Parallel algorithm for convection diffusion system based on least squares procedure

Transcription

1 Zang et al. SpringerPlus (26 5:69 DOI.86/s RESEARCH Parallel algoritm for convection diffusion system based on least squares procedure Open Access Jiansong Zang *, Hui Guo 2, Hongfei Fu 3 and Yanzen Cang 4 *Correspondence: jszang@upc.edu.cn Department of Applied Matematics, Cina University of Petroleum, Qingdao 26658, Cina Full list of autor information is available at te end of te article Abstract Combining subspace correction metod wit least-squares finite element procedure, we construct ew overlapping domain decomposition parallel algoritm for solving te first-order time-dependent convection diffusion system. Tis algoritm is fully parallel. We analyze te convergence of approximate solution, and study te dependence of te convergent rate on te spacial mes size, time increment, iteration number and subdomains overlapping degree. Bot teoretical analysis and numerical results suggest tat only one or two iterations are needed to reac to given accuracy at eac time step. Keywords: Overlapping domain decomposition, Parallel subspace correction, Least-squares, Convection diffusion system Matematics Subject Classification: 65M55, 65M6, 65M2, 65M5 Background In tis paper, we consider te following initial-boundary value problem for timedependent convection diffusion system: c u t + σ + qu = f, x Ω, < t T, σ + A u + bu =, x Ω, < t T, u =, x Γ D, ( σ ν =, x Γ N, t T, u(x, = u (x, x Ω, were Ω is an open bounded domain R d ( d 3, wit a Lipscitz continuous boundary Γ = Γ D Γ N ; and ν is te unit vector normal to Γ N ; te flow field b = (b, b 2,..., b d T ; te source term q = q(x, t and exterior flow function f = f (x, t are some given functions; te coefficient c = c(x is positive function and te diffusion coefficient matrix A = (a(i, j d d is a symmetric uniformly positive definite matrix, i.e., tere exist some positive constants c and a suc tat d d a ξi 2 a ij (xξ i ξ j, c c(x, ξ R d, x Ω. (2 i= i,j= Tis type of partial differential equation arises in many important fields, suc as te matematical modeling of aerodynamics, porous medium fluid flow, fluid dynamics (e.g. 26 Te Autor(s. Tis article is distributed under te terms of te Creative Commons Attribution 4. International License (ttp://creativecommons.org/licenses/by/4./, wic permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to te original autor(s and te source, provide a link to te Creative Commons license, and indicate if canges were made.

2 Zang et al. SpringerPlus (26 5:69 Page 2 of 9 Euler equations, Navier-Stokes equations, meteorology, and semiconductor devices. Many numerical metods ave been establised to simulate tis problem, for example, finite element and finite difference metod, Eulerian Lagrangian localized adjoint metod Celia et al. (99. Te streamline diffusion finite element metod Huges and Brooks (979, least-squares mixed element metods Yang (999, 2, 22, Zang and Guo (22, Zang et al. (2 and Zang (29, and so on. Generally, tese numerical procedures result in a large scale of algebraic system, so it is very important and useful to develop effective parallel algoritms bot in engineering applications and matematical analysis. Recently domain decomposition parallel computation as become a powerful tool for solving a large scale system of partial differential equations. A lot of work as been done on domain decomposition parallel algoritms, for example, see Beilina (26, Bramble et al. (99, 99, Cai (989, Dolean et al. (28, 25, Dryja and Widlund (987, Lu et al. (99, Ma et al. (29, Tarek (28, Xu (989, 992, 2 and Yang (2. But many parallel algoritms based on overlapping domain decomposition are iterative algoritms so tat many iteration steps are needed to reac given accuracy, wic leads to muc more global amount of computational work. On te basis of te idea of te parallel subspace correction metod proposed by Xu (989, 992, 2, te first autor of tis paper and Yang establised ew parallel algoritm combined wit caracteristic finite element sceme, finite difference sceme and least-square sceme for one dimensional convection diffusion problem in Zang et al. (2 and Zang and Yang (2a, b, were bot teoretical analysis and numerical results suggest tat wen overlapping degree as a positive lower bound independent of mes size, only one or two iterative times is needed to reac te optimal convergence precision at eac time level. In tis paper, using te same tecnique as in Zang et al. (2, Zang and Yang (2 and Zang and Yang (2, we establis ew parallel algoritm for solving te convection diffusion system. Here te arbitrary dimensional problem is considered, unlike in Zang and Yang (2 only one dimensional model was studied. And te different least-squares finite element sceme from te one in Zang and Yang (2 is used to obtain te optimal L 2 -norm error estimate. Te partition of unity is applied to distribute te corrections in te overlapping domains reasonably in tis parallel algotitm. We analyze te convergence of approximate solution, and study te dependence of te convergent rate on te spacial mes size, time increment, iteration number and sub-domains overlapping degree. Bot teoretical analysis and numerical experiments indicate te full parallelization of te algoritms and very good approximate property. Parallel algoritm Trougout tis paper we use usual definitions and notations of Sobolev spaces as in Adams (975. Let W k,p (Ω (k, p be Sobolev spaces defined on Ω wit usual norms W k,p (Ω and H k (Ω = W k,2 (Ω. Define inner products as follows: (u, v = u(xv(xdx u, v L 2 (Ω, (σ, ω = Ω d (σ i, ω i σ, ω [L 2 (Ω] d, d 3. i=

3 Zang et al. SpringerPlus (26 5:69 Page 3 of 9 Introduce te spaces W ={ω [L 2 (Ω] d ; ω L 2 (Ω, ω ν = on Γ N } and V = {v H (Ω; v = on Γ D }. Make a time partition = t < t < < t M < t M = T and set τ n = t n t n and τ = max n M τ n. Let w n (x = w(x, t n. By use of te difference tecnique wit first-order accuracy to discretize te first-order system (, we can rewrite te system ( as follows [see Yang (999] c(x t u n (x + σ n (x + q n (xu n (x = f n (x + R n (x, x Ω, t σ n (x + A(x t u n (x + t (b n (xu n (x =, x Ω, u n (x =, x Γ D, σ n (x ν(x =, x Γ N, u (x = u (x, x Ω (3 were ( R n (x = c(x( t u n 2 u (x u t (x = O τ n t 2, t u n (x = (u n u n /τ n. To construct parallel subspace correction algoritm, we firstly make a domain decomposition. Assume tat {Ω i }N i= is on-overlapping domain decomposition of Ω. In order to obtain an overlapping domain decomposition, we extend eac subregion Ω i to a larger region Ω i suc tat Ω i Ω i Ω and dist( Ω i \ Ω, Ω i\ Ω H for eac i N, were H > is called as overlapping degree. Let T u and T σ be two families of quasiregular finite element partitions of te domain Ω suc tat te elements in te partitions ave te diameters bounded by u and σ, respectively. Assume tat T u,i = T u Ωi and T σ,i = T σ Ωi just are one finite element partition of Ω i for i N. Let W σ W, and V u V be piecewise r-degree and k-degree polynomial spaces defined on te partitions T σ and T u, respectively. Denote by à te inverse of A and define a bilinear form ( ((σ, w, (ω, v = c (cw ( σ + q n w, cv ( ω + q n v (Ã(σ + A w + b n w, ω + A v + b n v. Based on (3 and Yang (999, we get te standard least-squares finite element procedure: Least-squares sceme Given an initial approximation (, w W σ V u. For n =, 2,..., M, seek ( n, wn W σ V u suc tat (( n, wn, (ω, v ( = c (cwn f n, cv ( ω + q n v (Ã( n + A w n (ω, v W σ V u. + b n w n, ω + A v + b n v, (4 In te following part of tis section, we propose te parallel domain decomposition algoritm of te system (4. Define finite element sub-spaces:

4 Zang et al. SpringerPlus (26 5:69 Page 4 of 9 V i u = { v V u ; v = in Ω\Ω i }, i N and W i σ = { σ W σ ; σ = in Ω\Ω i }, i N. It is clear tat V u = V u + V 2 u + +V N u and W σ = W σ + W 2 σ + +W N σ. It is easily seen tat tere exists a finite open covering family {O i } N i= of te domain Ω suc tat O i Ω Ω i. We know tat tere exists a partition of unity {ϕ i } N i= (see Toselli and Widlund (25, Lemma 3.4 suc tat (a supp(ϕ i O i, ϕ i, ϕ i W r, CH r, i N; (b ϕ + ϕ 2 + +ϕ N = in Ω. Let ϕ i u and ϕ i σ be te nodal piecewise linear interpolation of ϕ i on te finite element meses T u and T σ, and I u and I σ be te interpolating operators on V u and W σ. Based on (4, we formulate te parallel subspace correction algoritm. Parallel algoritm Let m denote te iteration number at eac time step. Give an initial approximation (σ, u = (, w W σ V u. For n =, 2,..., M, seek (σ n, un W σ V u by four steps: Step. Set ( σ n, ũn n = (σ, u n and j :=. Step 2. For i =, 2,..., N, seek (εj i, ei j Wi σ V i u, in parallel, suc tat ((ε j i, ei j, (ω, v ( = c (cun f n, ci u (ϕ i u v +τ n ( I σ (ϕ i σ ω + q n I u (ϕ i u v n (Ã(σ + A u n + b n u n, I σ (ϕ i σ ω + A I u (ϕ i u v + b n I u (ϕ i u v ( ( σ j n, ũn j, (I σ (ϕ i σ ω, I u (ϕ i u v, Step 3. Set corrections (ω, v W σ V u. (5 σ n j N = σ j n + εj i, i= N ũn j =ũ n j + ej i. i= (6

5 Zang et al. SpringerPlus (26 5:69 Page 5 of 9 Step 4. If j < m, ten set j := j + and return te step 2; or set σ n = σ n m, un =ũn m and ten return back to te first step to start iteration at te next time step. Some lemmas and main result In te following sections, we denote by K and δ some general constants and small positive constants independent of te mes parameters H, σ u and τ, wic may be different at different occurrences. Let ( (ω, v 2 = c (cv ( ω + q n v, cv ( ω + q n v (Ã(ω + A v + b n v, ω + A v + b n v. In order to analyze te convergence of parallel algoritm, we introduce projection operators P i σ : W σ W i σ and Q i u : V u V i u suc tat ((P i σ ω, Q i u v, (ω, v = ((ω, v, (ω, v, (ω, v W i σ V i u i =, 2..., N. Now, we give some important lemmas wic are used to analyze te convergence of parallel algoritm. We assume tat finite element spaces W σ and V u ave te inverse property and approximate properties [see Ciarlet (978] tat tere exist some integers r, r, k >, suc tat, for q and ω H(div; Ω [W r+,q (Ω] d, inf ω W σ ω ω [L q (Ω] d r+ σ ω [W r+,q (Ω] d, inf ω W σ (ω ω L q (Ω r σ ω W r,q (Ω, inf v v L v q (Ω k+ u v V W k+,q (Ω, v L2 (Ω W k+,q (Ω. u Based on Teorem 3.3 in Yang (999, te following result can be read: Lemma Let (σ, u and ( n, wn be te solutions of ( and least-squares sceme, respectively. Ten tere olds te a priori error estimate max n u n w n L 2 (Ω + max σ n n n [L 2 (Ω] d {k+ u + r σ + τ}. (7 Lemma 2 [See Yang (2] For any function ϕ W, (Ω and ω W σ, we ave te following estimate ϕω I σ (ϕω [L 2 (Ω] d σ min( ϕ W, (Ω ω [L 2 (Ω] d, ϕ H (Ω ω [L (Ω] d, were d =, 2, 3.

6 Zang et al. SpringerPlus (26 5:69 Page 6 of 9 Lemma 3 For i N, we ave (I I σ (ϕ i σ ω [L 2 (Ω] d σ H ω [L 2 (Ω] d, ω W σ, (I I u (ϕ i u v L 2 (Ω u H v L 2 (Ω, v V u. (8 Proof Using Lemma 2, we know tat (I I σ (ϕ i σ ω [L 2 (Ω] d σ ϕ i σ W, (Ω ω [L 2 (Ω] d σ H ω [L 2 (Ω] d. Tis is te first inequality of (8. In addition, by using te tecnique of Teorem 3. in Yang (2, we can easily obtain (I I u (ϕ i u v L 2 (Ω u min( ϕ i u W, (Ω v L 2 (Ω, ϕi u H (Ω v L (Ω u ϕ i u W, (Ω v L 2 (Ω u H v L 2 (Ω. Tat is te second inequality of (8. Te proof of Lemma 3 is completed. Lemma 4 Te following estimate ( N (ψ, w, (ω, v ((ψ, w, (I σ (ϕ i σ P i σ ω, I u (ϕ i u Q i u v i= ( τ H + (ψ, w an (ω, v an H (9 olds for eac (ψ, w and (ω, v in W σ V u. Proof It is easily seen tat ((ψ, w, (I σ (ϕ i σ P i σ ω, I u (ϕ i u Q i u v = ((ψ, w, (ϕ i σ P i σ ω, ϕ i u Q i u v + ((ψ, w, ((I σ I(ϕ i σ P i σ ω, (I u I(ϕ i u Q i u v, and

7 Zang et al. SpringerPlus (26 5:69 Page 7 of 9 ((ψ, w, (ϕ i σ P i σ ω, ϕ i u Q i u v = ((P i σ (ϕ i σ ψ, Q i u (ϕ i u w, (ω, v [( c (cw ( ψ + q n w, (P i σ ω ϕ i σ ( c ψ ϕi σ, cq i u v ( (P i σ ω + q n Q i u v + (Ã(ψ + A w + b n w, A ϕi Qi u v ] ( ϕ i u w, P i σ ω + A (Q i u v + b n Q i u v and ((ψ, w, (ω, v = N i= ((ϕ i σ ψ, ϕ i u w, (ω, v. Hence we ave ((ψ, w, (ω, v = N ((ψ, w, (I σ (ϕ i σ P i σ ω, I u (ϕ i u Q i u v i= N (((I P i σ (ϕ i σ ψ, (I Q i u (ϕ i u w, (ω, v i= N ((ψ, w, ((I σ I(ϕ i σ P i σ ω, (I u I(ϕ i u Q i u v i= N [( τ n c (cw ( ψ + q n w, (P i σ ω ϕ i σ i= ( c ψ ϕi σ, cq i u v ( (P i σ ω + q n Q i u v + (Ã(ψ + A w + b n w, A ϕ i u Q i u v ] ( ϕ i u w, P i σ ω + A (Q i u v + b n Q i u v. ( Noting tat (I σ I(ϕ i σ P i σ ω, (I u I(ϕ i u Q i u v an { c(i u I(ϕ i u Q i u v L 2 (Ω i (I σ I(ϕ i σ P i σ ω L 2 (Ω i q n (I u I(ϕ i u Q i u v L 2 (Ω i + [ τ n (I σ I(ϕ i σ P i σ ω L 2 (Ω i ]} + A ((I u I(ϕ i u Q i u v L 2 (Ω i + bn (I u I(ϕ i u Q i u v L 2 (Ω i { ( u H Qi u v L 2 (Ω i τ σ n σ H Pi σ ω [L 2 (Ω i ] d + [ ]} σ τ n ( H Pi σ ω [L 2 (Ω i ] d + u u H Qi u v L 2 (Ω i + u H Qi u v L 2 (Ω i τ { H + Q i H u v L 2 (Ω i + ]} τ n [ P i σ ω [L 2 (Ω i ] d + Qi u v L 2 (Ω i ( τ H + (P i H σ ω, Q i u v an,ω i,

8 Zang et al. SpringerPlus (26 5:69 Page 8 of 9 we ave and N a n ((ψ, w, ((I σ I(ϕ i σ P i σ ω, (I u I(ϕ i u Q i u v i= N i= N = ( H + τ H [ N ] /2, (ψ, w an (P i σ ω, Q i u v 2,Ω i i= (((I P i σ (ϕ i σ ψ, (I Q i u (ϕ i u w, (ω, v i= ( H + N τ n i= (((I I σ (ϕ i σ ψ, (I I u (ϕ i u w, ((I P i σ ω, (I Q i u v τ H [ N ] /2 (ψ, w an ((I P i σ ω, (I Q i u v 2,Ω i i= [ ( c (cw ( ψ + q n w, (P i σ ω ϕ i σ ( + c ψ ϕi σ, cq i u v ( (P i σ ω + q n Q i u v + (Ã(ψ + A w + b n w, A ϕ i u Q i u v ] + ( ϕ i u w, P i σ ω + A (Q i u v + b n Q i u v [ N ] /2. τ n (ψ, w an (P i H σ ω, Q i u v 2,Ω i i= Substituting tese estimates into ( leads to (9. Tis ends te proof of Lemma 4. For parallel algoritm, we ave te following convergence result: Teorem Let (σ, u and (σ n, un are te solutions of te system ( and parallel algoritm, respectively. If 2m = O(τ, ten tere olds te following a priori error estimate max n u n u n L 2 (Ω + max σ n σ n n [L 2 (Ω] d { ( 2 H 2 + τ m } 2 H 2 + k+ u + r σ + τ, ( were = max( σ, u. Proof of Teorem It is easily seen tat parallel algoritm is also equivalent to use an iteration wit initial values (σ n, u n to solve te following equation: ( ˆσ n, ûn W σ V u suc tat for any (ω, v W σ V u

9 Zang et al. SpringerPlus (26 5:69 Page 9 of 9 (( ˆσ n, ûn, (ω, v = From (2 we ave ((σ n, un, (ω, v = ( c (cun f n, cv ( ω + q n v (Ã(σ n + A u n + b n u n, ω + A v + b n v. (2 ( c (cun f n, cv ( ω + q n v (Ã(σ n + A u n + ((σ n ˆσ n, un ûn, (ω, v. + b n u n, ω + A v + b n v (3 Let θ n = u n wn, ρn = w n un, π n = σ n n and ηn = n σ n. Subtracting (4 from (3, we can get ((π n, θ n, (ω, v = ( θ n, cv ( ω + q n v (Ã(π n + A θ n + b n θ n, ω + A v + b n v + ((σ n ˆσ n, un ûn, (ω, v. (4 Lemma 5 For parallel algoritm, we ave te estimate ( (σ n ˆσ n 2, un ûn H 2 + τ m 2 H 2 (σ n ˆσ n, un û n. (5 Proof From (5, we ave ((ε i j, ei j, (ω, v = ((ε i j, ei j, (Pi σ ω, Q i u v In addition, from parallel algoritm we can obtain te following equation Taking (ω, v = ( σ n j = (( ˆσ n σ n j, ûn ũ n j, (I σ (ϕ i σ P i σ ω, I u (ϕ i u Q i u v. (( σ n j ˆσ n, ũn j û n, (ω, v ( σ n j (( = (( σ j n ˆσ n, ũn j ûn, (ω N N, v + εj i, i= i= i= ˆσ n, ũn j û n in (7 and using Lemma 4, we ave e i j, (ω, v = (( σ j n ˆσ n, ũn j ûn, (ω, v ( N + (( ˆσ n σ j n, ûn ũn j, N I σ (ϕ i σ P i σ ω, I u (ϕ i u Q i u v. ( ˆσ n, ũn j û n 2 2 H 2 + τ H 2 ( σ j n ˆσ n, ũn j ûn 2. i= (6 (7 (8

10 Zang et al. SpringerPlus (26 5:69 Page of 9 Tus, we ave ( ( σ m n ˆσ n 2, ũn m ûn 2 H 2 + τ m H 2 ( σ n ˆσ n, ũn ûn 2. Tat is te inequality (5. Tis ends te proof of Lemma 5. (9 Hence, we need to estimate te bounds of σ n ˆσ n and un û n. Lemma 6 For parallel algoritm, we ave te following estimate ( ˆσ n σ n, û n un 2 a { n t n ( τ n t n t, w } 2 dt [ π n 2 t L 2 (Ω + θ n 2 L 2 (Ω ]. (2 Proof From (4 we ave (( ˆσ n σ n, û n un, (ω, v = (( n n, w n wn, (ω, v ( τ n c ( π n + q n θ n, cv ( ω + q n v τ n (Ã(b n b n θ n, ω + A v + b n v (2 Taking (ω, v = ( ˆσ n σ n, û n un in (2 and using te inequality ab δ a2 + δb 2, we can obtain ( ˆσ n σ n, û n un 2 a { n t n ( τ n t n t, w } 2 dt [ π n 2 t L 2 (Ω + θ n 2 L 2 (Ω ] + δ ( ˆσ n σ n, û n un 2 Hence, wen we coose sufficiently small δ, we can obtain te estimate (2. Tis ends te proof of Lemma 6. Finally, we prove Teorem. Proof Let (ω, v = (π n, θ n θ n in (4, we ave ((π n, θ n θ n, (π n, θ n θ n = τ n (Ã(π n π n, π n + A (θ n θ n + b n (θ n θ n ( τ n c ( π n + q n θ n, c(θ n θ n ( π n + q n (θ n θ n (Ã(b n b n θ n, π n + A (θ n θ n + b n (θ n θ n + ((σ n ˆσ n, un ûn, (π n, θ n θ n.

11 Zang et al. SpringerPlus (26 5:69 Page of 9 Since ((π n, θ n θ n, (π n, θ n θ n τ n (Ã(π n π n, π n = (c(θ n θ n, θ n θ n [(A (θ n θ n, (θ n θ n ( + (Ãb n (θ n θ n, θ n θ n c π n, π n ( ] c qn (θ n θ n, q n (θ n θ n (Ãπ n, π n 2 [(Ãπ n, π n (Ãπ n, π n + (Ã(π n π n, π n π n ] ( + 2τ n c (c(θ n θ n π n, q n (θ n θ n + 2τ n (Ã + (θ n θ n, b n (θ n θ n, we ave (c(θ n θ n, θ n θ n [(A (θ n θ n, (θ n θ n + (Ãb n (θ n θ n, θ n θ n ( c π n, π n ( ] c qn (θ n θ n, q n (θ n θ n (Ãπ n, π n 2 [(Ãπ n, π n + (Ã(π n π n, π n π n ] = τ n 2 (Ãπ n, π n (Ã(π n π n, A (θ n θ n + b n (θ n θ n ( τ n c ( π n + q n θ n, c(θ n θ n ( π n + q n (θ n θ n (Ã(b n b n θ n, π n + A (θ n θ n + b n (θ n θ n + ((σ n ˆσ n, un ûn, (π n, θ n θ n. (22 Next, we estimate te terms on te rigt-and side of te error equation (22. It is clear tat and τ n (Ã(π n π n, A (θ n θ n τ n [(Ã(π n π n, π n π n + (A (θ n θ n, (θ n θ ] n 2

12 Zang et al. SpringerPlus (26 5:69 Page 2 of 9 τ n (Ã(π n π n, b n (θ n θ n + τn ( c ( π n + q n θ n, c(θ n θ n ( π n + q n (θ n θ n (Ã(b n b n θ n, π n + A (θ n θ n + b n (θ n θ n + ((σ n ˆσ n, un ûn, (π n, θ n θ n {( 2 τ n H 2 + τ t n ( H 2 m σ t n t, w 2 dt [ θ n 2 t L 2 (Ω + (θ n θ n 2 [L 2 (Ω] d + π n 2 [L 2 (Ω] d + π n 2 [L 2 (Ω] d + π n 2 L 2 (Ω ] } + δ[ θ n θ n 2 L 2 (Ω + τ 2 n π n 2 L 2 (Ω ]. Substituting te above estimates into (22 and ten summing it up from to n, we get π n 2 [L 2 (Ω] d + n ] τ j [ t θ j 2 L 2 (Ω + τ j t θ j 2 [L 2 (Ω] d + τ j π j 2 L 2 (Ω j= ( 2 H 2 + τ m t n ( σ H 2 t, w 2 dt + t + θ j 2 L 2 (Ω + τ 2 j t θ j 2 [L 2 (Ω] d ]}. n τ j [ π j 2 [L 2 (Ω] d j= (23 Applying a known inequality n θ n 2 L 2 (Ω θ 2 L 2 (Ω + δ τ j t θ j 2 L 2 (Ω + K j= n τ j θ j 2 L 2 (Ω j= and discrete Gronwall s lemma to (23, we derive tat max n ( θ n L 2 (Ω + max π n 2 n [L 2 (Ω] d H 2 + τ m 2 H 2. (24 Using Lemma, we can obtain te estimate (. Te proof of Teorem is complete. Numerical results As in Zang and Yang (2, we first consider te one dimensional convection diffusion problem: u t u a 2 x 2 b u + u = f x [, ], t T. x We divide [ te domain ] [, ] into tree sub-domains: Ω = Ω 2 = 3 H 2, H 2 and Ω 3 = Fig.. (25 [ ], 3 + H 2, [ 23 H 2, ], were H is te overlapping degree (see

13 Zang et al. SpringerPlus (26 5:69 Page 3 of 9 Ω Ω 2 Ω 3 Fig. Te sub-domains of Ω We use piecewise linear polynomial spaces, set u = σ = and take te linear unit decomposition functions as in Zang et al. (2. We define te L 2 -norm error as follows: (e, E 2 2 = max n and te L -norm error (e, E = max n ( en, E n. u n u n L 2 (Ω + max σ n σ n n L 2 (Ω, Experiment I In tis experiment, te exact solution is cosen as u = e t sin 2 πx. Set T =, and b =. For different parameters a,, τ and te iterative number m at eac time step, we give L 2 -norm errors and te L -norm errors in Tables, 2 and 3. Tese numerical results suggest tat we can get a good result for convection diffusion problem using parallel algoritm, even iterating only one or two cycle at eac time step. Moreover, tese numerical results also imply tat te errors caused by decomposing domain decrease as te discretization parameters and τ decrease and increase as te overlapping degree H becomes small, wic are coincided wit our teoretical result. Table H = 6, = τ m a = a = e 2 a = e e 2.925e 8.68e e e 3.542e 2 4.9e e e e e 3.33e e 2.75e e e e e e 2.77e 8.643e e e e e e 8.643e e e e 2.e e e e e e 3.753e 2.523e 4.935e e e e e e e e e e e e e e e e e e e e e e e e e 3 2.6e 2.823e e e e e e e e e e e e 2.824e e e 3 4.7e e e 2.824e e e e e e 2.824e e 3 * Te numerical results by least-squares algoritm

14 Zang et al. SpringerPlus (26 5:69 Page 4 of 9 Table 2 H = 2, = τ m a = a = e 2 a = e e 2.925e 8.68e e e 3.542e e e 8.774e e e 3.664e e 2.939e e e e 3.537e e e e e e e e e e e e e 2.e e e e e e e 2.595e e e e e e e 2 4.9e e e e e e 2 4.9e e 2 3.6e e e e 2 4.9e e 2 3.6e e e e e 3 2.6e 2.823e e 3.368e e e e 2.825e e e e e e e e e e e e e e e e e e e e 3 * Te numerical results by least-squares algoritm Table 3 H = 24, = τ m a = a = e 2 a = e e 2.925e 8.68e e e 3.542e e 4.68e 9.4e e e e e 4.58e e e e 3.534e e e 9.836e e e e e e 9.64e e e e 2.e e e e e e 3.58e e 4.866e e e e e 2.75e e e e e e e e e e e e e e e e e e e 3 2.6e 2.823e e e 2.2e e e 2.825e e e 2.444e 2.35e e e e e e 2.4e e 2.826e e e 2.427e 2.4e e 2.826e e 3 * Te numerical results by least-squares algoritm

15 Zang et al. SpringerPlus (26 5:69 Page 5 of 9 Experiment II As in Zang and Yang (2, we select te rigt-and side function wit complex structure and te initial condition as follows: { f (x, t = e t 5 x cos(8πxt sin 2 (7πx, u (x =. Coosing H = /2, = τ = /, b =, and a = e 4, we observe numerical results at different time (see Figs. 2, 3. We use * to denote u and σ, te values of te parallel algoritm and use - to denote w and, te values of least-squares algoritm. Tese figures clearly sow tat u, σ approximate to w and at different time, respectively, wic is coincided wit our teoretical analysis. Next, we consider te two dimensional convection diffusion problem: u + σ + u = f, x Ω, < t < T, t (26 σ + A u + bu =, x Ω, < t < T, were Ω =[, ] [, ], A = ae, E is te unit matrix, and b = (, T. We divide Ω into four sub-domains: Ω =[,.6] [,.6], Ω 2 =[.4, ] [,.6], Ω 3 =[,.6] [.4, ], Ω 4 =[.4, ] [.4, ], see Fig. 4. In tis section, we use piecewise linear polynomial spaces. And We take te linear unit decomposition functions {ϕ i } 4 i= as follows:, (x, y [,.4] [,.4], 3 5y, (x, y [,.4] [.4,.6], ϕ (x, y = 3 5x, (x, y [.4,.6] [,.4], (x + y, oterwise, 4, (x, y [.6, ] [,.4], 3 5y, (x, y [.6, ] [.4,.6], ϕ 2 (x, y = 5x 2, (x, y [.4,.6] [,.4], 4 5 (x y, oterwise, 4, (x, y [,.4] [,.4], 5y 2, (x, y [,.4] [.4,.6], ϕ 3 (x, y = 3 5x, (x, y [.4,.6] [.6, ], (y x, oterwise, 4, (x, y [.6, ] [.6, ], 5y 2, (x, y [.6, ] [.4,.6], ϕ 4 (x, y = 5x 2, (x, y [.4,.6] [.6, ], 5 (x + y, oterwise. 4 Experiment III Here we still select te same rigt-and side function wit complex structure and te initial condition as in Zang and Yang (2, { f (x, t = e y 3 x 2 2t sin(3πx 6y + t 2 cos(4πyt, u (x =.

16 Zang et al. SpringerPlus (26 5:69 Page 6 of 9 H=/2,= τ =/, m=2,t=. 6 H=/2,= τ=/,m=2,t=.5 w 5 4 w u u ρ ρ 5 4 σ σ Fig. 2 Numerical results at time T =., H=/2,=τ=/,m=2,T=2 4 H=/2,=τ =/,m=2,t=2.5 w w 2 u 2 u ρ 2 σ 2 ρ σ Fig. 3 Numerical results at time T = 2., Set H =.2, = τ = /4, and a = e 2, T =., m =. We can get Figs. 5, 6 and 7. Tese results suggest tat te values u, σ = (σ, σ 2 by parallel algoritm approximate to w and te values ρ = (ρ, ρ2 by least-squares sceme respectively, wic implies tat our metod is valid for two-dimensional problem.

17 Zang et al. SpringerPlus (26 5:69 Page 7 of 9 (, Ω 3 Ω 4 (,.6 (,.4 Ω Ω 2 (, Ω (.4, (.6, (, Fig. 4 Te sub-domains of Ω t=., w t=., u Fig. 5 Te values of w and u at T = Conclusions In tis paper, combined subspace correction metod wit least-squares mixed element procedure, ew class of parallel domain decomposition algoritm is proposed to solve convection diffusion problem. Te convergence of approximate solution, and te dependence of te convergent rate on te spacial mes size, time increment, iteration number and sub-domains overlapping degree are studied. Bot teoretical analysis and numerical experiments indicate te full parallelization of te algoritms and very good approximate property.

18 Zang et al. SpringerPlus (26 5:69 Page 8 of 9 t=.,ρ t=.,σ Fig. 6 Te values of ρ and σ at T = t=., ρ 2 t=., σ Fig. 7 Te values of ρ 2 and σ 2 at T = In fact, toug we consider te convection diffusion problem in tis paper, we can extend our metod to oter complex problems, e.g. saltwater intrusion problem, aerodynamic problems, nuclear waste disposal, etc., wic are our future work. Autors contributions JSZ designed te algoritm and wrote te main manuscript text. HFF and HG did some te experiments and prepared all te figures. All te autors contributed to te discussion of te results. All autors read and approved te final manuscript. Autor details Department of Applied Matematics, Cina University of Petroleum, Qingdao 26658, Cina. 2 Department of Computational Matematics, Cina University of Petroleum, Qingdao 26658, Cina. 3 Department of Pure Matematics, Cina University of Petroleum, Qingdao 26658, Cina. 4 Department of Matematics, Beijing University of Cemical Tecnology, Beijing 29, Cina. Acknowledgements Jiansong Zang s work was partially supported by te National Nature Science Foundation of Cina (2684, 4588, Cina Scolarsip Council, and te Natural Science Foundation of Sandong Province (ZR24AQ5, te Fundamental Researc Funds for te Central Universities and te Young Talent Attraction fellowsip from te Brazilian

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