Mixed time discontinuous space-time finite element method for

Appl. Math. Mech. -Engl. Ed., 2008, 29(12):1579–1586 DOI 10.1007/s10483-008-1206-y c Shanghai University and Springer-Verlag 2008

Applied Mathematics and Mechanics (English Edition)

Mixed time discontinuous space-time finite element method for convection diffusion equations ∗ LIU Yang (

), LI Hong (

),



HE Siriguleng (

)

(School of Mathematical Sciences, Inner Mongolia University, Huhhot 010021, P. R. China) (Communicated by ZHOU Zhe-wei)

Abstract A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method. Key words convection diffusion equations, mixed finite element method, time discontinuous space-time finite element method, convergence Chinese Library Classification O242.21 2000 Mathematics Subject Classification

65N30, 65M60

Introduction In a space-time finite element method, the time variable and space variable are considered uniformly. It takes full advantages of the virtues of the finite element method in both time and space directions. The accuracy of the numerical solution is improved. The space-time finite element approach has three types[1−9] : continuous in both space and time, discontinuous in either time or space direction, and discontinuous in space and time simultaneously. The original discontinuous finite element method was introduced by Reed and Hill[1] to solve the neutron transport equation in 1973. It had not been well studied and applied untill the techniques of combining the discontinuous finite element method and the Runge-Kutta method were studied for one and high dimensional conservation laws by Cockburn & Lin[2] and Cockburn et al[3] . at the end of the 1980s. With the development of the method, the space-time finite element method, continuous in space but discontinuous in time, was discussed for the equation ut +Au = f by Thom´ee[5]. The uniqueness of the discrete solution was given. Also, different proof ideas of convergence were presented at the same time. Furthermore, the space-time mixed discontinuous finite element method was studied for elliptic and parabolic equations in Ref. [6-7]. The time discontinuous space-time finite element method for parabolic equations was discussed in Ref. [8], and the existence and convergence of the week solution were proved. Two methods, the fully discrete continuous space-time formulation and the space continuous but time discontinuous ∗ Received Mar. 31, 2008 / Revised Oct. 21, 2008 Project supported by the National Natural Science Foundation of China (No. 10601022), NSF of Inner Mongolia Autonomous Region of China (No. 200607010106), 513 and Science Fund of Inner Mongolia University for Distinguished Young Scholars (No. ND0702) Corresponding author LIU Yang, Doctor, E-mail: [email protected]

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LIU Yang, LI Hong and HE Siriguleng

method, for nonlinear Schr¨ odinger equation were considered respectively in Ref. [9]. Also, the conservation in the energy integration was proved for the continuous method. Meanwhile, the near conservation feature of the electron was given for the time discontinuous method. The higher order error estimates were obtained in two approaches. In this paper, the technique of combining the mixed finite element method and the discontinuous finite element method were studied for the following convection-diffusion equations: ⎧ ⎪ ⎨ ut − Δu(x, t) + b · ∇u(x, t) + cu(x, t) = f (x, t), Ω × J, ¯ u(x, t) = 0, ∂Ω × J, (1) ⎪ ⎩ ¯ u(x, 0) = u0 (x), Ω.  where Ω ∈ R (d = 1, 2, 3), J = (0, T ], b ∈ [L d

∞

1 (0; T, W∞ (Ω))]d ,

|b| =

d 

i=1

b2i (x, t) ≤

1 2,

1 ¯ c(x, t) ∈ [L∞ (0; T, W∞ (Ω))], 1 ≤ c(x, t), f ∈ L2 (0; T, L2 (Ω)), u0 ∈ L2 (Ω). Papers focusing on the mixed time discontinuous finite element method to deal with the second-order convection-diffusion problems have not been seen so far. In this paper, the mixed time discontinuous space-time finite element formulation for the convection-diffusion equation is constructed. The stability, uniqueness, existence and convergence of the finite element solution are proved. Finally, the numerical results are provided to illustrate the efficiency of the mixed time discontinuous space-time finite element method discussed in this paper.

1

Notations, definitions and lemmas

To introduce the mixed time discontinuous space-time finite element method for equation (1), we discretize the time interval [0, T ] by 0 = t0 < t1 < · · · < tN = T first. Let In = (tn , tn+1 ), time step kn = tn+1 − tn , n = 0, 1, 2, · · · , N − 1. Th is the regular partition of Ω and the partition unit is τ . Define the space-time domain Q := Ω × J, the space-time slab S n := Ω × In . Suppose Th,n is the regular partition of S n and the partition unit is K = τ × In . Let hn = max (hK ), n = 0, 1, 2, · · · , N − 1, h = max hn . Define discrete approximate spaces

K∈Th,n

n

Vh,n = {v : v|K ∈ Pk (τ ) × Pk (In ), ∀K ∈ Th,n }, Qh,n = {v : v|K ∈ Pm (τ ) × Pm (In ), ∀K ∈ Th,n }, Hh,n = {ϕ : ϕ|K ∈ (Qh,n )d , ∇ · ϕ|K ∈ Qh,n , ∀K ∈ Th,n }, Vh = {v : R+ × R2 → R, v|(tn ,tn+1 ) ∈ Vh,n }, where Pk denotes the polynomial space of degree at most k. We introduce definitions and lemmas which will be used in this paper. n  Definition 1 Define the inner product of space-time slab S by (ω, v)n = (ω, v)S n = (ω, v)ds, where (ω, v) is the inner product in Ω, and the corresponding norm is ||v||n = In  1 1/2 (v, v)n = ( In ||v||2Ω ds) 2 .

Definition 2 At the time level t = tn (n = 0, 1, · · · , N − 1), define the inner product of 1/2 L by ω, vn = ω(·, tn ), v(·, tn )Ω , and the corresponding norm is |v|n = v, vn . 2

Definition 3

Define the left and right limits by v± (x, t) = lim± v(x, t + s), and the jump s→0

term at discontinuous nodes in time by [v] = v+ − v− . Define norm |||v|||2 = 12 (|v|2N + |v|20 + N −1 |[v]|2n ). n=1

Definition 4

Define the norm of space L2 (J, L2 (Ω)) by ||v||2Q =

 tN 0

||v||2Ω dt.

Mixed time discontinuous space-time finite element method

Definition 5

¯ L2 (Ω)) by Define the norm of space L∞ (J,

max

¯ t∈J=[0,T ]

1581

· Ω , where · Ω

denotes the corresponding norm of Sobolev space L2 (Ω). Lemma 1 There exists an elliptic projection Ph : H01 (Ω) → Vh,n such that (∇ · X, u − Ph u, ) = 0,

∀X ∈ Hh,n ,

and the corresponding error estimate[10] ||u − Ph u|| ≤ Chk+1 ||u||k+1 . Lemma 2

There exists an operator Πh : (H 1 (Ω))d → Hh,n such that (∇ · (v − Πh v), w) = 0,

∀w ∈ Vh,n

holds, then[10] ||v − Πh v|| ≤ Chm+1 ||v||H m+1 (divΩ) .

2

The stability, existence and uniqueness of discrete solutions

First, we construct the mixed time discontinuous space-time finite element schemes for Eq. (1), let ψ = ∇u, then equation (1) can be rewritten as follows: ⎧ ut − ∇ · ψ + b · ψ + cu(x, t) = f, Ω × J, ⎪ ⎪ ⎪ ⎨ ψ − ∇u = 0, Ω × J, (2) ⎪ u = 0, ∂Ω × J, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), Ω. The corresponding weak formulation is ⎧ N tN tN N −1 t

⎪ ⎪ ⎪ ⎪ (u , w)dt − (∇ · ψ, w)dt + (b · ψ, w)dt + [u], w+ n + u+ , w+ 0 t ⎪ ⎪ ⎪ 0 0 0 n=1 ⎪ ⎪ ⎪ ⎪ tN tN ⎪ ⎪ ⎨+ (cu, w)dt = u, w  + (f, w)dt, w ∈ H 1 (Q), + 0

0 0 ⎪ ⎪ tN tN ⎪ ⎪ ⎪ ⎪ (ψ, v)dt + (u, ∇ · v)dt = 0, ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 , x ∈ Ω.

0

v ∈ (L2 (Q))d ,

That is, in interval In = (tn , tn+1 ) it holds ⎧ ⎪ ⎨ (ut , w)n − (∇ · ψ, w)n + (b · ψ, w)n + [u], w+ n + (cu, w)n = (f, w)n , (ψ, v)n + (u, ∇ · v)n = 0, v ∈ (L2 (S n ))d , ⎪ ⎩ (u(0), w) = (u0 , w), w ∈ H01 (S n ).

w ∈ H01 (S n ), (3)

So the discrete formulation of the original problems can be written as: Finding (uh , ψ h ) ∈ Vh,n × Hh,n , such that ⎧ h h h h h h h h h h ⎪ ⎨ (ut , w )n − (∇ · ψ, w )n + (b · ψ , w )n + [u ], w+ n + (cu , w )n = (f, w )n , (4) (ψ h , v h )n + (uh , ∇ · v h )n = 0, ⎪ ⎩ h h h (u (0), w ) = (u0 , w ). For the above schemes, we have

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LIU Yang, LI Hong and HE Siriguleng

The solution (uh , ψ h ) ∈ Vh,n × Hh,n for the problems (4) satisfies

Theorem 1

max (||uh ||Ω + ||ψ h ||Ω ) ≤ M (||uh0 ||Ω + ||f ||Q ),

0≤t≤T

where M is a constant independent of hn and kn . Proof Choosing wh = uh and v h = ψ h in (4) and summing from n = 1 to N , we have

tN

0

(uht , uh )dt

−

tN

h

(u , ∇ · ψ )dt + h

h

0



+ uh+ , uh+ 0 +

tN

tN

0

(b · ψ , u )dt + h

0



tN

(ψ , ψ )dt +

h

N −1

[uh ], uh+ n

n=1



(cuh , uh )dt = uh , uh+ 0 +

h

0

tN

tN

(f, uh )dt,

0

(uh , ∇ · ψ h )dt = 0.

(5)

0

Adding the two equations, we have

tN 0

(uht , uh )dt +

+ uh+ , uh+ 0 +

tN

(ψ h , ψ h )dt +

0 tN

0

tN

(b · ψ h , uh )dt +

0

[uh ], uh+ n

n=1



(cuh , uh )dt = uh , uh+ 0 +

N −1

tN

(f, uh )dt.

(6)

0

After integration by parts in the first term, we have

tN

|||uh |||2 +

(ψ h , ψ h )dt +

0

0

tN

(cuh , uh )dt = uh , uh+ 0 +

tN

(f, uh )dt +

0

tN

(b · ψ h , uh )dt.

0

(7)

Using the Cauchy-Schwarz inequality and Young’s inequality, we obtain |||u ||| + h

2

0

tN

||ψ h ||2Ω dt

+ 0

tN

||uh ||2Ω dt

≤

||uh0 ||2Ω

tN

+ 0

||f ||2Ω dt.

(8)

Taking the max norm for time, we get max (||uh ||2Ω + ||ψ h ||2Ω ) ≤ M (||uh0 ||2Ω + ||f ||2Q ).

0≤t≤T

(9)

The conclusion of Theorem 1 is proved. Theorem 2

The solution (uh , ψ h ) ∈ Vh,n × Hh,n of the problems (4) exists and unique.

Proof In fact, since (4) is linear, it suffices to show that the associated homogeneous system ⎧ h h h h h h h h h h ⎪ ⎨ (ut , w )n − (∇ · ψ , w )n + (b · ψ , w )n + u+ , w+ n + (cu , w )n = 0, (ψ h , v h )n + (uh , ∇ · v h )n = 0, ⎪ ⎩ h (u (0), wh ) = 0. has only the zero trivial solution. From the conclusion of Theorem 1, we can get uh = 0 and ψ h = 0 immediately. So the existence and uniqueness of the solution for the problems (4) have been demonstrated.

Mixed time discontinuous space-time finite element method

3

1583

The proof of convergence Theorem 3

For the discrete solution, we have error estimates

||ψ − ψ h ||Q + ||u − uh ||Q ≤ M hmin(k+1,m+1) (||u||k+1,Q + ||ψ||H m+1 (divΩ),Q +

max

¯ t∈J=[0,T ]

u Ω ).

Proof We now decompose the errors as u − uh = u − Ph u + Ph u − uh = ζ1 + ζ2 , ψ − ψ h = ψ − Πh ψ + Πh ψ − ψ h = η1 + η2 . Noticing (3) and (4), we have

tN

(ut −

0 N −1

+



tN

−

uht , wh )dt

h

0

  h h [u − uh ], w+ + (u+ − uh+ ), w+ + n 0

n=1

 h = u − uh , w+ 0



tN

tN

(∇ · (ψ − ψ ), w )dt + h



tN

(ψ − ψ h , v h )dt +

0

(b · (ψ − ψ h ), wh )dt

0



tN

(c(u − uh ), wh )dt

0

(u − uh , ∇ · v h )dt = 0.

(10)

0

Choosing wh = ζ2 and v h = η2 in (10) and summing the two expressions, we have

tN

(ut −

0 N −1

+

uht , ζ2 )dt

tN

−

tN

+

(∇ · (ψ − ψ ), ζ2 )dt +

0

  [u − uh ], ζ2+ n + (u+ − uh+ ), ζ2+ 0 +

n=1



h

(ψ − ψ h , η2 )dt +

0

tN

0



tN

(b · (ψ − ψ h ), ζ2 )dt

0 tN

(c(u − uh ), ζ2 )dt

0

 (u − uh , ∇ · η2 )dt = (u − uh ), ζ2+ 0 .

(11)

Using lemmas, we get



tN

(ζ1t , ζ2 )dt + 0



tN

(ζ2t , ζ2 )dt + 0



tN

+



+ ζ2+ , ζ2+ 0 +

N −1

n=1

tN

(cζ1 , ζ2 )dt 0

[ζ1 ], ζ2+ n +

N −1

[ζ2 ], ζ2+ n + ζ1+ , ζ2+ 0

n=1



tN

(η1 , η2 )dt + 0

tN

(cζ2 , ζ2 )dt + 0

(b · (η1 + η2 ), ζ2 )dt +

0



tN

0

 (η2 , η2 )dt = (u − uh ), ζ2+ 0 .

(12)

By the definition of |||v|||, we have |||ζ2 |||2 + N −1

n=1



tN

(ζ1t , ζ2 )dt + 0

+



tN

(cζ2 , ζ2 )dt + 0

[ζ1 ], ζ2+ n + ζ1+ , ζ2+ 0 +





tN

(cζ1 , ζ2 )dt + 0

tN



(η1 , η2 )dt + 0

0

tN

(b · (η1 + η2 ), ζ2 )dt

0 tN

 (η2 , η2 )dt = (u − uh ), ζ2+ 0 .

(13)

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LIU Yang, LI Hong and HE Siriguleng

After integration by parts for the second term of (13), we get



tN

|||ζ2 ||| + 2

(cζ2 , ζ2 )dt + 0



= ζ1 + ζ2 , ζ2+ 0 + −

tN

tN

(η2 , η2 )dt 0

tN

(ζ2,t , ζ1 )dt + 0



(η1 , η2 )dt −

0

N −1

ζ1− , [ζ2 ]n − ζ1− , ζ2− N

n=1 tN



(cζ1 , ζ2 )dt −

0

tN

(b · (η1 + η2 ), ζ2 )dt.

(14)

0

Using the Cauchy-Schwarz inequality, Young’s inequality, the triangle inequality, and the definition of |||v|||, we obtain |||ζ2 |||2 +

1 2



tN

0

||η2 ||2Ω dt +

1 2

0

tN

N

1 |||ζ2 |||2 + M ( |ζ1− |2 + ||ζ1 ||2Q + ||η1 ||2Q ). 2 n=1

||ζ2 ||2Ω dt ≤

(15) That is, |||ζ2 |||2 + ||η2 ||2Q + ||ζ2 ||2Q ≤ M (

N

|ζ1− |2 + ||ζ1 ||2Q + ||η1 ||2Q ).

(16)

n=1

By Lemmas 1 and 2, we get ||η2 ||2Q + ||ζ2 ||2Q ≤ M h2 min(k+1,m+1) (||u||2k+1,Q + ||ψ||2H m+1 (divΩ),Q +

max

¯ t∈J=[0,T ]

u 2Ω ).

(17)

Applying the triangle inequality can complete the proof.

4

Numerical examples

To illustrate the efficiency of the mixed time discontinuous space-time finite element method presented in this paper, we consider the following equations: ⎧ ⎪ ⎨ ut (x, t) − uxx (x, t) + u(x, t) = f (x, t), (x, t) ∈ (0, 1) × (0, 1], u(0, t) = 0, t ∈ [0, 1], (18) ⎪ ⎩ u(x, 0) = sin(πx), x ∈ [0, 1], where (x, t) ∈ [0, 1] × [0, 1], f (x, t) = π 2 e−t sin(πx). It is not difficult to verify that the exact solution is u(x, t) = e−t sin(πx), the middle item is ψ(x, t) = πe−t cos(πx). Let Vh,n and Hh,n denote the finite dimensional subspaces of [0, 1] × H01 and [0, 1] × L2 , respectively. The corresponding basis functions are tn+1 − t xi+1 − x 1 x − xi 1 t − tn xi+1 − x 2 x − xi 2 ( u1 + u2 ) + ( u1 + u ), Δt Δx Δx Δt Δx Δx 2 tn+1 − t xi+1 − x 1 x − xi 1 t − tn xi+1 − x 2 x − xi 2 ψ h (x, t) = ( ψ1 + ψ2 ) + ( ψ1 + ψ ). Δt Δx Δx Δt Δx Δx 2

uh (x, t) =

The errors in L2 (J, L2 (Ω)) norm and the accuracy of the approximate solutions uh and ψ h are given in Tables 1 and 2.

Mixed time discontinuous space-time finite element method Δt L2 (L2 )-errors and numerical orders of accuracy ( Δx = 4)

Table 1 Δt Δx

=4

1585

||u − uh ||L2 (L2 )

Order of ||u − uh ||L2 (L2 )

||ψ − ψh ||L2 (L2 )

Order of ||ψ − ψh ||L2 (L2 )

(20,5)

8.016 5E-004

(40,10)

2.070 9E-004

1.952 7

0.001 5

1.793 5

(80,20)

5.487 6E-005

1.916 0

3.929 0E-004

1.932 7

(160,40)

1.427 0E-005

1.943 2

1.016 3E-004

1.950 8

(320,80)

3.647 2E-006

1.968 1

2.585 2E-005

1.975 0

(640,160)

9.235 0E-007

1.981 6

6.692 1E-006

1.949 7

Δt L2 (L2 )- errors and numerical orders of accuracy ( Δx = 3)

Table 2 Δt Δx

=3

0.005 2

||u − uh ||L2 (L2 )

Order of ||u − uh ||L2 (L2 )

||ψ − ψh ||L2 (L2 )

Order of ||ψ − ψh ||L2 (L2 )

(18,6)

0.001 1

(36,12)

2.820 6E-004

1.963 4

6.298 6E-004

0.002 1 1.737 3

(72,24)

7.172 9E-005

1.975 4

1.734 8E-004

1.860 3

(144,48)

1.815 9E-005

1.981 9

4.550 5E-005

1.930 7

(288,96)

4.573 3E-006

1.989 4

1.165 2E-005

1.965 4

(576,192)

1.149 6E-006

1.992 1

3.114 4E-006

1.903 6

We can see from the above data that the convergence orders obtained in numerical simulations agree with the results obtained in theoretical analysis when the time step or the spatial Δt step ratio is 3 or 4 (that is, Δx = 3 or 4). It is easy to see that the introduced middle variable does not reduce the precision of the formulation. Moreover, the simulation results of the middle item(ψ = ux ) is fairly satisfied. Meanwhile, Figs. 1 and 2 give the comparison between the exact solution (u, ψ) and the numerical solution (uh , ψ h ) with the partition (80, 20) (N = 20, Δt = 1/20, Δx = 1/80) at time t = 0.25, 0.5, 0.75, 1. The numerical results show that the mixed time discontinuous space-time finite element method introduced in this paper is efficient for second-order convection diffusion equations. u(x, t) uh(x, t)

3 t = 0.25

0.6

t = 0.5

0.4

t = 0.75 t = 1.0

ψ(x, t), ψh(x, t)

u(x, t), uh(x, t)

0.8

0.2 0 0

Fig. 1

2

t=0.25 t=0.5 t=0.75

1

t=1.0

0 −1

ψ(x, t) ψh(x, t)

−2 0.2

0.4

x

0.6

0.8

1.0

Comparison between the exact solution u and the approximate solution uh with N =20, Δt = 1/20, Δx = 1/80 at time t = 0.25, 0.5, 0.75, 1

−3 0

Fig. 2

0.2

0.4

x

0.6

0.8

1.0

Comparison between the exact solution ψ and the approximate solution ψ h with N = 20, 1/Δt = 1/20, 1/Δx = 1/80 at time t=0.25, 0.5, 0.75, 1

References [1] Reed N H, Hill T R. Triangle mesh methods for the Neutron transport equation[R]. Report LA2 UR-73-479, Los Alamos Scientific Laboratory, 1973.

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LIU Yang, LI Hong and HE Siriguleng

[2] Cockburn B, Lin S Y. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems[J]. J Comp Phys, 1989, 84(1):90–113. [3] Cockburn B, Hou S C, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin method for conservation laws IV: the multidimensional case[J]. J Comp Phys, 1990, 54(3):545– 581. [4] Yan Jue, Shu Chi-wang. A local discontinuous galerkin method for KdV-type equation[R]. NASA/CR-2001-211026 ICASE, Report No.2001-20. [5] Thom´ee Vider. Galerkin finite element methods for parabolic problems[M]. New York: SpringerVerlag, 1997. [6] Li Hong, Guo Yan. The discontinuous space-time mixed finite element method for fourth order parabolic problems[J]. Acta Scientiarum Naturalium Universitatis NeiMongal, 2006, 37(1):20–22 (in Chinese). [7] Brezzi F, Hughes T J R, Marini L D, Masud A. Mixed discontinuous Galerkin methods for Darcy flow[J]. Journal of Scientific Computing, 2005, 22(2):119–145. [8] Li Hong, Liu Ruxun. The space-time finite element methods for parabolic problems[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(6):687–700. DOI 10.1023/A:1016314405090 [9] Tang Qiong, Chen Chuanmiao, Liu Luohua. Space-time finite element method for Schr¨ odinger and its conservation[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(3): 335–340. DOI 10.1007/s10483-006-0308-2 [10] Zhangxin chen. Finite element methods and their applications[M]. Berlin: Springer-Verlag, 2005.