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Two-grid scheme for characteristics finite-element solution of nonlinear convection diffusion problems
Applied Mathematics and Computation 165 (2005) 419–431 www.elsevier.com/locate/amc
Two-grid scheme for characteristics finite-element solution of nonlinear convection diffusion problems q Xin-qiang Qin
a,b,*
, Yi-chen Ma
a
a
b
School of Sciences, Xi’an Jiaotong University, Xi’an 710049, China School of Sciences, Xi’an University of Technology, No. 5 Jinhua Nanlu, Xi’an 710048, China
Abstract A two-grid scheme for characteristics finite-element solution of nonlinear convection-dominated diffusion equations was constructed. The L2 and H1 error estimates of the scheme were derived. A numerical example was also presented. The scheme involves solving one small, nonlinear problem on the coarse-grid system, one linear problem on the fine-grid system. The error estimates and the numerical example show that the new scheme is efficient to the nonlinear convection-dominated diffusion equations. 2004 Elsevier Inc. All rights reserved. Keywords: Convection–diffusion equations; Characteristics finite-element; Two-grid scheme
q
This work supported by National Science Foundation of China grants 10371096. Corresponding author. Address: School of Sciences, XiÕan University of Technology, No. 5 Jinhua Nanlu, XiÕan 710048, China. E-mail address: [email protected] (X.-q. Qin). *
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.021
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1. Introduction Convection–diffusion equations occur in numerous applications, noteworthy among them being mathematical models of environment pollution and neutron transportation. In this paper we discuss the nonlinear form 8 ou ou o ou > < cðxÞ ot þ bðxÞ ox ox aðxÞ ox ¼ f ðu; x; tÞ; ðx; tÞ 2 I ð0; T ; ð1:1Þ uðx; tÞ ¼ 0; ðx; tÞ 2 oI ð0; T ; > : uðx; 0Þ ¼ u0 ðxÞ; x 2 I; we use the modified method of characteristics finite-element developed by Douglas–Russell [1], to discrete in space with equal-order accuracy in u. To linearize the resulting discrete equations, we use a two-grid scheme, which allows us to iterate on a grid much coarser than that used for the final solution. We owe the impetus for using a two-grid approach to Xu [2,3,8], who demonstrates the possibility of mapping a fine-grid Galerkin finite-element problem onto a coarse grid. In that context, one solves the nonlinear problem via Newton-like iterations. After convergence on the coarse grid, one then extrapolates back to the fine grid using a Taylor expansion. Dawson and Wheeler [4] extend the method to nonlinear reaction–diffusion equations. Li Wu and Myron [5] extend it to another nonlinear reaction–diffusion equations. The remainder of the paper is organized as follows: Section 2 describes the algorithm. Section 3 gives an error analysis establishing the methodÕs convergence. Section 4 presents some computational results confirming the methodÕs utility, and Section 5 draws conclusions. We assume that the coefficients a, b and c are bounded and that d bðxÞ (H1) 0 < a0 6 aðxÞ 6 1; 0 < c0 6 cðxÞ 6 1; j bðxÞ j þ j dx ðcðxÞÞj 6 C; x 2 I cðxÞ 2 o f (H2) j of j þ j j 6 C; x 2 I 2 ou ou
Throughout this paper, we shall use the letter C to denote a generic positive constant which may for different values at its different occurrences. Let Wm,p(X) denote the Sobolev space on I. And the norm is defined by 8 1=p > P > p < kDa vkLp ðXÞ ; 1 6 p < 1; a6m kvkW m;p ¼ > > : max esssupjDa vj; p ¼ 1: a6m
m,2
We write W (X) = Hm(X), kvkHm(X) = kvkm, and kvkL2(X) = kvk0 = kvk and also assume that the solution u of (1.1) satisfies: (H3) u 2 L1(0,T;Hr + 1(I)), 2 L2 ð0; T ; H rþs Þ; r ¼ 1 if s = 1 and r P 2 if s = 0, (H4) ou ot 2 (H5) oot2u 2 L2 ð0; T ; L2 ðIÞÞ.
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2. Description of the method 2.1. Characteristics finite-element discretization We begin by briefly reviewing the characteristics finite-element discretization of the problem (1.1). Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðxÞ ¼ c2 ðxÞ þ b2 ðxÞ ð2:1Þ and let the characteristic direction associated with the operator cut + bux be denoted by s = s(x,t), where o cðxÞ o bðxÞ o ¼ þ : ð2:2Þ os wðxÞ ot wðxÞ ox Then, Eqs. (1.1) can be put in form 8 ou o ou > > aðxÞ ¼ f ðu; x; tÞ; x 2 I; t 2 ð0; T ; > wðxÞ < os ox ox ð2:3Þ uðx; tÞ ¼ 0; x 2 oI t 2 ð0; T ; > > > : uðx; 0Þ ¼ u0 ðxÞ; x 2 I: We introduce the Sobolev space V ¼ H 10 ðIÞ, and the notation (u,v) = Iu(x)v(x)dx,"u,v2V. Let Aðu; vÞ ¼ ða ou ; ovÞ. Then, multiplying the first ox ox equation of (2.1) by any v 2 V and applying Green formula, (2.3) can be written in the equivalent variational form 8 < w ou ; v þ Aðu; vÞ ¼ ðf ðuÞ; vÞ; v 2 V ; os ð2:4Þ : Aðuð0Þ u0 ; vÞ ¼ 0; v 2 V : We consider a time step Dt and approximate the solution at times tn = nDt, n = 1, 2, . . ., N, Dt ¼ NT . The characteristic derivative is approximated in the following way at t = tn n ou uðx; tn Þ uðx; tn1 Þ uðx; tn Þ uðx; tn1 Þ : ð2:5Þ w wðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cðxÞ os Dt 2 ðx xÞ þ Dt2 Namely, a backtracking algorithm is used to approximate the characteristic derivative. x ¼ x bðxÞ Dt is the foot (at level tn1) of the characteristic correcðxÞ sponding to x at the head (at level tn) (see Fig. 1).
Fig. 1. The direction of characteristic.
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Let the mesh size of the grid be h, we denote this grid by Dh and call it the fine grid. Vh be a finite-element subspace of V ˙ W1,1(I). The characteristics finite-element method for (2.4) is defined: For n = 1, 2, . . ., N, find unh 2 V h such that 8 n > un1 < c uh h ; v þ Aðunh ; vÞ ¼ ðf ðunh Þ; vÞ; v 2 V h ; Dt ð2:6Þ > : 0 Aðuh u0 ; vÞ ¼ 0; v 2 V h ; where
bðxÞ Dt; tn1 : uhn1 ðxÞ ¼ uh ðx; tn1 Þ ¼ uh x unh ¼ uh ðtn Þ; cðxÞ
ð2:7Þ
The initial approximation u0h in Vh can be defined as any reasonable approximation of u0 such as the interpolation of u0 in Vh. The (2.6) determines funh g uniquely in terms of the data u0. 2.2. Linearization via the two-grid method To solve the system (2.6), we use the Newton-like iterations. The idea is to devote all of the effort of nonlinear iteration to coarse-grid problems. Thus the iterations yield approximate finite-element solutions unH on a coarse subgrid DH having mesh size DH Dh,H > h and corresponding space VH. To approximate unh on Dh, we then execute iteration of the fine-grid system, using the converged coarse-grid solution unH together with the heuristic n unh ¼ unH þ enh ; uh ^ ð2:8Þ f ðunh Þ f ðunH Þ þ f 0 ðunH Þenh : This tactic yields the following system of equations for the fine-grid solution: 8 n n eh uH un1 > H n 0 n n n > ;v > c ; v þ Aðeh ; vÞ ðf ðuH Þeh ; vÞ ¼ ðf ðuH Þ; vÞ c < Dt Dt n > > AðuH ; vÞ; 8v 2 V h ; > : Aðu0h u0 ; vÞ ¼ 0; v 2 V h : ð2:9Þ Namely, the two-grid method has two stages: Stage 1: On coarse-grid DH, solve the following nonlinear system for unH 2 V H : 8 n un1 < c uH H ; v þ Aðun ; vÞ ¼ ðf ðun Þ; vÞ; v 2 V H; H H Dt ð2:10Þ : 0 AðuH u0 ; vÞ ¼ 0; v 2 V H :
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Stage 2: On the fine-grid Dh, solve the following linear system for ^unh 2 V h ! 8 n1 n > ^ ^ u u < c h h ; v þ Aðun ; vÞ ¼ ðf ðun Þ þ f 0 ðun Þð^un un Þ; vÞ; h
Dt
> :
Aðu0h u0 ; vÞ ¼ 0;
H
H
h
H
423
v 2 V h;
v 2 V h: ð2:11Þ
This linearization is efficient if the coarse-grid DH be coarser than the finegrid Dh without sacrificing the accuracy associated with the fine-grid solution unh . The analysis in Section 3. demonstrate that, as with the Galerkin-based scheme proposed by Xu [2], one can choose H = O(h4/7) and still retain the error estimates associated with the characteristics finite-element scheme. Numerical examples in Section 4. confirm the efficiency to solve the nonlinear problems with two-grid method. 3. Convergence analysis The two-grid method affords a remarkably efficient linearization. It is possible to execute all of the Newton-like iterations on very coarse grids, then use the heuristic (2.9) to obtain an accurate fine-grid solution in one additional step. We devote the rest of this section to the analysis of the two-grid scheme. We begin by describing the discretization space and associated projection operator. We then give a convergence analysis for the characteristics finite-element method, using the key parts of this argument in the analysis of the two-grid scheme. In what follows, we consider (1.1) on I. For v 2 Hr + 1 the following approximation property holds: inf ðkv vh k þ hkv vh k1 Þ 6 Chrþ1 kvkrþ1 ;
vh 2V h
ð3:1Þ
where r P 1. Let wh: [0,T] ! Vh, satisfy Aðu wh ; vÞ ¼ 0;
v 2 V h:
ð3:2Þ
Set g = u wh, n = uh wh, then u uh = gn. It is well known that [1], for p = 2,1 and 1 6 s 6 r + 1, kgkLp ð0;T ;L2 ðIÞÞ þ hkgkLp ð0;T ;H 1 ðIÞÞ 6 Chs kukLp ð0;T ;H s ðIÞÞ for r P 2 and 1 6 s 6 r + 1 og og s ou þ h 6 Ch : ot 2 ot 2 ot L2 ð0;T ;L2 ðIÞÞ L ð0;T ;H 1 ðIÞÞ L ð0;T ;H s1 ðIÞÞ
ð3:3Þ
ð3:4Þ
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The inverse property on Vh holds [6], namely, for vh 2 Vh, kvh k1 6 h1=2 kvh k:
ð3:5Þ
And we also have the approximation property [7]: kun wnh k1 6 kun krþ1;1 hrþ1 :
ð3:6Þ
3.1. Error estimates for the characteristics finite-element We now estimate the errors. From (3.3) and (3.4), to obtain error bounds for uuh, it suffices to estimate n. Theorem 1. Let u and uh be the respective solutions of (2.4) and (2.6). Under assumptions (H1–H5). We have the error estimate " o2 u n n max ku uh k 6 C Dt ot2 2 16n6N L ð0;T ;L2 Þ !# ð3:7Þ ou rþ1 þh kukL1 ð0;T ;H rþ1 ðIÞÞ þ os L2 ð0;T ;H rþs ðIÞÞ 6 CðDt þ hrþ1 Þ for sufficient small Dt, where s = 1 if r = 1 and s = 0 if r > 1. Proof. At t = tn, subtracting (2.4) from (2.6) and let v = nn to give c
nn nn1 n oun un un1 n ; n þ Aðnn ; nn Þ ¼ w c ;n Dt os Dt ! n n1 n1 n1 n1 g g g g nn1 n n n n þ c ;n þ c ;n c ;n Dt Dt Dt þ ðf ðun Þ f ðunh Þ; nn Þ:
ð3:8Þ
At any point x2I, by the Taylor theorem, un ðxÞÞðun ðxÞ unh ðxÞÞ f ðun ðxÞÞ f ðunh ðxÞÞ ¼ f 0 ð~
ð3:9Þ
~n ðxÞ. Therefore for some value u un ðxÞÞðun ðxÞ unh ðxÞÞ; nn Þ ðf ðun Þ f ðunh Þ; nn Þ ¼ ðf 0 ð~ un ðxÞÞgn ; nn Þ ðf 0 ð~un ðxÞÞnn ; nn Þ: ¼ ðf 0 ð~
ð3:10Þ
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425
Consequently, from Eq. (3.8), n n nn1 n oun un un1 n ; n þ Aðnn ; nn Þ ¼ w c ;n c Dt os Dt ! n n1 n1 n1 n1 g g g g nn1 n n n n þ c ;n þ c ;n c ;n Dt Dt Dt þ ðf 0 ð~ un ðxÞÞgn ; nn Þ ðf 0 ð~ un ðxÞÞnn ; nn Þ T 1 þ T 2 þ T 3 þ T 4 þ T 5 þ T 6 : ð3:11Þ We can estimate the left terms of (3.11) by aða bÞ P 12 ða2 b2 Þ and (H1) n n nn1 n 1 a0 2 ½ðcnn ; nn Þ ðcnn1 ; nn1 Þ þ knn k1 ; n þ Aðnn ; nn Þ P c 2Dt Dt 2 ð3:12Þ and use the results provided in [1] to the right terms T1 T4, we have 2 2 2 2 o u o u n 2 n 2 jT 1 j 6 C Dt þ ekn k 6 C os2 2 n1 n 2 os2 2 n1 n 2 Dt þ ekn k1 ; L ðt ;t ;L Þ L ðt ;t ;L Þ ð3:13Þ C og 2 jT 2 j 6 þ eknn k1 ; Dt ot L2 ðtn1 ;tn ;H 1 Þ
ð3:14Þ
2
2
ð3:15Þ
2
2
ð3:16Þ
jT 3 j 6 Ckgn1 k þ eknn k1 ; jT 4 j 6 Cknn1 k þ eknn k1 for any positive constant e. And from (H2), we have 2
2
jT 5 j 6 Cðkgn k þ knn k Þ; 2
jT 6 j 6 Cknn k :
ð3:17Þ ð3:18Þ
Choosing proper e, inequalities (3.12)–(3.18) can be combined with (3.11) to give the recursion relation 1 a0 2 ½ðcnn ; nn Þ ðcnn1 ; nn1 Þ þ knn k1 2Dt " 2 2 2 o u 1 og n 2 n1 2 6 C kn k þ kn k þ Dt 2 þ os L2 ðtn1 ;tn ;L2 Þ Dt ot L2 ðtn1 ;tn ;H 1 Þ # 2
2
þkgn k þ kgn1 k :
ð3:19Þ
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It follows from (2.6) and (3.2) that n0 = 0. If we multiply (3.19) by 2Dt, sum over n, and apply the discrete Gronwall lemma, it follows that " 2 # o2 u og max kn k 6 C Dt 2 þ þ kgkL1 ð0;T ;L2 Þ 16n6N os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H 1 Þ " !# o2 u ou rþ1 ; þh kukL1 ð0;T ;H rþ1 ðIÞÞ þ 6 C Dt 2 os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H rþs ðIÞÞ n
ð3:20Þ
which, together with un unh ¼ gn nn , (3.3) and (3.4) yields the desired result (3.7). h An optimal order estimate for u uh in H1(I) can be derived in a similar fashion. We can give the theorem below: Theorem 2. Let u and uh be the respective solutions of (2.4) and (2.6). Under assumptions (H1–H5), We have the error estimate n
max ku
16n6N
unh k1
" 2 o2 u 2 6 C Dt þ hr kukL1 ð0;T ;H rþ1 ðIÞÞ os2 2 2 L ð0;T ;L Þ !# ou þ 6 CðDt þ hr Þ ot L2 ð0;T ;H r ðIÞÞ
ð3:21Þ
for sufficient small Dt and r P 2. In the proof, the embedding theorem was used.
3.2. Error estimates for the two-grid method We now analyze the two-grid method for iteratively solving the nonlinear system (2.6). This scheme is attractive because it requires nonlinear iteration only on the coarse grid, give the fine-grid solution in a single, linear, Newton-like step. The following theorem gives the error: Theorem 3. Let u and ^ uh be the respective solutions of (2.4) and (2.11). Under assumptions (H1–H5), we have unh k 6 CðDt þ hrþ1 þ H 2rþ3=2 Þ max kun ^
16n6N
for Dt sufficiently small, where r P 1.
ð3:22Þ
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427
n Proof. To estimate the error, set gn ¼ un wnh ,^ n ¼ ^unh wnh . At t = tn. Subtractn ing Eqs. (2.11) from itÕs respective counterparts (2.4) and choosing v ¼ ^n , we have ! n n1 n n n1 ^ n n n ^ ^n ; n ^ þ Aðn ^n Þ ¼ w ou c u uh ; ^nn ;n c Dt os Dt 0 1 n1 n ^nn1 ^n n g gn1 ^n gn1 gn1 ^n ;n þ c ; n @c ; ^n A þ c Dt Dt Dt n þ ðf ðun Þ f ðunH Þ f 0 ðunH Þð^ un unH Þ; ^ n ÞÞ:
A Taylor expansion of f about
ð3:23Þ
unH
yields 1 2 uÞðun unH Þ ð3:24Þ f ðun Þ ¼ f ðunH Þ þ f 0 ðunH Þðun unH Þ þ f 00 ð~ 2 for some function ~ u. From (3.22) and (3.23), we get ! n n1 ^ ^ n n n oun un uhn1 ^n n n ^ ^ ^ c ; n þ Aðn ; n Þ ¼ w c ;n Dt os Dt 0 1 n1 n n1 ^nn1 ^n n g gn1 ^n g gn1 ^n ;n þ c ; n @c ; ^n A þ c Dt Dt Dt n n 1 00 2 ðf ð~uÞðun unH Þ ; ^n þ ðf 0 ðunH Þgn ; ^ n Þ ðf 0 ðunH Þnn ; nn Þ þ 2 7 X T 0i : ð3:25Þ i¼1
Then arguments similar to those used for the characteristics finite-element error estimate (Theorem 1) yield n n n1 n1 n 2 n1 2 1 a0 n 2 ½ðc^ n ;^ n k1 6 C k^n k þ k^n k n Þ ðc^ n ;^ n Þ þ k^ 2Dt 2 2 2 o u 1 n 2 n1 2 og þ Dt os2 2 n1 n 2 þ Dt ot 2 n1 n 1 þ kg k þ kg k L ðt ;t ;H Þ L ðt ;t ;L Þ þ kðun unH Þ2 k2 : ð3:26Þ Now multiplying both sides of (3.26) 2Dt and summing over n and apply the discrete Gronwall lemma, it follows that: " 2 og o2 u n ^ þ max kn k 6 C Dt 2 16n6N os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H 1 Þ # 2
þkgkL1 ð0;T ;L2 Þ þ max kðun unH Þ k ; 16n6N
^0 ¼ 0 is used. where the n
ð3:27Þ
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Let h = H in (3.7) and (3.20) and using (3.5) and (3.6), we have 2
kðun unH Þ k 6 kun unH k1 kun unH k 6 ðkun wnH k1 þ kwnH unH k1 Þkun unH k 6 Cðkun krþ1;1 H rþ1 þ H 1=2 kwnH unH kÞkun unH k
ð3:28Þ
6 CðH rþ1 þ H 1=2 ðH rþ1 þ DtÞÞðH rþ1 þ DtÞ 2
6 CH 1=2 ðH rþ1 þ DtÞ 6 CðH 2rþ3=2 þ DtÞ: n
Combining un ^ unh ¼ gn ^ n together with (3.3) and (3.4) and (3.27) and (3.28) yields the results (3.22). h An optimal order estimate for u ^ uh in H1(I) can be derived in a similar fashion. We can give the theorem below: Theorem 4. Let u and uh be the respective solutions of (2.4) and (2.11).Under assumptions (H1H5). We have the error estimate unh k1 6 CðDt þ hr þ H 2r Þ max kun ^
16n6N
ð3:29Þ
for sufficient small Dt and r P 2. Theorems 1 and 3 demonstrate a remarkable fact about two-grid scheme: when we iterate on a very coarse-grid DH and still get good approximations by taking one iteration on the fine-grid Dh. Specifically, for h = H7/4 as r = 1, the two-grid scheme solution ^ uh has the same accuracy compared with that of the nonlinear iteration solution uh. 4. Numerical example To illustrate the effectiveness of the two-grid linearization, we examine the following simple test problem: 8 ou ou o ou > 2 > > < ot þ ox ox aðxÞ ox ¼ u þ Gðx; tÞ; x 2 ½0; 1; t 2 ð0; T ; ð4:1Þ uð0; tÞ ¼ 1; uð1; tÞ ¼ 0; t 2 ð0; T ; > > > : uðx; 0Þ ¼ 1 x; x 2 ½0; 1; where a(x) = 0.001. The G(x,t) is determined by the exact solution u(x,t) = (1 x)ext. We solve (4.1) by Dt = 0.125 · 105 from t = 0 to 0.20. For H = 24, h = 7/4 H = 27, we use the two-grid scheme presented in this paper. First, we give the coarse-grid partition DH with H = 24 (Fig. 2) and get the nonlinear itera-
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429
Fig. 2. The coarse-grid DH with H = 24.
Fig. 3. The fine-grid Dh with h = H7/4 = 27.
tion solution uH on DH according to (2.10). Then we give the fine-grid partition Dh with h = H7/4 (Fig. 3) and obtain the fine-grid solution ^uh in one additional step on Dh to use the converged coarse-grid solution uH according to (2.11). To check up the numerical accuracy of the two-grid method, we obtain the nonlinear iteration solution uh on the fine-grid Dh according to (2.6). The numerical results of the two-grid method and the nonlinear iteration method and the exact solution are presented in Fig. 4. From Fig. 4 we can conclude that the two-grid method solution ^uh has the same accuracy as that of the nonlinear iteration solution uh. To confirm the efficiency of the two-grid method, we obtain the CPU time and errors of the numerical solution of the two-grid scheme (2.10) and (2.11) and the nonlinear iteration scheme (2.6). Table 1 shows the L2 errors of ^ uh and the CPU time for two-grid scheme of characteristics finite-element
1 exact solution twogrid solution nonlinear iteration solution
0.9 0.8 0.7
u(x,t)
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x Fig. 4. Three kind of solutions at t = 0.20.
0.9
1
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Table 1 L2 errors for ^ uh at t = 0.20 h 7
2
k^uh uk kuk
H 4
CPU time 4
1.212 · 10
2
00
93
Table 2 L2 errors for uh at t = 0.20 h 4
2
kuh uk kuk
CPU time 4
1.228 · 10
184
00
method when h = H7/4. Table 2 shows the L2 errors of uh and the CPU time for characteristics finite-element nonlinear iteration method over the range of fine grids Dh. Tables 1 and 2 also show that the two-grid solution ^uh and the nonlinear iteration solution uh have the same order of accuracy. And the computational time of two-grid scheme is less than nonlinear iteration method. The convergence speed increases by about two times. Although the CPU time reported here provides only a rough measure, the efficiency of the two-grid approach is obvious.
5. Conclusion Two-grid linearization offers an attractive way to solve the nonlinear problems involving convection–diffusion equations. The key feature of the two-grid method is that it allows one to execute all of the nonlinear iterations on a system associated with a coarse spatial grid, without sacrificing the order of accuracy of the fine-grid solution. The two-grid scheme combined with the characteristics finite-element method, cannot only decrease the numerical oscillation caused by dominated convections, but also save much more computational time for solving the nonlinear convection-dominated diffusion problems.
References [1] J. Douglas Jr., T.F. Russell, Numerical method for convection-dominated diffusion problem based on combining the method of characteristics with finite element or finite difference procedures[J], SIAM J. Numer. Anal. 19 (5) (1982) 871–885. [2] J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1) (1994) 231–237.
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[3] J. Xu, Two grid finite element discretization techniques for linear and nonlinear PDEs[J], SIAM J. Numer. Anal. 33 (5) (1996) 1759–1777. [4] C.N. Dawson, M.F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations[J], Contemp. Math. 180 (1994) 191–203. [5] W. Li, M.B. Allen III, Two-grid methods for mixed finite-element solutions of reaction– diffusion equations, Numer. Meth. Partial Differential Eqs. [J] 15 (5) (1999) 589–604. [6] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, SpringerVerlag, Berlin, New York, 1996. [7] Ph.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978. [8] J. Xu, Local and parallel finite element algorithms based on two-grid discretizations[J], Math. Comput. 69 (231) (1999) 881–909.
Two-grid scheme for characteristics finite-element solution of nonlinear convection diffusion problems q Xin-qiang Qin
a,b,*
, Yi-chen Ma
a
a
b
School of Sciences, Xi’an Jiaotong University, Xi’an 710049, China School of Sciences, Xi’an University of Technology, No. 5 Jinhua Nanlu, Xi’an 710048, China
Abstract A two-grid scheme for characteristics finite-element solution of nonlinear convection-dominated diffusion equations was constructed. The L2 and H1 error estimates of the scheme were derived. A numerical example was also presented. The scheme involves solving one small, nonlinear problem on the coarse-grid system, one linear problem on the fine-grid system. The error estimates and the numerical example show that the new scheme is efficient to the nonlinear convection-dominated diffusion equations. 2004 Elsevier Inc. All rights reserved. Keywords: Convection–diffusion equations; Characteristics finite-element; Two-grid scheme
q
This work supported by National Science Foundation of China grants 10371096. Corresponding author. Address: School of Sciences, XiÕan University of Technology, No. 5 Jinhua Nanlu, XiÕan 710048, China. E-mail address: [email protected] (X.-q. Qin). *
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.021
420
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1. Introduction Convection–diffusion equations occur in numerous applications, noteworthy among them being mathematical models of environment pollution and neutron transportation. In this paper we discuss the nonlinear form 8 ou ou o ou > < cðxÞ ot þ bðxÞ ox ox aðxÞ ox ¼ f ðu; x; tÞ; ðx; tÞ 2 I ð0; T ; ð1:1Þ uðx; tÞ ¼ 0; ðx; tÞ 2 oI ð0; T ; > : uðx; 0Þ ¼ u0 ðxÞ; x 2 I; we use the modified method of characteristics finite-element developed by Douglas–Russell [1], to discrete in space with equal-order accuracy in u. To linearize the resulting discrete equations, we use a two-grid scheme, which allows us to iterate on a grid much coarser than that used for the final solution. We owe the impetus for using a two-grid approach to Xu [2,3,8], who demonstrates the possibility of mapping a fine-grid Galerkin finite-element problem onto a coarse grid. In that context, one solves the nonlinear problem via Newton-like iterations. After convergence on the coarse grid, one then extrapolates back to the fine grid using a Taylor expansion. Dawson and Wheeler [4] extend the method to nonlinear reaction–diffusion equations. Li Wu and Myron [5] extend it to another nonlinear reaction–diffusion equations. The remainder of the paper is organized as follows: Section 2 describes the algorithm. Section 3 gives an error analysis establishing the methodÕs convergence. Section 4 presents some computational results confirming the methodÕs utility, and Section 5 draws conclusions. We assume that the coefficients a, b and c are bounded and that d bðxÞ (H1) 0 < a0 6 aðxÞ 6 1; 0 < c0 6 cðxÞ 6 1; j bðxÞ j þ j dx ðcðxÞÞj 6 C; x 2 I cðxÞ 2 o f (H2) j of j þ j j 6 C; x 2 I 2 ou ou
Throughout this paper, we shall use the letter C to denote a generic positive constant which may for different values at its different occurrences. Let Wm,p(X) denote the Sobolev space on I. And the norm is defined by 8 1=p > P > p < kDa vkLp ðXÞ ; 1 6 p < 1; a6m kvkW m;p ¼ > > : max esssupjDa vj; p ¼ 1: a6m
m,2
We write W (X) = Hm(X), kvkHm(X) = kvkm, and kvkL2(X) = kvk0 = kvk and also assume that the solution u of (1.1) satisfies: (H3) u 2 L1(0,T;Hr + 1(I)), 2 L2 ð0; T ; H rþs Þ; r ¼ 1 if s = 1 and r P 2 if s = 0, (H4) ou ot 2 (H5) oot2u 2 L2 ð0; T ; L2 ðIÞÞ.
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2. Description of the method 2.1. Characteristics finite-element discretization We begin by briefly reviewing the characteristics finite-element discretization of the problem (1.1). Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðxÞ ¼ c2 ðxÞ þ b2 ðxÞ ð2:1Þ and let the characteristic direction associated with the operator cut + bux be denoted by s = s(x,t), where o cðxÞ o bðxÞ o ¼ þ : ð2:2Þ os wðxÞ ot wðxÞ ox Then, Eqs. (1.1) can be put in form 8 ou o ou > > aðxÞ ¼ f ðu; x; tÞ; x 2 I; t 2 ð0; T ; > wðxÞ < os ox ox ð2:3Þ uðx; tÞ ¼ 0; x 2 oI t 2 ð0; T ; > > > : uðx; 0Þ ¼ u0 ðxÞ; x 2 I: We introduce the Sobolev space V ¼ H 10 ðIÞ, and the notation (u,v) = Iu(x)v(x)dx,"u,v2V. Let Aðu; vÞ ¼ ða ou ; ovÞ. Then, multiplying the first ox ox equation of (2.1) by any v 2 V and applying Green formula, (2.3) can be written in the equivalent variational form 8 < w ou ; v þ Aðu; vÞ ¼ ðf ðuÞ; vÞ; v 2 V ; os ð2:4Þ : Aðuð0Þ u0 ; vÞ ¼ 0; v 2 V : We consider a time step Dt and approximate the solution at times tn = nDt, n = 1, 2, . . ., N, Dt ¼ NT . The characteristic derivative is approximated in the following way at t = tn n ou uðx; tn Þ uðx; tn1 Þ uðx; tn Þ uðx; tn1 Þ : ð2:5Þ w wðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cðxÞ os Dt 2 ðx xÞ þ Dt2 Namely, a backtracking algorithm is used to approximate the characteristic derivative. x ¼ x bðxÞ Dt is the foot (at level tn1) of the characteristic correcðxÞ sponding to x at the head (at level tn) (see Fig. 1).
Fig. 1. The direction of characteristic.
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Let the mesh size of the grid be h, we denote this grid by Dh and call it the fine grid. Vh be a finite-element subspace of V ˙ W1,1(I). The characteristics finite-element method for (2.4) is defined: For n = 1, 2, . . ., N, find unh 2 V h such that 8 n > un1 < c uh h ; v þ Aðunh ; vÞ ¼ ðf ðunh Þ; vÞ; v 2 V h ; Dt ð2:6Þ > : 0 Aðuh u0 ; vÞ ¼ 0; v 2 V h ; where
bðxÞ Dt; tn1 : uhn1 ðxÞ ¼ uh ðx; tn1 Þ ¼ uh x unh ¼ uh ðtn Þ; cðxÞ
ð2:7Þ
The initial approximation u0h in Vh can be defined as any reasonable approximation of u0 such as the interpolation of u0 in Vh. The (2.6) determines funh g uniquely in terms of the data u0. 2.2. Linearization via the two-grid method To solve the system (2.6), we use the Newton-like iterations. The idea is to devote all of the effort of nonlinear iteration to coarse-grid problems. Thus the iterations yield approximate finite-element solutions unH on a coarse subgrid DH having mesh size DH Dh,H > h and corresponding space VH. To approximate unh on Dh, we then execute iteration of the fine-grid system, using the converged coarse-grid solution unH together with the heuristic n unh ¼ unH þ enh ; uh ^ ð2:8Þ f ðunh Þ f ðunH Þ þ f 0 ðunH Þenh : This tactic yields the following system of equations for the fine-grid solution: 8 n n eh uH un1 > H n 0 n n n > ;v > c ; v þ Aðeh ; vÞ ðf ðuH Þeh ; vÞ ¼ ðf ðuH Þ; vÞ c < Dt Dt n > > AðuH ; vÞ; 8v 2 V h ; > : Aðu0h u0 ; vÞ ¼ 0; v 2 V h : ð2:9Þ Namely, the two-grid method has two stages: Stage 1: On coarse-grid DH, solve the following nonlinear system for unH 2 V H : 8 n un1 < c uH H ; v þ Aðun ; vÞ ¼ ðf ðun Þ; vÞ; v 2 V H; H H Dt ð2:10Þ : 0 AðuH u0 ; vÞ ¼ 0; v 2 V H :
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Stage 2: On the fine-grid Dh, solve the following linear system for ^unh 2 V h ! 8 n1 n > ^ ^ u u < c h h ; v þ Aðun ; vÞ ¼ ðf ðun Þ þ f 0 ðun Þð^un un Þ; vÞ; h
Dt
> :
Aðu0h u0 ; vÞ ¼ 0;
H
H
h
H
423
v 2 V h;
v 2 V h: ð2:11Þ
This linearization is efficient if the coarse-grid DH be coarser than the finegrid Dh without sacrificing the accuracy associated with the fine-grid solution unh . The analysis in Section 3. demonstrate that, as with the Galerkin-based scheme proposed by Xu [2], one can choose H = O(h4/7) and still retain the error estimates associated with the characteristics finite-element scheme. Numerical examples in Section 4. confirm the efficiency to solve the nonlinear problems with two-grid method. 3. Convergence analysis The two-grid method affords a remarkably efficient linearization. It is possible to execute all of the Newton-like iterations on very coarse grids, then use the heuristic (2.9) to obtain an accurate fine-grid solution in one additional step. We devote the rest of this section to the analysis of the two-grid scheme. We begin by describing the discretization space and associated projection operator. We then give a convergence analysis for the characteristics finite-element method, using the key parts of this argument in the analysis of the two-grid scheme. In what follows, we consider (1.1) on I. For v 2 Hr + 1 the following approximation property holds: inf ðkv vh k þ hkv vh k1 Þ 6 Chrþ1 kvkrþ1 ;
vh 2V h
ð3:1Þ
where r P 1. Let wh: [0,T] ! Vh, satisfy Aðu wh ; vÞ ¼ 0;
v 2 V h:
ð3:2Þ
Set g = u wh, n = uh wh, then u uh = gn. It is well known that [1], for p = 2,1 and 1 6 s 6 r + 1, kgkLp ð0;T ;L2 ðIÞÞ þ hkgkLp ð0;T ;H 1 ðIÞÞ 6 Chs kukLp ð0;T ;H s ðIÞÞ for r P 2 and 1 6 s 6 r + 1 og og s ou þ h 6 Ch : ot 2 ot 2 ot L2 ð0;T ;L2 ðIÞÞ L ð0;T ;H 1 ðIÞÞ L ð0;T ;H s1 ðIÞÞ
ð3:3Þ
ð3:4Þ
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The inverse property on Vh holds [6], namely, for vh 2 Vh, kvh k1 6 h1=2 kvh k:
ð3:5Þ
And we also have the approximation property [7]: kun wnh k1 6 kun krþ1;1 hrþ1 :
ð3:6Þ
3.1. Error estimates for the characteristics finite-element We now estimate the errors. From (3.3) and (3.4), to obtain error bounds for uuh, it suffices to estimate n. Theorem 1. Let u and uh be the respective solutions of (2.4) and (2.6). Under assumptions (H1–H5). We have the error estimate " o2 u n n max ku uh k 6 C Dt ot2 2 16n6N L ð0;T ;L2 Þ !# ð3:7Þ ou rþ1 þh kukL1 ð0;T ;H rþ1 ðIÞÞ þ os L2 ð0;T ;H rþs ðIÞÞ 6 CðDt þ hrþ1 Þ for sufficient small Dt, where s = 1 if r = 1 and s = 0 if r > 1. Proof. At t = tn, subtracting (2.4) from (2.6) and let v = nn to give c
nn nn1 n oun un un1 n ; n þ Aðnn ; nn Þ ¼ w c ;n Dt os Dt ! n n1 n1 n1 n1 g g g g nn1 n n n n þ c ;n þ c ;n c ;n Dt Dt Dt þ ðf ðun Þ f ðunh Þ; nn Þ:
ð3:8Þ
At any point x2I, by the Taylor theorem, un ðxÞÞðun ðxÞ unh ðxÞÞ f ðun ðxÞÞ f ðunh ðxÞÞ ¼ f 0 ð~
ð3:9Þ
~n ðxÞ. Therefore for some value u un ðxÞÞðun ðxÞ unh ðxÞÞ; nn Þ ðf ðun Þ f ðunh Þ; nn Þ ¼ ðf 0 ð~ un ðxÞÞgn ; nn Þ ðf 0 ð~un ðxÞÞnn ; nn Þ: ¼ ðf 0 ð~
ð3:10Þ
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425
Consequently, from Eq. (3.8), n n nn1 n oun un un1 n ; n þ Aðnn ; nn Þ ¼ w c ;n c Dt os Dt ! n n1 n1 n1 n1 g g g g nn1 n n n n þ c ;n þ c ;n c ;n Dt Dt Dt þ ðf 0 ð~ un ðxÞÞgn ; nn Þ ðf 0 ð~ un ðxÞÞnn ; nn Þ T 1 þ T 2 þ T 3 þ T 4 þ T 5 þ T 6 : ð3:11Þ We can estimate the left terms of (3.11) by aða bÞ P 12 ða2 b2 Þ and (H1) n n nn1 n 1 a0 2 ½ðcnn ; nn Þ ðcnn1 ; nn1 Þ þ knn k1 ; n þ Aðnn ; nn Þ P c 2Dt Dt 2 ð3:12Þ and use the results provided in [1] to the right terms T1 T4, we have 2 2 2 2 o u o u n 2 n 2 jT 1 j 6 C Dt þ ekn k 6 C os2 2 n1 n 2 os2 2 n1 n 2 Dt þ ekn k1 ; L ðt ;t ;L Þ L ðt ;t ;L Þ ð3:13Þ C og 2 jT 2 j 6 þ eknn k1 ; Dt ot L2 ðtn1 ;tn ;H 1 Þ
ð3:14Þ
2
2
ð3:15Þ
2
2
ð3:16Þ
jT 3 j 6 Ckgn1 k þ eknn k1 ; jT 4 j 6 Cknn1 k þ eknn k1 for any positive constant e. And from (H2), we have 2
2
jT 5 j 6 Cðkgn k þ knn k Þ; 2
jT 6 j 6 Cknn k :
ð3:17Þ ð3:18Þ
Choosing proper e, inequalities (3.12)–(3.18) can be combined with (3.11) to give the recursion relation 1 a0 2 ½ðcnn ; nn Þ ðcnn1 ; nn1 Þ þ knn k1 2Dt " 2 2 2 o u 1 og n 2 n1 2 6 C kn k þ kn k þ Dt 2 þ os L2 ðtn1 ;tn ;L2 Þ Dt ot L2 ðtn1 ;tn ;H 1 Þ # 2
2
þkgn k þ kgn1 k :
ð3:19Þ
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It follows from (2.6) and (3.2) that n0 = 0. If we multiply (3.19) by 2Dt, sum over n, and apply the discrete Gronwall lemma, it follows that " 2 # o2 u og max kn k 6 C Dt 2 þ þ kgkL1 ð0;T ;L2 Þ 16n6N os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H 1 Þ " !# o2 u ou rþ1 ; þh kukL1 ð0;T ;H rþ1 ðIÞÞ þ 6 C Dt 2 os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H rþs ðIÞÞ n
ð3:20Þ
which, together with un unh ¼ gn nn , (3.3) and (3.4) yields the desired result (3.7). h An optimal order estimate for u uh in H1(I) can be derived in a similar fashion. We can give the theorem below: Theorem 2. Let u and uh be the respective solutions of (2.4) and (2.6). Under assumptions (H1–H5), We have the error estimate n
max ku
16n6N
unh k1
" 2 o2 u 2 6 C Dt þ hr kukL1 ð0;T ;H rþ1 ðIÞÞ os2 2 2 L ð0;T ;L Þ !# ou þ 6 CðDt þ hr Þ ot L2 ð0;T ;H r ðIÞÞ
ð3:21Þ
for sufficient small Dt and r P 2. In the proof, the embedding theorem was used.
3.2. Error estimates for the two-grid method We now analyze the two-grid method for iteratively solving the nonlinear system (2.6). This scheme is attractive because it requires nonlinear iteration only on the coarse grid, give the fine-grid solution in a single, linear, Newton-like step. The following theorem gives the error: Theorem 3. Let u and ^ uh be the respective solutions of (2.4) and (2.11). Under assumptions (H1–H5), we have unh k 6 CðDt þ hrþ1 þ H 2rþ3=2 Þ max kun ^
16n6N
for Dt sufficiently small, where r P 1.
ð3:22Þ
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427
n Proof. To estimate the error, set gn ¼ un wnh ,^ n ¼ ^unh wnh . At t = tn. Subtractn ing Eqs. (2.11) from itÕs respective counterparts (2.4) and choosing v ¼ ^n , we have ! n n1 n n n1 ^ n n n ^ ^n ; n ^ þ Aðn ^n Þ ¼ w ou c u uh ; ^nn ;n c Dt os Dt 0 1 n1 n ^nn1 ^n n g gn1 ^n gn1 gn1 ^n ;n þ c ; n @c ; ^n A þ c Dt Dt Dt n þ ðf ðun Þ f ðunH Þ f 0 ðunH Þð^ un unH Þ; ^ n ÞÞ:
A Taylor expansion of f about
ð3:23Þ
unH
yields 1 2 uÞðun unH Þ ð3:24Þ f ðun Þ ¼ f ðunH Þ þ f 0 ðunH Þðun unH Þ þ f 00 ð~ 2 for some function ~ u. From (3.22) and (3.23), we get ! n n1 ^ ^ n n n oun un uhn1 ^n n n ^ ^ ^ c ; n þ Aðn ; n Þ ¼ w c ;n Dt os Dt 0 1 n1 n n1 ^nn1 ^n n g gn1 ^n g gn1 ^n ;n þ c ; n @c ; ^n A þ c Dt Dt Dt n n 1 00 2 ðf ð~uÞðun unH Þ ; ^n þ ðf 0 ðunH Þgn ; ^ n Þ ðf 0 ðunH Þnn ; nn Þ þ 2 7 X T 0i : ð3:25Þ i¼1
Then arguments similar to those used for the characteristics finite-element error estimate (Theorem 1) yield n n n1 n1 n 2 n1 2 1 a0 n 2 ½ðc^ n ;^ n k1 6 C k^n k þ k^n k n Þ ðc^ n ;^ n Þ þ k^ 2Dt 2 2 2 o u 1 n 2 n1 2 og þ Dt os2 2 n1 n 2 þ Dt ot 2 n1 n 1 þ kg k þ kg k L ðt ;t ;H Þ L ðt ;t ;L Þ þ kðun unH Þ2 k2 : ð3:26Þ Now multiplying both sides of (3.26) 2Dt and summing over n and apply the discrete Gronwall lemma, it follows that: " 2 og o2 u n ^ þ max kn k 6 C Dt 2 16n6N os L2 ð0;T ;L2 Þ ot L2 ð0;T ;H 1 Þ # 2
þkgkL1 ð0;T ;L2 Þ þ max kðun unH Þ k ; 16n6N
^0 ¼ 0 is used. where the n
ð3:27Þ
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Let h = H in (3.7) and (3.20) and using (3.5) and (3.6), we have 2
kðun unH Þ k 6 kun unH k1 kun unH k 6 ðkun wnH k1 þ kwnH unH k1 Þkun unH k 6 Cðkun krþ1;1 H rþ1 þ H 1=2 kwnH unH kÞkun unH k
ð3:28Þ
6 CðH rþ1 þ H 1=2 ðH rþ1 þ DtÞÞðH rþ1 þ DtÞ 2
6 CH 1=2 ðH rþ1 þ DtÞ 6 CðH 2rþ3=2 þ DtÞ: n
Combining un ^ unh ¼ gn ^ n together with (3.3) and (3.4) and (3.27) and (3.28) yields the results (3.22). h An optimal order estimate for u ^ uh in H1(I) can be derived in a similar fashion. We can give the theorem below: Theorem 4. Let u and uh be the respective solutions of (2.4) and (2.11).Under assumptions (H1H5). We have the error estimate unh k1 6 CðDt þ hr þ H 2r Þ max kun ^
16n6N
ð3:29Þ
for sufficient small Dt and r P 2. Theorems 1 and 3 demonstrate a remarkable fact about two-grid scheme: when we iterate on a very coarse-grid DH and still get good approximations by taking one iteration on the fine-grid Dh. Specifically, for h = H7/4 as r = 1, the two-grid scheme solution ^ uh has the same accuracy compared with that of the nonlinear iteration solution uh. 4. Numerical example To illustrate the effectiveness of the two-grid linearization, we examine the following simple test problem: 8 ou ou o ou > 2 > > < ot þ ox ox aðxÞ ox ¼ u þ Gðx; tÞ; x 2 ½0; 1; t 2 ð0; T ; ð4:1Þ uð0; tÞ ¼ 1; uð1; tÞ ¼ 0; t 2 ð0; T ; > > > : uðx; 0Þ ¼ 1 x; x 2 ½0; 1; where a(x) = 0.001. The G(x,t) is determined by the exact solution u(x,t) = (1 x)ext. We solve (4.1) by Dt = 0.125 · 105 from t = 0 to 0.20. For H = 24, h = 7/4 H = 27, we use the two-grid scheme presented in this paper. First, we give the coarse-grid partition DH with H = 24 (Fig. 2) and get the nonlinear itera-
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429
Fig. 2. The coarse-grid DH with H = 24.
Fig. 3. The fine-grid Dh with h = H7/4 = 27.
tion solution uH on DH according to (2.10). Then we give the fine-grid partition Dh with h = H7/4 (Fig. 3) and obtain the fine-grid solution ^uh in one additional step on Dh to use the converged coarse-grid solution uH according to (2.11). To check up the numerical accuracy of the two-grid method, we obtain the nonlinear iteration solution uh on the fine-grid Dh according to (2.6). The numerical results of the two-grid method and the nonlinear iteration method and the exact solution are presented in Fig. 4. From Fig. 4 we can conclude that the two-grid method solution ^uh has the same accuracy as that of the nonlinear iteration solution uh. To confirm the efficiency of the two-grid method, we obtain the CPU time and errors of the numerical solution of the two-grid scheme (2.10) and (2.11) and the nonlinear iteration scheme (2.6). Table 1 shows the L2 errors of ^ uh and the CPU time for two-grid scheme of characteristics finite-element
1 exact solution twogrid solution nonlinear iteration solution
0.9 0.8 0.7
u(x,t)
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x Fig. 4. Three kind of solutions at t = 0.20.
0.9
1
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Table 1 L2 errors for ^ uh at t = 0.20 h 7
2
k^uh uk kuk
H 4
CPU time 4
1.212 · 10
2
00
93
Table 2 L2 errors for uh at t = 0.20 h 4
2
kuh uk kuk
CPU time 4
1.228 · 10
184
00
method when h = H7/4. Table 2 shows the L2 errors of uh and the CPU time for characteristics finite-element nonlinear iteration method over the range of fine grids Dh. Tables 1 and 2 also show that the two-grid solution ^uh and the nonlinear iteration solution uh have the same order of accuracy. And the computational time of two-grid scheme is less than nonlinear iteration method. The convergence speed increases by about two times. Although the CPU time reported here provides only a rough measure, the efficiency of the two-grid approach is obvious.
5. Conclusion Two-grid linearization offers an attractive way to solve the nonlinear problems involving convection–diffusion equations. The key feature of the two-grid method is that it allows one to execute all of the nonlinear iterations on a system associated with a coarse spatial grid, without sacrificing the order of accuracy of the fine-grid solution. The two-grid scheme combined with the characteristics finite-element method, cannot only decrease the numerical oscillation caused by dominated convections, but also save much more computational time for solving the nonlinear convection-dominated diffusion problems.
References [1] J. Douglas Jr., T.F. Russell, Numerical method for convection-dominated diffusion problem based on combining the method of characteristics with finite element or finite difference procedures[J], SIAM J. Numer. Anal. 19 (5) (1982) 871–885. [2] J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1) (1994) 231–237.
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[3] J. Xu, Two grid finite element discretization techniques for linear and nonlinear PDEs[J], SIAM J. Numer. Anal. 33 (5) (1996) 1759–1777. [4] C.N. Dawson, M.F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations[J], Contemp. Math. 180 (1994) 191–203. [5] W. Li, M.B. Allen III, Two-grid methods for mixed finite-element solutions of reaction– diffusion equations, Numer. Meth. Partial Differential Eqs. [J] 15 (5) (1999) 589–604. [6] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, SpringerVerlag, Berlin, New York, 1996. [7] Ph.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978. [8] J. Xu, Local and parallel finite element algorithms based on two-grid discretizations[J], Math. Comput. 69 (231) (1999) 881–909.