Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions

M2AN. MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS

- MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

A. S ZEPESSY

Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions

M2AN. Mathematical modelling and numerical analysis - Modéli- sation mathématique et analyse numérique, tome 25, n^o6 (1991), p. 749-782

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MOOÉUSATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE (Vol. 25, n° 6, 1991, p. 749 à 782)

CONVERGENCE OF A STREAMLINE DIFFUSION FINITE ELEMENT METHOD FOR SCALAR CONSERVATION LAWS WITH BOUNDARY CONDITIONS (*)

A . SZEPESSY O Communicated by R. TEMAM

Abstract. — A higher order accurate shock-capturing streamline diffusion fïnite element method for gênerai scalar conservation laws is analysed ; convergence towards the unique solution is proved for several space dimensions with initial and boundary conditions, using a uniqueness theorem for measure valued solutions. Furthermore, some numerical results are given.

Résumé. — On démontre, pour des solutions approchées par la méthode d'éléments finis

« streamline diffusion », la convergence vers la solution exacte dans le cas d'une loi de conservation scalaire générale dans un domaine borné Cl de Rd, d s* 1, avec des conditions aux limites sur la frontière de Cl. On utilise un théorème d'unicité de la solution qui peut être une mesure. Finalement, quelques exemples numériques sont considérés.

0. INTRODUCTION

In this note convergence of a higher-order accurate shock-capturing streamline diffusion fînite element method (SC-method) is proved for gênerai scalar conservation laws in several space dimensions with initial and boundary conditions, using the uniqueness theorem for measure valued solutions in [Sz III]. This theorem, which is a generalization of the corresponding resuit for the pure initial value problem by DiPerna [Di], yields convergence in Lp> l=s/>===oo, towards the unique solution, for approximate solutions of a scalar conservation law provided they are

(A) uniformly bounded in the Z^-norm,

(B) weakly consistent with ail entropy inequalities, (C) strongly consistent with the initial condition.

(*) Received December 1989.

O IN A DA, Royal Institute of Technology, 10044 Stockholm, Suède.

M2AN Modélisation mathématique et Analyse numérique 0764-583X/91/06/749/34/S 5.40

In Section 3 the iSC-method is proved to satisfy (A) and in Section 4 the conditions (B) and (C) are verifîed. We note that the convergence proof does not require estimâtes of the total variation, which are usualiy used together with classical compactness arguments to prove convergence of finite différence schemes. The only previous results for fini te différence methods applied to scalar conservation laws with boundary conditions are given in [LR I] and [LR II], where convergence was obtained by classical total variation estimâtes. We also remark that in the case of scalar conservation laws in an unbounded domain, i.e. without boundary con- ditions, one can replace the assumption (A) by « uniformly bounded in Z^-norm, 1 === q ^p » if the flux involved in the conservation law growths as a polynomial of degree not more than p in infînity, see [Sz II].

We now give some background material on the SC-method. The streamline diffusion method is a gênerai finite element method for hyperbolic problems which may be viewed as a certain combination of the standard Galerkin method and a least squares method giving added stability through the weighted least squares control of the residual. In the shock- capturing variant artificial viscosity is added with the viscosity coefficient depending locally on the residual and the mesh parameter h. The effect of the shock-capturing term is to add substantial artificial viscosity locally where the solution is non smooth, which improves the quality of the approximations near shock fronts compared to the streamline method. The shock-capturing streamline diffusion method combines &{hk+xl2) accuracy for smooth solutions approximated by polynomials of degree k9 with good stability obtained through the least squares control of the residual and the shock-capturing artificial viscosity.

The resuit presented hère extends to higher order éléments the analysis of SC-methods with piecewise linear éléments initialized in [JSz I, JSz III] and continued in [Sz I]. In [JSz III] numerical results were presented for the Euler équations in two space dimensions and convergence was established for a Cauchy problem in one dimension for Burgers' équation using the theory of compensated compactness. Further, in [Sz I] convergence to the unique solution was proved for a SC-method applied to a Cauchy problem for a scalar conservation law in two space dimensions by using the uniqueness resuit for measure valued solutions of DiPerna [Di],

In Section 5 the resuit of some numerical experiments are presented.

1. MEASURE VALUED SOLUTIONS WTTH BOUNDARY CONDITION

In this section we recall some results for measure valued solutions of scalar conservation laws with initial and boundary conditions given in [Sz III]. The proof of convergence of the finite element solutions will be based on Theorem 1.1 below.

Let H be a bounded open set of Rd with smooth boundary F = dû with

MAN Modélisation mathématique et Analyse numérique

outward unit normal n. We consider for u : Cl x R+ -• R the conservation law

(1.1) ut+ Y,fj(u)Xj = 0 i n O x i î + ,

j = \

with initial condition

(1.2) w(.,0) =u⁰ in Cl,

and boundary condition : for ail k e R, (x, t) e Y x R+

(1.3)

(sgn («(S, t) - k) - sgn (a(S, /) - *))(/•(«(*, r)) - ƒ ( * ) ) . « (X) » 0 , where ƒ = (ƒ „ ..., f d) : R -> Rd, uo:Cl->R and a : T x /î+ -> iî are given smooth functions and the function sgn :/?->/? is defîned by

- G

We remark that i f / is linear, then (1.3) requires u to be equal to the given boundary data a on the inflow boundary (where f'.n^ 0), but does not impose any boundary condition on outflow boundaries (where ƒ ' . n => 0).

More generally, in [BLN] it is proved that solutions ue of parabolic regularizations of (1.1) defined by

d

u\ + £ fj(u^e)^Xj = e àu^e in fl x i?⁺ ,

J = i

«e(., 0 ) = «0 on H , ue = a on r x i ?+,

converge a.e. to a function w e 5 V (Q x R+) as s -> 0 + , which is the unique weak entropy satisfying solution (the i?F-solution) of (1.1-1.3).

Moreover, the trace of u satisfies (1.3). Dubois and Le Floch [DL] have pointed out that (1.3), in the case of one space dimension, is equivalent to u being a solution of a certain Riemann problem on the boundary.

Let us now recall the concept of measure valued solution of (1.1-3) given in [Sz III] which will be used to prove convergence of the finite element method. Let then {uj} be a uniformly bounded séquence in Lo0(fl x R+ ) , Le.,

(in the applications the Uj will be approximate solutions of (1.1-1.3)). Then àccoï&irig-' to Youngs theorem there exists, cf. [Di], [Ta], a subsequence, which we still label {«,-}, and an associated measurable measure valued mapping v^{( )}: î l x i ?⁺ -+ Prob (R), such that

(1.5) suppv^{( j C f O}c {k: \k\ ^K} , a.e., (x, t) e Cl x R⁺ ,

and Vgr G *ë (R) the L^iO, x i ?+) weak star limit (1.6a) g(uj(,)) g(.) as j -> oo , exists, where

(1.66)

0(x,t)=\ g(K)dv

f

Hère Prob (R) is the space of ail positive Borel measures on R of unit mass and v( ) is measurable means that (vyig) is measurable with respect to y G n x i?⁺ for each continuous function #. The mapping y is a Young measure associated with the séquence {M,-}.

To define measure valued solutions satisfying boundary conditions we shall associate with a given Young measure v: fl x R± -• Prob (R) satisfy- ing (1.5), in gênerai in a non unique way, a Young measure yv^y. F x R+ -> Prob (R), which we consider to be a «trace» of v on F x R⁺. For this purpose we introducé the change of coordinates x -+ (5c, y ) for x in a neighbourhood of F :

(1.7) x = x-yn(x)9

where (x, y) G F x (0, e ), for some e > 0.

L E M M A 1.1 [Sz III] : Let vMx R+ ^ Prob (iî) be a Young measure associated with a séquence {uj} satisfying (1.4). Then there is a séquence {yj G (0, e ) } , where j>y —> 0, ö«öf ?/î^re is « measurable Young measure

yv:F x R+ -y P r o b ( i ? ) JMCA

s u p p 7 v( 5 f 0c {X: | \ | ^ i ^ } a.e., (x, t) G F x R+ , rf, /or everj g e ^(R), the ^ ( F x R+ ) vvea/c 5?ar /z'mzï

<v(x(.,^)5.> ff (M) -^ fl(-3 • )> as 7 -• oo , exists, i.e.

(1.8a) l i m f (vix{x )tt)9g(\))<pdsdt= \ g(x, t) <p ds dt,

j^OD JTxR+ JTXR+

for ail <p G Lr(T x R+ ), where ds is the Lebesgue measure on F, and (1.86) 0(x, 0 = f » ( X ) r f y vt t 0 S <7vft0, jf(X)> ,

/or a.e. (x, r) ^G F xR+.

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DEFINITION [SZ III] : A Young measure v, associated to a séquence {uj} which satisfies (1.4), is a measure valued solution (mv-solution) to (1.1- 1.3) if for ail <|> e ^ ( f i x R+ ), * ^ 0, and for all k e R, we have

(1.9a) f (<v(Xj/), |X-*|><|>,+ <v(X)?), ( s g n ( X - * ) ) x

V*)dx A - f <7v(x- 0, ƒ (X) - ƒ(*)>

J rXi ?+

x «(x) c() sgn (a — k) ds dt ^ 0 , and

H m f < v( X ) 0, | X -

Remark 1.1 : In gênerai Lemma 1.1 associâtes with v a trace yv in a non- unique way. However, in the proof of Theorem 1.1 in [Sz III] it is seen that the expected value (yvç),/(X)), which appears in the définition (1.9&), is in fact uniquely defîned. D The following uniqueness resuit for mt?-solutions is proved in [Sz III].

THEOREM 1.1 [Sz III] : Suppose that a Young measure v associated with the séquence {uj} is a mv-solution to (1.1-1.3) and let w dénote the unique BV-solution o/(l.l-1.3). Then

i.e., V(x,t) reduces a.e. to the Dirac measure concentrated at w{x, t), and the séquence {uj} converges strongly in L}OC(IÎ X R+) to w.

2. FORMULATION OF THE FINITE ELEMENT METHOD

In Sections 2-4 we shall use Theorem 1.1 to prove strong convergence in L\oc towards the unique £F-solution of (1.1-1.3) for the SC-method. First we consider a one-dimensional problem, i.e. we let H = (0, 1) and T = 3H = {0, l } . The generalization to several dimensions is straight forward, see Remark 2.2. We shall assume that there is a constant C such that the smooth function ƒ also satisfies

(2.1) sup \f»(y)\ * s C .

y€R

This is no essential restriction since the exact solution is bounded and thus f (y) may be modifled for large \y\ so as to satisfy (2.1). Below we dénote

by C a positive constant independent of h, not necessarily the same at each occurrence.

Let us now introducé the flnite element spaces of the SC-method for (1.1- L3). Let 0 = tö <=: tx < t2 ... < t $ = T be a séquence of time levels, set In = (tn> tn+ i) a n (i introducé the « slabs » Sn = Q x In and sets fln = n x {tn}. For h>0 and n = 0, 1, 2, ... let 7^ be a quasi-uniform triangulation of 5„ into triangles K of diameter h^K~h with smallest angle uniformly bounded away from zero and let K have one right angle if K H F # 0 (in the case of several space dimensions this is generalized to mean that the set of space coordinates, of the vertices oî Ke T% such that

^ n r # 0 , are the same for t = tn + l and t == r j . Defîne for a given natural number & 2= 1

Vnh= {v e H\Sn):v\Ke Pk(K), Ke Tnh,v\TxR+ = o} ,

r*= n ^'

n s 0

where Pk(K) dénotes the set of polynomials of degree at most k on K. In other words, VI consists of continuous piecewise polynomials on the slab Sn. Typically, tn + ï - tn~h with the slab Sn one element wide. To defîne the shock-capturing modification also for k ^ 1, we divide each KeT^ into similar triangles Kh i = 1, 2, 3, ..., k2 and introducé 7£ = {jf,- : z =

1, 2, 3, ..., k2, KeT%},

VI - e ? , ( ! , - ) , z = 1, ..., k2, Ke Tnh, v | r ^ = o } ,

We also introducé the usual nodal interpolation operators

where the degrees of freedom of Vh and FA are the values at the vertices of

&i ^ il.

We will seek an approximate solution uh - Uh + â ,

MAN Modélisation mathématique et Analyse numérique

where Uh e Vh and â : Ù x R+ is a smooth extension of a. Note that the functions in Vh and Vh are zero on F x R+ and continuous in x and possibly discontinuous in / at the discrete time levels tn.

We shall need the following standard interpolation error estimate (2.2),

«super-approximation» resuit (2.3) and inverse estimate (2.4) (a proof of (2.3) is given in the appendix).

L E M M A 2.1 : There are constants C such thatfor w e Ws>p(a>) H <%(Sn)9

veV^h9n = O⁹ 1, 2, ...

(2.2a) ||w-7rvt;||^too(ü))^Ch5-r||M;||^Co(üï), s = 1, ..., k + 1 ,

r=0, 1 , p = oo , (2.2Ô) \\w-™\\rM*Ch*+l-^w^+lM,

r = 0, 1 , 1 =s k^k , p = 2 , (2.3fl) ||t7^W-7T(i7^W)||^«^(œ)^Ch¹-n|v|L^w(M)||M;||^ï^.«^((B),

r = 0, 1 , p = oo , (2.3Ô) ||vw - ^ ( ^ ) | | ^( w ) ^ C | | I ; | |L Û O ( W/ £ AI"-r||w||rf(-),

i = i

r = 0, 1 , /» = 2 ,

(2.3c) ||tw - T r ( ^ ) | |L 2 ( n j ^ C||t)||L œ ( S n ) £ A'"5 l l w l l ^ ^ , , /» = 2 ,

(2Aa) ||^

( 2 . 4 6 ) I I ^ L ^ ^ ^

where o> = Q,n, Sn, K n O,n or K n Sn for K e Th and W**p is the usual Sobolev space (hère dot dénotes semi norm and W"¹ = H*). The same estimâtes hold if we replace ir by TT, Vh by Vh and k by 1.

The SC method for (1.1-1.3) can now be formulated as foliows : Find u^k = U^h + â where U = U^h G V^h such that for n = 0, 1, 2, ..., (2.5) |Js

Jsn

r f -

E²(U) U^xv^xdx dt

L

where

w = Ttw , VH> G V h ,

L(U) ^ Ut + f'(U + â) Ux+ g(U, x, t) , g(U,x,t)=f'(U+â)âx+ât3

dxdt, K

=E f

JK

f \U

-U_\dxf \ dx,

J K n an J K n nn

e if ï n ( r x i+) ^ 0

0 otherwise

i?± (x, r) = lim u(x, f + s) , C/_ (., 0) = «o • Further ô, ô, 8, e are positive parameters satisfying (2.6) 8 = Ch , 8 = Ch" 8 = Ch"2,

(h + 8 + E/h + 8)/e where the are constants with -3 1

2, - < a2< l .

0 as h -> 0 , From now on U = Uh will dénote a solution of (2.5).

Existence of a solution to (2.5) follows from a variant of Brouwer's fixed point theorem as in [Li], [JS].

Remark 2.1 : Comparing with the SC-method for the pure initial value problem considered in [JSz III] and [Sz I], we consider hère an in- homogeneous problem with source term g ^ 0, and the shock-capturing term is accordingly modified to contain the total residual L(U). Further, we add extra diffusion (tyuxx) on the éléments next to the boundary F = an. In [Sz I] we proved that the method (2.5) with k = 1 is accurate of order & (h3/2) for a corresponding linear conservation law with smooth exact solution. This analysis easily extends to the method (2.5) in the nonlinear case with k > 1 if the exact solution is smooth, giving accuracy &(hk+ ^2) away from the boundary (and &(h^l+d^2) in a neighbourhood of the boundary) by using Lemma 3.4 below. D

The main resuit is the following.

THEOREM 2.1 : Thefunctions uh = Uh + à, where Uh e Vh satisfies (2.5), converge strongly in L\0C(Q X R+) to the unique BV-solution of (1.1-1.3) as h tends to zero.

M2AN Modélisation mathématique et Analyse numérique

The proof is divided into Lemma 2.2 and Lemma 2.3, which are proved in Sections 3 and 4, respectively.

LEMMA 2.2 : There is a constant C such that the functions uh in Theorem 2.1 satisfy

(2-7) K UL „ ( n x* + )^ , A ^ O .

LEMMA 2.3 : There is a subsequence of the functions uh given in Theorem 2.1 that generale an mv-solution v of (1.1-1.3).

Remark 2.2 : The proof of Theorem 2.1 extends with the following to a gênerai scalar conservation law in several dimensions yielding convergence of the SC-method applied to the conservation law (1.1-1.3) with d^ 1.

The only major modifications in the case with d > 1 occur in the proof s of Lemma 3.2 and Lemma 3.3. In [Sz I] the corresponding results are Lem- ma 4.1 and Lemma4.2, and they are proved for d=2 for a special triangulation of fi x {tn} into triangles i^with one right angle and dividing each prism K x (tⁿ, t^{n + l}) into three tetrahedrons. The proof of Lemma 4.2 in [Sz I] easily extends to an element given by the ^-simplex Ed =

d ï

{x G Rd: 0 =s xt, £ xt === h \. In particular this proves Lemma 3.2 below for éléments which at t = tn reduce to sets Ed_ x x {/„} (with the new constant chp"2 which does not change the analysis). For gênerai triangulations with ds* 19 Lemma 3.2 and 3.3 thus remain true for a SC-method if the shock- capturing terms are defined by

I

s2(U)V'xU.V'j>dxdt,

with e(- as before and where V' dénotes the gradient calculated in the orthonormal coordinates on Ed which are given by the linear transformation of K onto Ed and V^ is the space-gradient in orthonormal coordinates on E^d _ x which are given by the linear transformation of K O O,ⁿ onto Ed-i x {tn}.

The localization result in Lemma 4.1 for d > 1 foliows if we let the normal coordinate y, which is defined in (1.7), play the role of « x ». Further, in the super approximation result (4.12) requiring k + 1 :> d/2, we can instead use Lx and L^ bounds as in (4.10) combined with (2.3a) in which case no restrictions on d enter. •

3. PROOF OF LEMMA 2.2

First we state and prove the following basic L2 stability estimate.

L E M M A 3.1 : For N = 0, 1, 2, ..., we have

\ [ \ (U_)

dx+ £ f (U

-U_)

dx) +

+ 8 | f (U

+f' U

fdxdt\ + f e

uldx+\ g(0,x,t)2dxdt) ,

J ^ /

n

AT

where S^N = \^J Sⁿ and intégrais over S^N are interpreted as a sum of intégrais

n = 0

over the Sⁿ.

Proof: Taking v = U in (2.5) we obtain

(3.1) 0 = f L(U)(U+h{Ut+f'Ux)) dxdt+ f (U+ -U_)U+ dx + + | tx{U)\VÛ\2dxdt+ \ E2(U)Û2xdxdt

= ^ ( f (t/_)

^x- f (C/_)

^x+ f (U

-U_)

dx)

(Ut+f'Ux)2dxdt+ \ e{(U)\VÛ\2dxdt

c

&²(U)Ûldxdt+ \ if' U^vU+gU)dxdt

+ 5 f g(f' Ux+ Ut)dxdt.

Define now the function F^m for m s N by

(3.2) Fm(s,x,t)=

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CONVERGENCE OF A STREAMLINE DIFFUSION

We have by (2.1) (3.3)

UxUdx

759

) ff

Ja

| ^-Fl(U,x,t)dx- I | (f'(w+ a))xwdwdx Jn "x Ja Jo

- I (f'(w+a))xwdwdx =s C U2dx.

Further by (2.1) (3.4)

so that by summing over n = 0, 1, 2, ..., N in (3.1), we get

(3.5) ^ ( | ( £ O

< / x + £ J (U

-U_)

dx +

+ 8 f (Ut+f'Ux)2dxdt

f

JsN

^ 1 f uldx+C i f U2dxdt+ f (g(0,x,t))2dxdt ) .

2 Ja \ Js" J^^ /

Also, for tN <t' ^tN + x

| U2(x,t')dx= f U2_ dx-

Ja J% + i Jf'

U2_dx- \""2(Ut + f' Ux)Udxdt

2 (/'(v^+ a))xwdwdxdt

Jt Jo

T (Ut + f Ux)2dxdt +

JsN

Hence, by a Gronwall inequality

(3.6) f (72(x, r') dx ^ C I \ Uldx + b \ (Ut + ƒ ' t/x)2 dx dt\ .

Letting now

N r

Ul dx,

N Ç

^ Jo

we then have by (3.5)

WN + i ~WN^c[\ u$dx + I (g(fi9x9t)fdxdt\ + ChWN.

\Jft JS^N » J

Using a discrete Gronwall argument we get

(3.7) hWN + l * c t f u\dx+ \ (g(0,x,t))2dxdt) ,

\ J a JsN J

which together with (3.6) inserted int o (3.5) proves the lemma. D Now we turn to the proof of the L^-estimate (2.6). This resuit will follow by letting p tend to infînity in Z^-estimates obtained by multiplication with

TÏ(UP~1). First we give three preliminary results in Lemma 3.2-3A.

LEMMA 3.2 : There is a positive constant c, independent of p such that f or p — 2 m, m = 1, 2, 3, ..., and n = 0, 1, 2, ...

(3.8) ch £ f

K 7 i

rx(it(Û'-l))xdxdt.

L

The proof of this resuit is analogous to the proof of Lemma 4.2 in [JSz III]. Further we have.

LEMMA 3.3: There is a constant c :> 0 independent of p such that for p = 2 m, m — 1, 2, 3, ..., n ~ 0, 1, 2, ...

3.9) I

(3.9) I EX(U) VU.

Proof \ Considering a triangle K e f^h and the three associated heights, there is always one that stands on a side of the triangle. Choose this side and the orthogonal height as coordinate directions (cf. fig. 3.1).

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CONVERGENCE OF A STREAMLINE DIFFUSION 761

2'

Figure 3.1.

Since zu VU.V>n(Ûp-1) and \VÛ\ \\Û\\Al2 P . are constant on K, it is

i i il il L ( A )

enough to prove that VU .V%(lp-X)^ ~ \VÛ\2 \Û¥~^.. Defîne the

i i M M LKK)

p

function fp : S2\ {± 3~ 1/2(1, 1, 1 )} -• R by

00(A)

where yh y2, y3 and ^2 a r e the values associated with the vertices and to the point 2', respectively, i.e. there is a p G [0, 1 ] such that

and

We note that fp is continuous on ^S2 so that by the définition o f f^p we have VU. V%{(p- ^l) ^ min fJy) \VÛ\² \\Û\\^P~². . We shall now

1 LœiK)

estimate min fp. Let us first assume that \y2 \ = max ( |yx |, | y2 \, | y3 \ ) (we s²

may then assume that l ; ^ - . ) ^ ! ^ 1^2 — 7 i | ) - Iⁿ this case we have

,2 - ƒ 2 ' )

2

- J1)2 + (y2 - y ' i ) \ y2 ! (y+ — 3

"2 C

F u r t h e r , i n t h e c a s e w h e n |>>3| = m a x ( | > > i | , \yi\,

so t h a t if also \y² — y\\ =£ |>^i — J^31 »

yî

a n d if \y² — yi\ >• \y\ —y-$\ (we still have \y³\ = m a x ( | ^ - | ) , then

• + —

+ 0>2 - y ' i )2 \y* I Oi -y-if + O2 - y i ) 1 2 * p1'

This proves Lemma 3.3, since fp(y\,y2>y3) is symmetrie in yx and

LEMMA 3.4 : rAere are constants c, C => 0 independent ofp and h such that for we Vh and KeTh

(3.10) c\\Vw\\T _ ^ ||Vw

(3.12) ^ ^ ^ ^ H n|

^

/? = 1 , 2 , 3 , . . . , . The same inequalities holds with V replaced by — . K by K C\ fln and K^t by Kj n nⁿ.

Proof: First we note that w = TTW, SO by standard interpolation estimâtes we obtain the left inequality in (3.10). Further, the polynomial w\^K has coefficients which depend linearly on the nodal values of w | ^ i = 1, ..., k \ Therefore we can defïne w | K using one nodal value together with k' = (k + 1 ){k + 2)/2 - 1 linearly independent différences (Pt, ..., P jfcO s P of nodal values of vv|^.. These variables also détermine MAN Modélisation mathématique et Analyse numérique

.. In order to prove the right inequality in (3.10) we defme the function

I I^{V w} « 1⁰⁰

In the case when p = 0 we have max ||Vwj|^L ^ = 0 and then (3.10) is trivially true. Next, when max ||Vvv||^£ - =t 0 we can define the new variables ( p j , . . . , p 'k,) = p ' by pj = P7/ | P | . Since the coefficients of the polynomials Vw\^K and VvO|^. depend linearly on P^y- and P = 0 implies Vw \K= Vvv |je. = 0 the function ƒ ( . ) is homogeneous of degree zero, i.e.

/ ( P ) = / ( p ' ) . Further we easily see that Vw and 1/VvP are continuous on the compact set S^'~l. This implies that there is a constant C such that

| | W | |L {è) =s C max || Vw\\L Â), which proves (3.10).

The inequality (3.11) follows as above taking now p equal to the nodal values of w and defining / ( P ) = ||w||L w/ m a x || w||L ^ .

In order to prove (3.12) we first note that if max \\w\\L (è) = 0 then (3.12) holds for ail C > 0. Next, we consider the case when max || vO II . =fc 0. Dividing then the inequality (3.12) by h~2 max ( || w ||PL k ) we may in the following assume that max || w || ^ = 1 and c < diam (K() < 1 for ail Kt a K. Suppose now that

We shall prove that

(3.14) ^Lq)(je)

Together with (3.11), (3.14) proves (3.12). To prove (3.14), we first note that by (3.13)

I I l l ^ J l I I ^ ^ I I l l ^ ^ ) V*.

which, in the case when e 2= —-—, by (3.10) implies 4 kp

I IV WI I L .W * C m a x ( H ^ I I L . C * , ) ) * CeW2+1> « Cpe .

Next, for e < ——- , using the notation ef-= ||VM>||L ^ and a relevant numbering of Kh we have since Vvv is constant on 7^z

( 3 . 1 5 ) | | t f | |^{t ( j f )} * l - £ B , , ' = 2 , . . . , * ².

' J = l

(

7 = 1 /

i = 1, ..., k 2 which by (3.15) implies EL =£= E and E2 =S E ( 1 + 2 E J ^2 ^ E exp(E/>) ^ 2 E. Continuingin this way we get E, =S 2 E,7 = 1, ..., ^ 2, which by (3.10) proves (3.14) also for this case.

The inequalities (3.10-3.12) with V replaced by —, K by K n Cin and

^ j by K^t n O„ follows as above by noting that û is also an interpolant on

K C\ £ln . D

Taking now v — TT(UP~ l) in (2.5) where p is an even integer greater than 2, we get

0 = L(U) Uf p-X dxdt +

- f L(U)(Up-l--n(Up-l))dxdt

+ ô f - S f

MAN Modélisation mathématique et Analyse numérique

Using (2.2), the fact that Us Vh implies || U\\ ^+i . «m = 0 and (2.4a), we have

i-

^ , r = 0,

which combined with Lemma 3.4 yields

\E2n\ + \E6\

\L(U)\(i+\f>\)dxdt\\vu\\lœ(K)\\u\\pL-JK)

where by Lemma 3.3

^ j Vf/. VTT(ÛP l)dxdt,

8 ^s„

and by Lemma 3.1

£ |77n| ^Cppk+3hh/E . Further, using Lemma 3.1-3.4 we get as above

\U+ - U_\ dx\\Ux\\l^Knnn)\\U\\i-JKnün)

E2(U)\Ûx\2dxdt

766 A. SZEPESSY Choosing now p such that

(3.16)

and combining the above estimâtes we get by summation over n = 0, 1,2, ..., TV,

f V f

J n„ J cin

J (Ut+f'Ux)2U?-2dxdt

l2/b + /z/ô) + bp(p - f 10 | | £/, + ƒ ' t/x| | tf |' "2

JsN

f' UXUP~X dxdt+p \ gUp-xdxdt

Using the convexity of the function U-> Up, (3.4), (2.1) and the same argument as in (3.3) with F{ replaced by Fp_{, we have

6(p-l))l[u'dxdt+ { \g(O,x,t)\Pdxdt\

The next step is to obtain Lp estimâtes for ail t 6 (0, T). For ( „ < / ' we have

f

-p f

[ r(f'{w+â-)

J t' J n Jo

p ⁺ⁱ f

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t' Ja p n + 1

+ C f"" f UPdxdt^\\U_\\\[

a

UxfUp-2dxdt Js

f

so that by using Gronwalls inequality we obtain for tn < t' =s= tn+ x

(3.18) | | C / ( . , O | l i# (n)* C ( | | C / _ | | 'v n i i )

f {Ut + f Ux)1Up-2dxdt\ .

Further, using a discrete Gronwaîl inequality we have by (3.17) (3.19) WU.W^^ + bp^p-l)^ N{Ul+f'Ux)1UP-2dxdt^

This proves by (2.6), (3.18), and (3.16) the existence of positive constants c, C independent of p and h such that

s u p | | £ / ( . , O | |M n )« C , for' 4 * / > * c l n (1/A).

Finally, using an inverse estimate we have

. ra - c Ö * > » u> II v » « (o,

which proves the lemma. •

4. PROOF OF LEMMA 2.3

To prove Lemma 2.3 we first note that by Lemma 2.2 the solutions Uh of (2.5) are uniformly bounded in the Z^-norm, so that the séquence {uh} defïned by uh = Uh + â satisfïes (1.4). Then there exists according to Youngs theorema Young measure v^y.ftx R+ -• Prob (R) associated to a subsequence {uh.} hj -• 0 such that v satisfïes (1.5-1.6), and by Lemma 1.1 there exists an associated Young measure yv : T x R+ -+ Prob (R) satisfying (1.8). We shall prove the following propositions.

PROPOSITION 4.1 : The Young measure v associated to [u^h.} is a mv- solution in the interior domain, i.e. VA: G R

3 v .x 7 I \ 3 y , / v 1 \ \ S J" S\ \ /* • 7 \ \ \ f\

/ ij IX i^" I \ 1 / il I O ÉT"fl I A A ^ I llTlXl T / À?" I I \ -*^—* I 1

9? * ox '

in o£/ \\h /S. XV . J.

PROPOSITION 4.2 : 77z£ Young measure yv associated to v given in Proposition 4.1 satisfies VA: G R

(yv(x, 0' (sSn (X — A:) — sgn (a - k))(f(k) - f (k))) . /Î === 0,

in ^ ' ( T x i v+) . We postpone the proofs of Proposition 4.1 and 4.2 to the end of this section. First we prove that they imply that v and yv satisfy (1.9a). To this end we let <(> G <g\Çl x R+ ), <|> ^ 0 and

0 0 ^ v < Ô .

0 ƒ 3= 3 8 , where x, and y are defined in (1.7), and write

f (<v

,.,), ! \ - £ | > < h +

Jfix/J+

+ < v( j c > r ), ( s g n ( X -

< ( j ) ) « < t ) S g n (a-k)dsdt

R+

f (<v

,,

,

J f i x Jî⁺

<v(x><), ( s g n (X -

f (<»(,,o. |X-*:|>((1-X

)«I>),

Jfix»t

- f <7V(*,o./(X)-/(fc)>.»<l>sgii (a-k)dsdt JTxRTxR++

» +

MAN Modélisation mathématique et Analyse numérique

By Proposition 4.1 we get /§ s* 0 since Xs 4> £ ^ O ( Û X i?+ ). Further by letting 8 tend to zero we obtain as in [(1.16), Sz III] and by Proposition 4.1 and 4.2

lim ( 7 /s + III8) = (y

f

This proves that v and 7V satisfy (1.9a). The fact that v also satisfîes the initial condition (1.9è) follows as in [Lemma 3.35 Szl]. Hence, v is a mû- solution and by Theorem 1.1 this implies that uh converges strongly in L/oc to the unique UF-solution of (1.1-1.3) as h -> 0.

It remains to prove Propositions 4.1 and 4.2. Let us introducé the notation

IntS;= {J K,

En (rxR+) = W

IntSN= (jïntSn,

n = 0

We start by taking in (2.5) v = TTW with w=j\(U+â — k) <&x where k E R and for ti > 0, j ^ = sgn * co^ e ^^œ(R) is a standard mollification of

f

sgn, where <*> € ^ Q ((— 1, 1 ) ) , o> ^ 0, ( o ^ = l , co ( W e n o t e t h a t

. , . _ f 1 , if j 2s -n , (4-1) ^ ^W " i - 1 , if s*-<*⁹

Further, 4> e ^^°(Ö x (0, r » , ^ O a n d x e ^ with x lmear and xlints^ = ^> s o ^ ^ *n particular %t\s =0* We have (4.2) 0 = L(U)wdxdt+ (U+ -U_)w+ dx-

L(U)(w-iTw)dxdt -

Int SM J lut On

j

JlntSB

- ô | L(U)((w-Trw)t+ff(w->nw)x) dxdt

+ Er rl(U)VU.Vitwdxdt+ e2(U)Ux(7tw)xdxdt

J Int Sn J Int Sn

r r

L(U)(w-<nw)dxdt - (U+ -U_)(w-*nw)dx

J Bdr Sn J Bdr Cln

+ ô f L(U)((<irw)t+f'('nw)x)dxdt

JBdrSn

c r

+ EX(U)VÛ.V>hwdxdt+

J Bdr £„ J Bdr Sn

L e t n o w

l3

^ ^ Eln

,k)= j\(s -k) ds ,Çy Jk

Note that by (4.1) ^ ( . , h ) is convex. Integrating by parts in &x and summing over w, we get from (4.2)

(J^(uk+, k) - (uh+ - uh_)j^(uh+ - k))

J

iV 13 13

n = 0 i = 3 i = 3

By the convexity of J^., k ), we see that the sum over n on the left hand side of (4.3) is nonnegative. By arguments similar to those used in the proof of Proposition 5.1 in [Sz I] using now also Lemma (3.4), we further have

MAN Modélisation mathématique et Analyse numérique

PROPOSITION 4.3 :

lim inf Y R1=== 0 .

13

Taking <f> e ^^(Ü x R+ ), cf> 5= 0 in (4.3) we note that £ i?' = 0 and

ƒ = 9

letting then fy tend to zero using Proposition 4.3 and (1.6), we get

>J dxdt^O,

L

which proves Proposition 4.1 by dominated convergence when r\ -• 0 + . In order to prove Proposition 4.2 we shall first give two preliminary lemmas in which we estimate the solution U = Uh near the boundary. Let CM be a constant such that || C/A || =s CM and defme p by

(4.4) P = l / [ sup ±\Fl(w,x,t)\

where Fx is deflned in (3.2). Further let us introducé the direction

and for n — 0, 1,2, ..., the point (x, i) :

(x, i). p = max (x, t). p .

{0} x /Ï+ n K^(y .

For « = 0, 1, 2, ... we also defme the cut-off function \\t : Sn -^ [R+.

- - fJ. (x - x, t - t )

e T , P . (x - x, r - 0 > 0 , 1 , P . (X - X, t - t) =£: 0 ,

where T = (h 4- ô/A + 6) a, and a is a suffïciently large constant. Note that i|i is equal to one on the part of Bdr Sn at x = 0 and decays exponentially in Int Sn.

We then have the following local stability result. In the case of linear

convection problems with et = e2 = 0 (no shock-capturing) such results were proved in [JNP].

LEMMA 4.1 : Under the above assumptions we have for ail h sufficiently small

\ \ (U_?tydx +

+ j El(U)\VÛ\2tydxdt + f z2(U)(Ûxf ty dx dt

Jsn Jsn

^ C ( ty dx + \\fi dx dt

An analogous resuit holds for localization near x = 1.

Proof : First we note that

(4.5a) max *( x+ y ) < e ,

(4.5*) Vi|> = - ^ i|/.

Taking v = TT(EA|O in (2.5) and using that TT(CA|;) = TT(Î/^), we get (4.6) f L(U)Utydxdt + b f

Jsn Jsn

+ | (*7⁺ - C/_)£/⁺ <|iöfcc+ f + f £i(U)(Ûx)2 ty dxdt = f

•+ô f L(U)((Uty - TT(U$)))t + f' (Uty - ir(U*))xdx dt

- 8 f I(t/)(^ + / ' W C / à ^ + f (t/+ - t/_)(t/i|;-^(^))+ dx

Jsn Jan

+ | e!(l7)Vl7. V ( L t y - T T ( W ) dx A - | ^(U) VU .Vty Û dx dt + | 82(t7)Öx(C/i|i-'Jr(t7i|i))jerfx:A- f e2(CO # * * * # « * * * = f £ ' •

J^„ J^n i - i

MAN Modélisation mathématique et Analyse numérique

CONVERGENCE OF A STREAMLINE DIFFUSION 773 Combining now the super approximation (2.3a), with o> = Ks Th, s = 0, 1, together with the Lm estimate (2.6), the inverse estimate (2.4), Lemma 3.4 and (4.5a), we fînd the following bounds for the terms on the right hand side of (4.6)

\L(U)\

T

dx dt Ch2

dx dt .

Hère and below, c is a constant to be chosen sufficiently small. Further,

<\>dx,

|

« c

r

|L i K ) dxdt

dxdt + C - \ z2{U){Ûxf ty dx dt

T J

where in the last inequality we used that ty(.)\K= 1 if Ke T\ and K n Supp vji ^t 0 . Integrating by parts in the first term on the left hand side of (4.6) and using the above estimâtes, we obtain

(4.7)

_ )² i|* dx + I ,+ f' Ux)2

sx(U)\VÛ\2tydx+ j e2(U)(Ûx)2x\fdxdt

Js„ w *'

'^z * '* U n /

Js,

C/2|VI(J| dxdt . Finally by (4.4), (4.5ô)

dx dt which together with (4.7) proves the lemma. D

We have the following estimate of U near the boundary.

LEMMA 4.2 : There is a constant C such that for h sufficiently small

which in particular implies that

Proof : S i n c e £ / ( . , r ) j = O, w e h a v e

Û(x, t) =z Ûx(s9 t) ds , Jo

so that by Lemma 3.4

Further, by Lemma 4.1 Ûldxdt «si

f

JBdr

C

which proves the lemma. D

MAN Modélisation mathématique et Analyse numérique

Let <t> = <j>Xç where 0 =s <j> G # O ° ( Ö X R⁺ ), and for £ > 0

# ç = 1 + £ (sên (• ~ 3 £/4) ~ sên( - + 3 S/4)) * <«>c/4 >

Then we have x^£ e * °°(fi x R⁺ ), Range x^£ s [0, 1 ] and 0 if | f l ( j c , / ) - f c | < £ / 2

1 if | â ( x , ?) — k | > £ . We thus obtain by Lemma 4.2

for h sufficiently small (i.e. such that C \/ï~jl < Ç/3 - TI). With this choice of 4> in (4.3), we have

PROPOSITION 4.4 :

lim £ Rl = — sgn (0 — k) (yv^ ^, ƒ (a) — ƒ (X)) . n<)> ds dt .

/i -> 0 i = 9 JY x R+

Before proving this proposition we shall show that it implies Prop-

8

osition 4.2. Letting h -+ 0 in (4.3) we have as above lim inf £ R^l 2= 0, and

h - 0 / = 3

using the fact that supp \x <= Bcir S^, the continuity of Q^ and Lemma 4.2, we get

(4.9')

Hence, by letting hj -• 0 in (4.3), using Proposition 4.4, (1.6) and Lemma (2.2) we get

- Q^ia, k) <$>. n ds dt + sgn (a - k) x

J r x R+ JTxR+

Using dominated convergence when ri tend to zero, we obtain (1.9a) with 4> = <j>xç- As in [(1,21a), Sz III] this implies

[

TxR+

. n lim x^ ^ ^5 ^ ^ 0 •

1^0 +

where we have used dominated convergence once more, now when Ç -> 0 + - Since (sgn (X - . ) - sgn ( a ( s , t) - . ))<ƒ (M - ƒ ( . ) ) is locaUy Lipschitz continuous on R\ {a(x, t)}, this yields as in [Sz III, (1.21a)].

(yv^t> (sgn (X - k) - sgn (a(x, t) - k))(f(k)-f(k))) .

.WSBO Vk^a (je, O

a.e. on T x i î+. Letting then &-+a(x, /) ± , we finally obtain Prop- osition 4.2.

It remains to prove Proposition 4.4 by the super approximation (2.3) and (4.9) we get

(4.10) | *9| ^ £ f \L(U)\ x

(K)dxdt

\L(U)\ dxdt^Ch ^ ^

| c/

- c/_ i x

^G r M HAT

x II sgn ( â - y c ) 4 > x - T r ( s g n ( â - fc) * x ) | |L (^n n ) dx

N r

^ Ch ^ | u⁺ - U_ | dx ^ C

We shall now estimate the terms Rn, Rn and Rn by using the équation (2.5) with v = ir(sgn (â-k) <|>(x - XP))> w h e r e XP e C5°(n), 0 ^ \P ^ 1 and Xp(x) = 1 for ail x e fl, such that dist (x, T) > p. To this end we add to i?11 + Rn + i?13 the following sum

1.H

•ƒ.

L(U)

Int5„

Vf7. VTr(sgn ( â -

MAN Modélisation mathématique et Analyse numérique

+ B2(U) £/x(Tr(sgn {â - k) <j>x))x dx dt

JlntS„

- 8

VU. ViT(sgn (â - k) $\p) dx dt

- j e2(t7) f/x(*(sgn (S-k)$X,))xdxdt and by the stability Lemma 3.1 we then have (4.11) G1« C ( p ) (N/ i + V 8 / h +

By using (2.5) with t; = Tr(sgn (â - k) <$>(x - XP)) we get

+ f s

(U)Û

(*(sgn(S-k)4>(x-X

)))

dxdt

+ 8 I.(t/)((ir(sgn(ff-*)<|)(x-Xp)))i (â - Â:) 4>(x " Xp)))J ^ A]

f

N o w , by (2.3) and (2.2) we have

(4.12a) | | ( x - XP) s g n (S- k) <|> - i r ( ( x - XP) sgn ( â -

(4.12A) II (x - XP) sgn (â- k) * - 7r((x - xPP) sgn (â - k))

Let now

| [ L(U)(X - XP) sgn (ff - k) Çdx dt + s„

- X p ) sgn ( f f -

Then it follows from (4.12) and Lemma 3.1 that (4.13) \Rn + Rl2+Rn + Gl + G2\

Integrating by parts and using the fact that <|> V sgn (a - k) = O, we have

(4.14) G

= £

- {u^h+-u^h+ -f K^A_ ) ( x - X^P)sgn (â - & ) c(>^x w*((x-X^P)sgn (a-A:)<())^r Jx Jr

/ ( « A ) ( ( X - XP) sgn ( a - f c ) ^ ^ * ]

= - I (w/,(|>, + ƒ(«/,)<(>J(x-X^P) sgn (â-k)dxdt - - f(uh)(x - Xo)x$ ssn ( ^ - £) ö&c^ -

Letting now h -> O using Lemma 2.2 and 4.2, (4.10) and (4.11-4.14), we find as in (4.9')

x sgn ( â -

\ (v^{x

- f(^a) n§ sgn {a - k) ds dt, JVxR+

MAN Modélisation mathématique et Analyse numérique

and by letting p -• 0 we then obtain as in [Sz III, (1.13)]

£*''->- f (yv^

f(a)-f(\))n<$>sgn(a-k)dsdt,

i = 9 JTxR+

which finally proves Proposition 4.4 and the lemma. • 5. NUMERICAL RESULTS

In figure 5.1 we give numerical results for the SC-method on a uniform mesh with k = 1 applied to Burgers' équation in (0,1) x R⁺. More precisely, figure 5.1 shows the approximate solutions for the problem (1.1- 1.3) with rf=l, f(u)=fx(u) = u2/2,mitmldnteLU0= I1' ° J x * °-5 , and boundary values a(0) = 1, a(\) = 0.

Let Vj, j = 1⁵...,7V be a finite element basis for V^h. We base our numerical results for Burgers équation on the following slightly modified version of (2.5), Find u = U + 1 - x, Ue Vh such that for n = 0, 1, 2, ...

(5-1) Fn(u9Vj)=09 j = l,...,N, where

f f

Fn(u9v)= (utv-f(u)vx)dxdt+ (u+ -u_)v+ dx +

Jsn Jan

r r

+ BX(U)VÜ.VV dxdt + \ e2(w) ûxvxdx dt + Ô f {ut + f(u)x){vt+f'{ü)vx)dxdt, where

U\^K = udxdt KsT^h.

JK

The équation (5.1) was then solved iteratively on each time interval

(5.2) um + l=um-^YdvJ yFn(um, Vj)/(Fn(um + yvp Vj) - Fn(um, t>,)) ,

j

m = 1,2,...

with the relaxation parameter p = 0.4, y = 0.01, k = 1, h = 0.01, 8 = A, E = 03 h^7/4 and Ô = E/h.

780

1.4 . .

1.2 . .

1.0

0.8 . . 0.6 . .

«-4 . . 0.2 . .

0 . 0

0.Ö00 0 . 1 2 5 0 . 2 3 0 0 . 3 7 5 0 . 0 0 0 . S 2 5 0 . 7 9 0 0 , 8 7 3 l . q -0.2.

-0.4.

-0.6.

1.4 .

1.2 .

1 . 0

0.8 .

0.6 .

0.4 .

0.2 .

0.1 - 0 . 2 .

- 0 . 4 .

- 0 . 6 .

00 0.125 0.250 0.375 0.500 ( X

1 1 j j P>

.Ê25 0.750 0.875 ijfioo"

\ y

Figure 5.1a.—Method (5.1-5.2), one time step. Figure 5.1 b. — Method (5.1-5.2), 41 time steps.

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 X

Figure 5.1c —Method (5.1-5.2), 81 time steps. Figure S.ld. — Method (5.1-5.2), 201 time steps.

APPENDIX

We now give a proof of the super approximation results (2.3) : Let us consider the case (2.3a) with o> = Sn. It is then suffïcient to consider one triangle K G Tnh, i.e. o> = K.

Defming

f f

= <p dx dt/ \ JK JK

<p dx dt/ \ dx dt , <p | ^ e L

f

MAN Modélisation mathématique et Analyse numérique

(A.l) II " - ^ H I ^ J Q * C* II V" II£.,(*)•

Further

||vw - -n ( W ) \\Lx(K) « || vPw - -n (v&w) \\L^K) + + || (ƒ -

Now, since v e V%and ^ w isconstant onK,ir(v^w) = v@*w onKand thus Tx = 0. Further we have by (A.l) and (2.2b)

where we in the last step used the inverse estimate (2.4a). This proves (2.3a) with o) = Sn, r = 0. The case r = 1 and w = £!„ are similar.

The inequality (2.36) follows as above by (2.2Z>) and (2Aa) using that

Finally to prove (2.3c) we fïrst note that for ƒ e <^C0(5„) we have

\f(x,tn)\*± \tn + t \f{x,s)\ds+ f'**' \ft(x,s)\ds,

which gives

I | / ( x , / „ ) |2^ i [ ' " + A \f(x,s)\2ds + h f'"*" \f,(x,s)\2ds.

So that by a density argument we obtain for ƒ e Hl(S„)

Combining now this estimate for ƒ = (/ - ir)(wv) with (23b) we get (2.3c). •

ACKNOWLEDGEMENT

This work was supported by the National Swedish board for Technical Development and the Swedish Institute of Applied Mathematics.