The Effect of Numerical Integration in Finite Element ... - m-hikari

Nonlinear Analysis and Differential Equations, Vol. 5, 2017, no. 1, 1 - 15 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/nade.2017.6866

The Effect of Numerical Integration in Finite Element Methods for General Nonlinear Hyperbolic Equations Nguimbi Germain Ecole Nationale Supérieure Polytechnique Marien Ngouabi University, Brazzaville, Congo Pongui Ngoma Diogène Vianney and Likibi Pellat Rhoss Beauneur Department of Mathematics Marien Ngouabi University, Brazzaville, Congo c 2016 Nguimbi Germain et al. This article is distributed under the Creative Commons Copyright Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract We consider the effect of numerical integration in finite element methods for general nonlinear hyperbolic equations and give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order L2 -error estimates are derived.

Keywords: Finite element, Quadrature scheme, Elliptic projection, error estimates

2

Nguimbi Germain et al.

Introduction In this paper we analyze the effect of numerical integration for generalized nonlinear hyperbolic equations  n n X X
   c(x, u)u = a (x)p(x, u)u + bi (x, u)uxi + f (x, t, u), tt ij xj xi    i,j=1 i=1  u(x, t)     u(x, 0)

= 0,

(x, t) ∈ ∂Ω × [0, T ],

= 0,

ut (x, 0) = 0,

(x, t) ∈ Ω × (0, T ],

x ∈ Ω.

The treatment of Galerkin’s approximations to the solution of partial differential equations using finite element techniques results in a system of equations. But for computing the coefficients X Z of the resulting linear system, being of the form φ(x)dx, we must consider the use of K

K∈Th R the quadrature scheme which consists in replacing the integrals K φ(x)dx by finite sums of the L X form ωl,K φ(bl,K ) with weights ωl,K and nodes bl,K ∈ K which are derived from a single l=1

quadrature formula defined over a reference finite element. The effect of numerical integration in finite element methods for solving elliptic equations, parabolic and hyperbolic equations has been analyzed by Ciarlet and Raviart [2], Raviart [3], So-Hsiang Chou and Li Qian [4], Li Qian and Wang Daoyu [9], Nguimbi Germain [10, 11, 12] and others. The main purpose of this paper is to find some sufficient conditions on the quadrature scheme which insure that the order of convergence in the absence of numerical integration is unaltered by the effect of numerical integration.

1

A Problem formulation Consider the following nonlinear hyperbolic intial-boundary value problem

 n n X X
   c(x, u)u = a (x)p(x, u)u + bi (x, u)uxi + f (x, t, u), tt ij x  j xi   i,j=1 i=1  u(x, t)     u(x, 0)

= 0,

(x, t) ∈ ∂Ω × [0, T ],

= 0,

ut (x, 0) = 0,

x ∈ Ω,

where Ω is a bounded domain in Rn with smooth boundary ∂Ω,

(x, t) ∈ Ω × (0, T ], (1.1)

3

The effect of numerical integration in finite element methods x = (x1 , x2 , · · · , xn ) ∈ Rn , ∇=

∂
∂ ,··· , , ∂x1 ∂xn

uxi =

∂u , ∂xi

uxj =

∂u , ∂xj

ut =

∂u , ∂t

utt =

∂2u , ∂t2

c, p, bi , f

are

known functions. We make several assumptions which we will refer to as condition (C): (i) there exist positive constants c0 , c1 such that √

c0 k ζ

n X

k22 6

aij (x)ζj ζi 6

√

c1 k ζ k22 , ζ 6= 0 ∈ Rn ,

where A = (aij (x))

i,j=1

is a symmetric matrix. Also c0 6 c(x, ω) 6 c1 , (x, ω) ∈ Ω × R,

and

√

c0 6 p(x, ω) 6

(ii) c(x, ω), ctt (x, ω), p(x, ω), f (x, t, ω) and bi (x, ω), i = 1, 2, · · · , n

√ c1 , (x, ω) ∈ Ω × R.

are Lipschitz continuous

with respect to ω. (iii) c(x, ω), p(x, ω), f (x, t, ω) and bi (x, ω), i = 1, 2, · · · , n, ¯ In the following, we set Sh = {χ ∈ C(Ω),

χ|Ω ∈ Pk (K),

together with all their partial derivatives up to cert χ|∂Ω = 0}.

Let {Sh }0 0 and any number p satisfying 1 6 p 6 ∞, as the usual Sobolev and Lebesgue spaces on Ω respectively. The associated norms and semi-norms are denoted as follows: k . km,p =k . kW m,p (Ω) ; | . |m =| . |H m (Ω) ;

| . |m,p =| . |W m,p (Ω) ;

k . k=k . kL2 (Ω) ;

k . km =k . kH m (Ω) ;

k . kLp =k . kLp (Ω) .

4

Nguimbi Germain et al. Let X be a normed vector space with norm k . kX . For ϕ : [0, T ] → X, define Z T 2 k ϕ(t) k2X dt; k ϕ kL∞ (X) = sup k ϕ(t) kX . k ϕ kL2 (X) = 06t6T

0

The weak form of (1.1) is as follows: Find u : [0, T ] → H01 (Ω) such that  n X


   c(u)u , v + A(u; u, v) = bi (u)uxi , v + f (u), v , v ∈ H01 (Ω), tt    i=1      



u(0), v = ut (0), v = 0,

(1.2)

v ∈ H01 (Ω),

where Z ϕψdx,

(ϕ, ψ) =

A(ω; ϕ, ψ) ≡

Ω

n X i,j=1




aij (x)p(ω)ϕxj , ψxi ≡

Z X n

aij (x)p(x, ω)ϕxj ψxi dx,

Ω i,j=1

c(ω) = c(x, ω), p(ω) = p(x, ω), bi = bi (x, u), f (u) = f (x, t, u), u(0) = u(x, 0), ut (0) = ut (x, 0). When numerical integration is not used, problem (1.2) has been extensively studied by T. Dupont [5], Yuan Yirang and Wang Hong [8] where optimal L2 and H 1 -estimates are obtained. Following [1], we give a general description of the corresponding formulation of (1.2) when numerical integration is present. ¯ with elements (K, PK , P ) with diameters 6 h. Let Th be a triangulation of the set Ω K The following assumptions shall be made P (i) The family (K, PK , K ), ˆ ). ˆ Pˆ , P finite element (K,

K ∈ Th for all h is a regular affine family with a single reference

ˆ the set of polynomials of degree less than or equal to k. (ii) Pˆ = Pk (K), [ (iii) The family of triangulations Th satisfies an inverse hypothesis. h

(iv) Each triangulation Th is associated with a finite-dimensional subspace Sh of trial functions ¯ which is contained in H 1 (Ω) ∩ C 0 (Ω). 0

ˆ We now intrduce a quadrature scheme over the reference set K. R P L ˆ ˆbl ), where the points ˆbl ∈ K ˆ x)dˆ ˆ and the x is approximated by l=1 ω ˆ l φ( A typical integral Kˆ φ(ˆ numbers ω ˆ l > 0,

1 6 l 6 L are respectively the nodes and the weights of the quadrature.

ˆ → x ≡ FK (ˆ ˆ onto Let FK : x ˆ ∈ K x) ≡ BK x ˆ + bK be the invertible affine mapping from K

5

The effect of numerical integration in finite element methods

ˆ are related K with the Jacobian of FK , det(BK ) > 0. Any two functions φ and φˆ on K and K ˆ x) for all x = FK (ˆ ˆ as φ(x) = φ(ˆ x), x ˆ ∈ K. The induced quadrature scheme over K is Z Z L X ˆ x)dˆ φ(x)dx = det(BK ) φ(ˆ x= ωl,K φ(bl,K ), ˆ K

K

l=1

with ωl,K ≡ det(BK )ˆ ωl and bl,K ≡ FK (ˆbl ), 1 6 l 6 L. Accordingly, we introduce the quadrature error functionals

Z EK (φ) ≡

φ(x)dx − K

ˆ ≡ ˆ φ) E(

Z

L X

ωl,K φ(bl,K ),

(1.3)

l=1

ˆ x)dˆ φ(ˆ x−

ˆ K

L X

ˆ ˆbl ), ωˆl φ(

(1.4)

l=1

which are related by ˆ EK (φ) = det(BK )E(φ).

(1.5)

Using these quadrature formulas, we next approximate the problem (1.2) by the following:    Find a map U : [0, T ] → Sh such that    n  X


c(U )Utt , χ h + Ah (U ; U, χ) = bi (U )Uxi , χ h + f (U ), χ h ,   i=1     U (0), χ
= U (0), χ
= 0, χ ∈ S t h h h

0 < t 6 T, χ ∈ Sh

(1.6)

where (ϕ, ψ)h ≡

L X X

ωl,K (ϕψ)(bl,K ),

U (0) = U (x, 0), Ut (0) = Ut (x, 0),

K∈Th l=1

Ah (ω; ϕ, ψ) =

n X

aij (x)p(ω)ϕxj , ψxi

i,j=1


h

≡

L X X K∈Th l=1

ωl,K

n X


aij (x)p(x, ω)ϕxj ψxi (bl,K ).

i,j=1

Let the elliptic projection W (t) of u(t) solution of (1.2) be defined by

W (t) : [0, T ] → Sh

such that A(u; W − u, χ) = 0,

χ ∈ Sh

(1.7)

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Nguimbi Germain et al.

2

Lemmas In what follows we set e = W − u and C will denote generalized constants independent of h

and Sh , and C may have different values at different places. Lemma 2.1.

Lemma 2.2.

[9]

Let u et W be the solutions of (1.2) and (1.7) respectively, then

k e k + k et k + k ett k +h{k e k1 + k et k1 } 6 Chk+1 ,

(2.1)

k e kL∞ (L2 ) + k et kL∞ (L2 ) + k ett kL∞ (L2 ) + k ettt kL∞ (L2 ) 6 Chk+1 .

(2.2)

[1,Thm.4.1.2]

ˆ x)dˆ Let there be given a quadrature scheme Kˆ φ(ˆ x= R

L X

ˆ ˆbl ) with ω ˆ l φ(

l=1 ˆ ), for which there exists an integer ˆ Pˆ , P ω ˆ l > 0, 1 6 l 6 L, over the reference finite element (K,

k > 1 such that: ˆ (i) Pˆ = Pk (K), (ii) the union

L [

ˆ {ˆbl } contains a Pk−1 (K)-unisolvent subset and/or the quadrature scheme is

l=1

ˆ exact for the space P2k−2 (K). Then there exists a constant a0 > 0 independent of h such that, for all approximate bilinear forms Ah (W ; ., .) and all spaces Sh , Ah (ω; v, v) > a0 k v k21 , ω, v ∈ Sh . Lemma 2.3.

[1,Thm.4.1.2]

Assume that, for some integer k > 1.

ˆ (i) Pˆ = Pk (K), (ii) the union

L [

ˆ {ˆbl } contains a Pk (K)-unisolvent subset and/or the quadrature scheme is

l=1

ˆ Then exact for the space P2k (K). c2 k v kh 6k v k6 c3 k v kh ,

| (v, ω)h |6 C k v kh k ω kh , v, ω ∈ Sh

v ∈ Sh ,

where

k v k2 ≡ (v, v)h .

7

The effect of numerical integration in finite element methods Lemma 2.4.

[1]

Assume, g ∈ C 0 (K). Then for all v, w ∈ Sh ,

| EK (gvw) |6 C k g kL∞ (K) k v kL2 (K) k w kL2 (K) where EK (.) is the quadrature error functional in (1.3). Lemma 2.5.

[4]

ˆ and that Assume that for some integer k > 1, Pˆ = Pk (K) ˆ = 0, ˆ φ) E(

ˆ ∀φˆ ∈ P2k−1 (K).

Then there exists a constant C independent of K ∈ Th and h such that for any 0

p ∈ W k+1,∞ (K), q ∈ Pk (K), q ∈ Pk (K), 0

0

| EK (pqxi qxj ) |6 Chk+1 K k p kW k+1,∞ (K) k q kH k (K) | q |H 1 (K)

(2.3)

where hK = diam(K). Using the analysis similar to that in the proof of Lemma 2.5[4] and [1, T hm.4.1.5], we have the following. Lemma 2.6. Under the same hypotheses as in Lemma 2.5. Furthermore assume that there exists a number q satisfying k + 1 >

n q.

Then there exists a constant C independent of K ∈ Th and 1

2 h such that for any f ∈ W k+1,q (K) and any w ∈ Pk (K), | EK (f w) |6 Chk+1 K (meas(K))

− 1q

k

f kW k+1,q (K) k w kH 1 (K) . Lemma 2.7. u, ut , utt ∈

[4, 8]

Assume that u, ut , utt , ∇u, ∇ut , ∇utt are bounded, and

L∞ (H k+1 ).

If k > n2 , then

k W kL∞ + k Wt kL∞ + k Wtt kL∞ + k ∇W kL∞ + k ∇Wt kL∞ + k ∇Wtt kL∞ 6 C. Lemma 2.8.

[4]

Assume that u, ut , utt ∈ L∞ (H k+1 ). Then for 1 6 s 6 k + 1,

k W kL∞ (H s ) + k Wt kL∞ (H s ) + k Wtt kL∞ (H s ) + k Wttt kL∞ (H s ) + + k ∇W kL∞ (H s ) + k ∇Wt kL∞ (H s ) + k ∇Wtt kL∞ (H s ) 6 C.

3

Error Estimates Our aim in this section is to derive the error estimates for U − u. Under the hypotheses of

Lemmas 2.2 and 2.3, it is easy to see that the problem (1.6) has a unique solution U since (1.6) can be written as a nonlinear system of ordinary differential equations (cf. [8]).

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Nguimbi Germain et al.

Theorem 3.1. Let u, U and W satisfy (1.2), (1.6) and (1.7) respectively. Assume that ˆ (i) Pˆ = Pk (K), ˆ x)dˆ (ii) the quadrature scheme Kˆ φ(ˆ x = R

L X

ˆ ˆbl ), ω ˆ l φ(

ˆ ω ˆ l > 0 is exact for the space P2k (K)

l=1

ˆ and the union and/or exact for the space P2k−1 (K)

L [

ˆ {ˆbl } contains a Pk (K)-unisolvent

l=1

subset. If u, ut ∈ L∞ (W k+1,∞ ), utt , uttt ∈ L∞ (H k+1 ), then under the conditions (C) with k + 1 >

n 2

and

for h sufficiently small, there exists a constant C = C(u) independent of h such that k U − W kL∞ (H 1 ) + k (U − W )t kL∞ (L2 ) 6 Chk+1 . Theorem 3.2. Under the hypotheses of Theorem 3.1, we have k u − U kL∞ (L2 ) + k (u − U )t kL∞ (L2 ) 6 Chk+1 . Proof. of Theorem 3.1 Let θ = U − W, e = W − u, then U − u = θ + e, θ ∈ Sh and θ(0) = θt (0) = e(0) = et (0) = 0. Let E(φψ) = (φ, ψ) − (φ, ψ)h . By Lemma 2.1 we have k e k6 Chk+1 ,

k et k6 Chk+1

(3.1)

Subtracting (1.2) form (1.6) and applying (1.7) and setting χ = θt , we have

(c(U )Utt − c(W )Wtt , θt )h + Ah (U ; θ, θt ) = E(c(W )Wtt θt ) − (c(W )Wtt − c(u)utt , θt ) + A(u; W, θt )+

−Ah (U ; W, θt ) +

n X

{(bi (U )Uxi , θt )h − (bi (u)uxi , θt )} + {(f (U ), θt )h − (f (u), θt )}

i=1

Integrating (3.2) with respect to t from 0 to t and using these following results Z 0

t

1 Ah (U ; θ, θt )dτ = − 2

Z

t

n X

1 (aij (x)ptt (U )Ut θxj , θxi )h dτ + Ah (U ; θ, θ), 2

0 i,j=1

(3.2)

9

The effect of numerical integration in finite element methods

t

Z

(c(U )Utt − c(W )Wtt , θt )h dτ = − 0

1 2

t

Z 0

1 (cu (U )Ut θt , θt )h dτ + (c(U )θt , θt )h + 2

t

Z

((c(U ) − c(W ))Wtt , θt )h dτ,

+ 0

Z

t

Z

t

Z ((c(W ) − c(u))Wtt , θt )dτ +

(c(W )Wtt − c(u)utt , θt )dτ =

(c(u)ett , θt )dτ. 0

0

0

t

We have Z Z n 1 1 t X 1 1 t (cu (U )Ut θt , θt )h dτ + (c(U )θt , θt )h + Ah (U ; θ, θ) = (aij (x)pu (U )Ut θxj , θxi )h dτ 2 2 2 0 2 0 i,j=1 Z t Z t Z t E(c(W )Wtt θt )dτ − (c(u)ett , θt )dτ − ((c(U ) − c(W ))Wtt , θt )h dτ + 0 0 0 Z t Z t + ((c(u) − c(W ))Wtt , θt )dτ + {A(u; W, θt ) − Ah (U ; W, θt )}dτ 0

0

Z tX Z t n + {(bi (U )Uxi , θt )h − (bi (u)uxi , θt )}dτ + {(f (U ), θt )h − (f (u), θt )}dτ 0 i=1 9 X

=

0

Ri

(3.3)

i=1

By Lemma 2.2, we have a0 1 1 c0 (c(U )θt , θt )h + Ah (U ; θ, θ) > k θt k2h + k θ k21 2 2 2 2 Since Z 1 t (cu (U )θt θt , θt )h dτ + (cu (U )Wt θt , θt )h dτ 2 0 0 Z n 1 t X − [(aij (x)pu (U )Wt θxj , θxi )h − (aij (x)pu (U )θt θxj , θxi )h ]dτ 2 0

1 R1 + R2 = 2

Z

t

i,j=1

Then by Lemmas 2.1 and 2.3, we have

Z

t

Z

2

t

(k θt kL∞ k ∇θ k + k ∇θ k )dτ + C (k θt kL∞ k θt k2 + k θt k2 )dτ 0 0 Z t 6 C(1+ k θt kL∞ (0,t;L∞ (Ω)) ) {kθt k2 + kθk21 }dτ

| R1 | + | R2 |6 C

2

0

Integrating by parts with respect to t, we have Z R3 = E(c(W )Wtt θ) −

t

E((c(W )Wtt )t θ)dτ 0

(3.4)

10

Nguimbi Germain et al.

Hence by Lemmas 2.6 and 2.8, we have t

Z

Chk+1 k (c(W )Wtt )t kH k+1 k θ k1 dτ kWtt kH k+1 kθk1 + 0 Z t 6 C{h2(k+1) + kθk21 dτ } + εkθk21 k+1

| R3 |6 Ch

0

By Lemmas 2.3 and 2.8, we have

Z

t

C(k ett k + k θ kk Wtt kL∞ (H k+1 ) + k e kk Wtt kL∞ (H k+1 ) ) k θt k dτ Z t 6 C{h2(k+1) + (kθk2 + kθt k2 )dτ }

|R4 | + |R5 | + |R6 | 6

0

0

Z

t

R7 =

n X

[(aij (x)p(u)Wxj , θtxi ) − (aij (x)p(U )Wxj , θtxi )h ]dτ

0 i,j=1

By integrating by parts, we have n X

R7 = Z − Z −

n X

(aij (x)(p(u) − p(U ))Wxj , θxi ) +

i,j=1 n t X

i,j=1

Z

n X

E(aij (x)pu (U )Ut Wxj θxi )dτ

0 i,j=1 t

Z

(aij (x)(p(u) − P (U ))Wxj , θxi )dτ −

6 X

t

(aij (x)(pu (u)ut − pu (U )Ut )Wxj , θxi )dτ −

0 i,j=1 n t X

0 i,j=1

=

E(aij (x)p(U )Wxj θxi )

n X

E(aij (x)P (U )Wtxj θxi )dτ

0 i,j=1

Fl

l=1

And by using the following inequality 2

Z

kθk 6 C

t

(kθk2 + kθt k2 )dτ,

(3.5)

0

We have

2(k+1)

| F1 |6 C k u − U kk ∇W kL∞ k ∇θ k6 C{h

Z +

t

(kθk2 + kθt k2 )dτ } + εk∇θk2 .

0

Notice that by the interpolation theory in Sobolev spaces we havec kW − πuk 6 kW − uk + ku − πuk 6 Chk+1 ,

(3.6)

11

The effect of numerical integration in finite element methods

ku − πukL∞ 6 Chk+1 .

(3.7)

where πu denotes the Sh -interpolant of u. We have n X

F2 = +

E(aij (x)(p(U ) − p(W ))Wxj θxi ) +

i,j=1 n X

E(aij (x)(p(πu) − p(u))Wxj θxi ) +

i,j=1

=

4 X

n X i,j=1 n X

E(aij (x)(p(W ) − p(πu))Wxj θxi ) E(aij (x)p(u)Wxj θxi )

i,j=1

Kl .

l=1

By using Lemma 2.4 and (3.5), we have with Z

t

p(U ) − p(W ) =

pu (W + τ (U − W ))dτ (U − W ) 0

n X

| K1 |= |

Z

t

E(aij (x)

pu (W + τ θ)dτ Wxj θθxi )| 0

i,j=1

Z t 6Ck pu (W + τ θ)dτ ∇W kL∞ k θ kk ∇θ k 0 Z t 6C (k θ k2 + k θt k2 )dτ + ε k ∇θ k2 . 0

Similarly we have by using Lemma 2.4 and the inequalities (3.6) and (3.7) Z

t

| K2 | + | K3 |6 C k

pu (πu + τ (W − πu))dτ ∇W kL∞ k W − πu kk ∇θ k + 0

+C k (πu − u)∇W kL∞ k ∇θ k 6 Ch2(k+1) + ε k ∇θ k2 We have by using Lemmas 2.5 and 2.8 and the inequality X X
1
1 X | bK |2 2 . | aK bK |6 | aK |2 2 K

K

K

| K4 |6 Chk+1 k p(u) kW k+1,∞ (Ω)

X

k W kH k (K) k θ kH 1 (K)

K k+1

6 Ch

X

kW

k2H k (K)

K 2(k+1)

6 Ch

+εkθ


1 X 2

K

k21

.

k θ k2H 1 (K)


1 2

12

Nguimbi Germain et al. Hence | F2 |6 C{h2(k+1) +

Rt

2 0 (kθk

+ kθt k2 )dτ } + εkθk21 .

we have Z t X Z n −F3 = (aij (x)(pu (u) − pu (U ))ut Wxj , θxi )dτ + 0 i,j=1

n X

t

(aij (x)pu (U )(ut − Ut )Wxj , θxi )dτ

0 i,j=1

Thus t

Z |F3 | + |F5 | 6

Z

t

C{ket k + kθt k}k∇W kL∞ k∇θkdτ Z t Z t 2(k+1) (kθk21 + kθt k2 )dτ }. C{kek + kθk}k∇θkk∇W kL∞ dτ 6 C{h + + C{kek + kθk}kut kL∞ k∇W kL∞ k∇θkdτ +

0

0

0

0

We have Z

t

−F6 =

n X

E(aij (x)[p(U ) − p(W )) + (p(W ) − p(πu)) + (p(πu) − p(u)) + p(u)]Wtxj θxi )dτ

0 i,j=1

Similar as in estimation of F2 and by using Lemma 2.7, we have Z t 2(k+1) |F6 | 6 C{h + kθk21 dτ }. 0

Therefore |R7 | 6 C{h

2(k+1)

Z +

t

(kθk21 + kθt k2 )dτ } + εk∇θk2 .

0

We have R8 =

n Z X

t

[(bi (U )θxi , θt ) + ((bi (U ) − bi (u))Wxi , θt ) + (bi (u)exi , θt ) − E(bi (U )θxi θt ) +

0

i=1

−E((bi (U ) − bi (W ))Wxi θt ) − E((bi (W ) − bi (πu))Wxi θt ) − E((bi (πu) − bi (u))Wxi θt ) + 8 X 0 −E(bi (u)Wxi θt )]dτ = Fl . l=1

Thus 0

0

Z t C k ∇θ kk θt k dτ + C k U − u kk ∇W kL∞ k θt k dτ 0 0 Z t 2(k+1) 6 C{h + (kθk21 + kθt k2 )dτ }. Z

t

|F1 | + |F2 | 6

0

By Green’s formula and integrating by parts with respect to t, we have 0

F3 = −

n Z X i=1

0

t

(bi (u)xi e, θt )dτ −

n X i=1

(bi (u)e, θxi ) +

n Z X i=1

0

t

(bi (u)t e + bi (u)et , θxi )dτ

13

The effect of numerical integration in finite element methods

0

Z t C{kek + ket k}k∇θkdτ + Ckekk∇θk Ckekkθt kdτ + 0 0 Z t 2(k+1) 6 C{h + (k∇θk2 + kθt k2 )dτ } + εk∇θk2 . t

Z

|F3 | 6

0 t

Z

0

Z C k bi (U ) kL∞ k ∇θ k kθt kdτ 6 C

|F4 | 6

t

(k∇θk2 + kθt k2 )dτ

0

0

Similar as in estimation of F2 , we have 0

0

0

|F5 | + |F6 | + |F7 | 6 C{h

2(k+1)

t

Z

(kθk2 + kθt k2 )dτ }.

+ 0

By integrating by parts, we have n X

0

F8 = −

E(bi (u)Wxi θ) +

i=1

n Z X i=1

t

E((bi (u)Wxi )t θ)dτ

0

Similar as in estimation of K4 , we have 0

|F8 | 6

n X X

Chk+1 K k bi (u)Wxi kH k+1 (K) k θ kH 1 (K) +

i=1 K∈Th

+

n Z X i=1

t

X

0 K∈T h

Chk+1 K k (bi (u)Wxi )t kH k+1 k θ kH 1 (K) t

Z

2(k+1)

6 C{h

kθk21 dτ } + εkθk21 .

+ 0

And therefore 2(k+1)

|R8 | 6 C{h

Z +

t

(kθk21 + kθt k2 )dτ } + εkθk21

0

We have

Z

t

{(f (U ) − f (W ), θt )h − E(f (W )θt ) + (f (W ) − f (u), θt )}dτ

R9 = 0

And integrating by parts, we obtain Z t Z t Z t R9 = (f (U ) − f (W ), θt )h dτ − E(f (W )θ) + E((f (W ))t θ)dτ + (f (W ) − f (u), θt )dτ 0

0

0

Thus by Lemmas 2.3 and 2.6, we have Z t |R9 | 6 Ckθkkθt kdτ + Chk+1 kf (W )kH k+1 kθk1 + 0 Z t Z t k+1 + Ch k(f (W ))t kH k+1 kθk1 dτ + Ckekkθt kdτ 0 0 Z t 6 C{h2(k+1) + (kθk21 + kθt k2 )dτ } + εkθk21 . 0

14

Nguimbi Germain et al.

Substituting (3.4) and the above estimates Ri , 1 6 i 6 9, into (3.3) and choosing ε sufficiently small, kθk21 + kθt k2h 6 C{h2(k+1) + 1 + kθt kL∞ (0,t;L∞ (Ω))




Z

t

(kθk21 + kθt k2 )dτ }

(3.8)

0

Let us make the induction hypothesis that there exists a small h0 > 0 such that for 0 < h 6 h0 , kθt kL∞ (0,t;L∞ (Ω)) 6 1,

0 6 t 6 T.

(3.9)

By Gronwall’s lemma for (3.9), we conclude from Lemma 2.3 that kθk21 + kθt k2 6 Ch2(k+1) ,

0 6 t 6 T,

Thus kθkL∞ (H 1 ) + kθt kL∞ (L2 ) 6 Chk+1 , i.e, kU − W kL∞ (H 1 ) + k(U − W )t kL∞ (L2 ) 6 Chk+1 .

(3.10)

Similar as in the proof of the inequality (20) in [8] we can prove that (3.9) holds. This completes the proof of Theorem 3.1. Proof. of Theorem 3.2 Applying Lemma 2.1 together with the inequality (3.10) of Theorem 3.1, the interpolation theory in Sobolev spaces and the triangle inequality, it follows that ku − U kL∞ (L2 ) + k(u − U )t kL∞ (L2 ) 6 Chk+1 .

(3.11)

References [1] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. [2] P.G. Ciarlet and P.A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, Chapter in Mathematical Foundations of the Finite Element Methods with Application to Partial Differential Equations, Academic Press, New York, 1972. https://doi.org/10.1016/b978-0-12-068650-6.50020-4 [3] P.A. Raviart, The Use of Numerical Integration in Finite Element Methods for Solving Parabolic Equations, in Topics in Numerical Analysis, S.H. Miller, Ed, Academic Press, New York, 1973.

The effect of numerical integration in finite element methods

15

[4] So-Hsiang Chou and Li Qian, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Equations, Numerical Methods for Partial Differential Equations, 6 (1990), 263-274. https://doi.org/10.1002/num.1690060306 [5] T. Dupont, L2 -Estimates for Galerkin methods for second order hyperbolic equation, SIAM J. Numer. Anal., 10 (1973), 880-889. https://doi.org/10.1137/0710073 [6] M.F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), 723-759. https://doi.org/10.1137/0710062 [7] J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1970), 575-626. https://doi.org/10.1137/0707048 [8] Yi-rang and Hong Wang, Error estimates for the finite element methods of nonlinear hyperbolic equations, J. Systems. Sci. Math. Sci., 5 (1985), no. 3, 161-171. [9] Li Qian and Wang Daoyu, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Equations, Pure and Applied Mathematics, 2 (1991), 57-61. [10] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Equations, Appl. Math.-A J. Chinese Univ. Ser. B, 16 (2001), no. 2, 219230. https://doi.org/10.1007/s11766-001-0029-8 [11] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Integrodifferential Equations, Shandong University (Natural Science) Journal of Shandong University, 36 (2001), no. 1, 31-41. [12] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Sobolev Equations, Numerical Mathematics, Journal of Chinese Universities, 9 (2000), no. 2, 222-233.

Received: September 2, 2016; Published: January 10, 2017