (Ω) and T h(Ω) are nested. Here we use the following piecewise linear finite element spaces: n o H S H ðOÞ ¼ vA C 0 ðOÞ : v9tH A P 1tH ; 8tH (10) O A T ðOÞ : O

O

1.00

Y

0.500

Figure 3. A triangulation TH(Ω)

0.000 0.00000

0.500

1.00

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In our numerical implementation, we fix H ¼ 1/8, and choose sub-domain ωi to be the patch of layer 1 defined on each vertex, that is Li ¼ 1 for all i, see Figure 4. The linear Lagrangian element (P1) is used for spacial discretization and backeard Euler scheme is adopted for temporal discretization. Since our scheme can be regarded as one approximation to the scheme of (5), we expect the errors of u in L2 – norm and H1 – norm to be k+h2 and k+h. In all experiments, we set b ¼ (2.0, 1.0)T and T ¼ 1. In addition, the results are obtained by using the software package FreeFem++ (FreeFem++, n.d.), and all the numerical results are obtained by the same computer. 4.1 Numerical experiment 1 In this experiment, we take the exact solution: uðx; y; t Þ ¼ x2 ð1xÞ2 yð1yÞð12yÞ cos ð2pt Þ: Then the initial condition, boundary conditions and f(x, y, t) in (1) are given by the exact solution. To compute the convergent rates of h, we choose the time step length k ¼ 0.001 which is small enough such that the entire error will not improve when k becomes further smaller. While for computing the convergent rates of k, we find h ¼ 1/64 is small enough. In Table I, we list the errors between the exact solution and the solutions of the PPUM we proposed and the convergence rates, with varying spacing h but fixed time 1.00

Y

Figure 4. Th(ωi)

0.500

0.000 0.00000

Table I. Errors and convergence rates in space of PPUM

0.500

1.00

1/h

:uu h;N :0

Rate

:uu h;N :1

Rate

8 16 24 32

3.0854e-4 8.1153e-5 3.7002e-5 2.1397e-5

1.9268 1.9369 1.9039

6.8535e-3 3.4959e-3 2.3401e-3 1.7576e-3

0.9712 0.9900 0.9950

step k. Apparently, the errors for u in the L2-norm and H1-norm are of the order of O(h2) and O(h), respectively, which agree with the expected results. For a comparison, we list the errors between the exact solution and the solutions of standard Galerkin method (SGM) and convergence rates in Table II. We can find that two schemes have the same orders of convergence and almost the same accuracy. In Table III, we list the errors and convergence rates with varying time step k but fixed spacing h ¼ 1/64. The numerical results show that the orders of convergence in time is O(k), which agree with the expected result. Finally, to show the effiency of the PPUM we proposed, we give the CPU time of SGM and PPUM in the entire time interval with different mesh size in Table IV. To get the time of the PPUM, we first compute the maximum value of the time costed by all the sub-problems at every time step, then sum them together and finally obtain the total time in the entire time interval. It is clear that the PPUM can save a lot of computing time compared with SGM.

A parallel partition of unity method 1953

4.2 Numerical experiment 2 In this numerical experiment, we take the exact solution: uðx; y; t Þ ¼ sin 2 ðpxÞ cos ðpxÞ sin ðpyÞ cos ð2pt Þ: The initial condition, boundary conditions and f(x, y, t) in (1) follow the exact solution.

1/h

:uuh;N :0

Rate

:uuh;N :1

Rate

8 16 24 32

3.0971e-4 8.1289e-5 3.6836e-5 2.1112e-5

1.9298 1.9522 1.9348

6.8427e-3 3.4958e-3 2.3400e-3 1.7575e-3

0.9689 0.9900 0.9950

k

:uu h;N :0

Rate

:uu h;N :1

Rate

1.1321 1.4062

7.4047e-3 3.4273e-3 1.49767e-3

1.1114 1.1944

0.1 0.05 0.025

1.0357e-3 4.7254e-4 1.7829e-4

1/h

k

Interval

SGM(s)

PPUM(s)

16 24 32

0.001 0.001 0.001

[0, 1] [0, 1] [0, 1]

73.828 166.718 300.698

10.547 23.507 45.635

Table II. Errors and convergence rates in space of SGM

Table III. Errors and convergence rates in time of PPUM

Table IV. Comparing the CPU time of SGM and the PPUM

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Table V. Errors and convergence rates of PPUM

Table VI. Errors and convergence rates of SGM

Table VII. Errors and convergence rates of PPUM

We first investigate the orders of convergence with respect to h and list the results in Table V. As expected, the errors of the PPUM approximation in L2 – norm and H1 – norm exhibit the second-order and first-order convergence rate, respectively. For a comparison, we also present the corresponding results of SGM in Table VI. From Tables V and VI, We can obtain the same conclusion as that from Tables I and II. Next we list the errors and orders of convergence in time in Table VII. The results show that the convergence rates in time are all O(k). In a conclusion, we propose a PPUM for solving the time-dependent convection-diffusion problem. Numerical results demonstrate that the PPUM not only reaches the same accuracy as the SGM but also saves a lot of computing time.

1/h

:uu h;N :0

Rate

:uu h;N :1

Rate

8 16 24 32

2.2151e-2 5.4349e-3 2.4620e-3 1.4169e-3

2.0271 1.9529 1.9205

4.8103e-1 2.4592e-1 1.6467e-1 1.2369e-1

0.9679 0.9892 0.9947

1/h

:uuh;N :0

Rate

:uuh;N :1

Rate

8 16 24 32

2.0716e-2 5.4593e-3 2.4721e-3 1.4144e-3

1.9240 1.9539 1.9410

4.8035e-1 2.4591e-1 1.6466e-1 1.2369e-1

0.9660 0.9892 0.9947

k

:uu h;N :0

Rate

:uu h;N :1

Rate

6.1612e-2 2.8434e-2 1.1251e-2

1.1156 1.3376

4.4323e-1 2.0474e-1 9.4776e-2

1.1143 1.1112

0.1 0.05 0.025

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