A characteristic mixed method with dynamic finite-element space … · 2017. 2. 11. · transient problems with localized phenomena, such as fronts, shocks and boundary taycrs. Keywords:

Journal of Computational and Applied Mathematics 43 (1992) 343- 353 North-Holland

343

CAIM 1188

A characteristic mixed method with dynamic finite-element space for convection-dominated diffusion problems

Daoqi Yang Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

Rcccivcd 18 April 1991 Revised 23 September 1991

Abstract

Yang, D.Q., A characteristic mixed method with dynamic finite-element space for convection-dominated diffusion problems, Journal of Computational and Applied Mathematics 43 (1992) 343-353.

A method for numerically solving convection-dominated diffusion problems using a combination of the method of characteristics with the mixed finite-element method is presented and anaiyzed in this paper. Different numbers of elements and different basis functions are implemented at different time levels in order to accurately resolve time-changing critical features. This new method provides the ability to obtain approxi- mations of the solution and its gradient simultaneously, adopt large time steps, and make grid refinements and basis function adjustments at any time necessary. Convergence analysis and error estimates are established. Furthermore, comments are made on the efficiency and capability of the method for accurately solving transient problems with localized phenomena, such as fronts, shocks and boundary taycrs.

Keywords: Method of characteristics; mixed finite-element methods; convection diffusion problems.

1. Introduction

The numerical simulation of convection-dominated diffusion treatment. Standard finite-element or finite-difference methods

problems requires special usually have considerable

practical difficulties in obtaining satisfactory numerical results. Generally, they either smear sharp physical fronts with excessive numerical diffusion, or introduce nonphysical oscillations into numerical solutions.

The method of characteristics [4,6,7,9,10,14,15,17,24,26-28,33-351 has proved effective in treating convection-dominated diffusion problems. Error estimates [6,7,9,10,15,24,26,28,33-3S] and numerical experiments [4,14,17,27,35] have shown that this method permits the use of large

Correspondence to: Dr. D.Q. Yang, Department of F+4athematics, Purdue University, West Lafayette, IN 47907, United States.

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344 D.Q. Yang / The characteristic mixed FEM

time steps, and avoids or sharply reduces the usual numerical difficulties: numerical diffusion and nonphy&al oscillations. The method of characteristics has been combined with the finite-element procedure in [4,9,10,14,15,17,26-28,34,35]. In the error analysis and numerical experiments for the characteristic finite-element method, the approximate solution is required to lie in the So*bolev space H’(a); and under other assumptions, optimal error estimates and satisfactory numerical results are obtained.

Since convection-dominated diffusion problems involve time-changing localized phenomena, such as fronts, shocks and boundary layers, it is advantageous to apply the dynamic finite-ele- ment method [32]. This method has the capability for dynamically making local grid refine- ments or unrefinements and basis function adjustments. Numerical results for the method with local grid refinements ]1,5,11-13,16,18,20-231 have proved that it can accurately solve transient problems with localized phenomena. In considering the advantages of the method of character- istics and the dynamic finite-element method, it is natural to combine these two methods to treat convection-dominated diffusion problems. This combination is better suited to many fluid flow problems, since sharp fluid interfaces move along characteristic or near-characteristic directions. In these cases, we can move local refinements with the fronts.

Many results for the mixed finite-element method applied to second-order parabolic prob- lems have also been reported in [19,29-331. Via the mixed method we can obtain the approximations of the solution and its gradient simultaneously. Thus we can use the gradient as local refinement indicator. Within large gradient areas, we refine the grid and lower the order of corresponding interpolation polynomials; within smooth solution areas, we coarsen the grid and increase the order of the polynomials. So the combination of the dynamic finite-element method 1321 with the mixed method will be effective in tracking phenomena with large gradients. The dynamic mixed finite-el,, -2ent method has been discussed by the author in 1321.

In practical computations, it is reasonable to choose this process: (ai ealctilate large gradient areas; (b) make local refinements; (cl move local refinements along characteristic directions for several time steps; then repeat. Clearly, we may as well use a large refined area within which critical features maintain for several time steps, and change the grid and basis functions after every several time steps.

In this psper, we shall consider combining the method of characteristics with the dynamic mixed finite-element method to treat the model problem given by

au * at4

cOz + C bi(x)z - V-(d(x)Vu)=f(x,t), (x, t)-XJ,

i=l 1 (1 1) .

u(x, t)=O, (x, t)MxJ, (12) . u(x, 0) =g(x), x=0, (13) .

where In is a polygonal domain in II?* with boundary r and J = [0, T]. We shall discretize the transport terms along the characteristic direction, apply the mixed method to approximate the solution u and its gradient, and implement different grids and different basis functions at different time levels.

Letting $(x1 = [c(x)* + ] b(x) I 2]1/2, b(x) = [b,(x), b,(x)], we denote the characteristic di- rection associated with the operator cu, + b - Vu by T = r(x), where

$rl(c$+b-v).

D.Q. Yang / The characteristic mired FEM 345

Now (1.1) can be put into the form

au @( ) x ,,-V(d(x)Vu)=f(x, t), (x,t)MxJ. (14) .

Given a time step At = T/N, where N is a positive integer, we shall approximate the solution at times t, = n At, n = 0, 1 in the following manner:

, . . . , N. The characteristic derivative will be approximated

(x, tn+J = e(x) un+l-En U - ii, [ (x-i12+At2]1'2

=c(x) n+;t , (15) .

where qn = $Xx, tn), &n = +(X9 ?n) and

z=x- b(x) At

4x) l

(16) .

As in [l&28,35] we assume that the problem (1.114 1.3) is periodic with period a. Thus formulas (1.5) and (1.6) are well defined. It should be noted that the treatment of the periodic case avoids some technical questions that are of very little interest in some practical problems. For example, in reservoir engineering problems, the main interest in secondary and tertiary recovery processes is in the body of the reservoir, not at its edges. (Actually, when the problem (l.l)-( 1.3) is not periodic, the boundary problem can be handled this way: whenever X, defined by (1.61, goes beyond the boundary, let U, = u(X, tn) = u(x, t,) I X E r = 0. By condition (1.2), the error of doing this is O( A t ).)

I Ul bUllVti,111tiUUU . . ” “.I__- m- ---A=~~ lx~p chnll denote the norms in the Sobolev space H”(L!) by II l II s and omit the subscript 0 when s = 0. We shall also write La(I; IT) for Lp( I; H”(R)), where I is a time interval, and drop I from the notation when Z = [0, T]. Throughout, the symbols K and E will denote, respectively, a generic constant and a generic small positive constant.

We shall make the following assumptions on the coefficients c, b and d:

0<1~,~c(x)~K,, O<K,d(x)<K,, XEL!, (l.?a)

II b II w,‘(n) + 11 c 11 w,‘(n) < K,.

In particular, (1.7b) implies that I x - X I G K At. We shall describe our characteristic mixed method with a dynamic finite-element space in

the next section, and in the third section we shall establish the convergence analysis and make some comments on our method.

2. The approximation scheme

Let v= L*(n), H = H(div; In). We introduce w = d(x) Vu( x, t) as a new variable. In view of (1.4), we see that problem (l.l)-(1.3) is equivalent to solving the following variational form:

D.Q. Yang / Tlte characteristic mked FEN

find {u, wj : J --) V >c H satisfying

au 9F’l’ 1 -(div w, ~)=(f, 17)) CEV,

7 (2 1) .

(a(x)w, z) + (u, div z) = 0, z EH, (2 2) .

(N(X, O), V) = (g, i.‘), L’ E V, (2 3) .

where ( l , - ) denotes the inner product in either L*(O) or (L*(LJ))*, a(x) = d(x)-‘. In order to resolve critical features we apply different grids and basis functions at different

times [32]. Let h,* be the diameter parameter associated with the quasiuniform subdivision of 0 at time t,, and k, be the order of the interpolation polynomials. We introduce a Raviart- Thomas space [25] E’, X H, c V X H for approximating (u, w} at time t = t,. (If J2 is decom- posed into rectangles, other mixed finite elements [2,3] can be constructed.) Note that we can make not only local grid refinements or unrefinements, but also basis function adjustments at any time level necessary. In order to obtain uniform error estimates, we let

h = max (h,), k = min (k,). n n

For the error analysis we shall make use of the following elliptic projection (R,u, R,w} of (N, w) given by (R,u, R,w} : J + Vn X H, such that

(div R,w, r) = (div w, L’), u E Vn, (2.4)

(a(x)R,w, z) + (R,u, div z) = 0, z E H,. (2 5) .

We shall assume that the solution u to problem (l.l)-( 1.3) has the following regularity:

zl E L”( Hkt2), 3"U ---Q ELZ(L2). (2 6) .

Then, it can be shown as in [S] that the problem (2.4), (2.5) has a unique solution for each t E J, and

EL - R,u II + II w - R,w II < fi” ll 14 ll s+ 1, l<s&+l, t(zJ, n=o, l,...) N.

(2 7) .

We are now in a position to describe our approximation scheme for problem (l.l)-(1.3). Let U0 = R,g, IV0 = R,w, = R&d(x) Vg). The approximation {U, W} of the solution {u, wj is defined by: find {Un + 1, Wn + 1) E V,‘, + , x H, + 1 satisfying

(( c fi~-u,),c)=o, L’EV,,1, (2.8)

1 u II + 1

c -$ \

\ At ’ :‘I - (div witl, c) = (fn+,, L), v E c+,, (2 9) .

(W a n+l, z)+(U,+,,divz)=O, ZEH,~+~, n=O, l,..., N-l, (2.10)

where Un E Vn+,, an = U( X, t,), cn = fi( X, t,), i is defined by (1.6). We ne d to-give some explanations for our scheme (2.8)-(2.10). Equation (2.8) can be

thought of as Un being the weighted L’-projection of U,* onto the next finite-element space V n+l when Vn f Vntl. This enables us to obtain initial values for starting a new time step, and

D.Q. Yartg / The characteristic nriml FEM 347

guarantees the convergence of our method. If we choose initial values in_another way, our method might not converge. Since the problem is assumed to be periodic, U,, is well-defined. Upon writing (2.8)-(2.10) in matrix form, we easily know that this scheme has a unique solution at each time step.

3. Error estimates

We first state the following lemma which will be useful in our error analysis.

Lemma 3.1. Let 9 E L”( L2 ); then

II CT,, II: < (1 + K At) II 9,, IIf,

where 119 lif = (~9, 9), the constant K depends only on K, and K,.

This lemma can be easily proved by a change-of-variable argument. See [9,33] for details. Now we turn to the error estimation for scheme (2.8)-(2.10). Let

e,, = u,, - &,K,,, P,, = u,, - RJJ,,, r,, = w,, - R,w,,

5 = ri,, - 4,. I%? iz = 1% - L- 1%.

Combining (2.11, (2.4) and (2.9), we obtain

( e n+ 1 - e’,, C

At ,L’- I Cd iv rn+], 4

II n+l - ii, P = Cxy tn+l) -c At ,u L: E vn+,.

Combining (2.5) and (2.10), we obtain

( ar,, + 1’ z)+(e,,+.,divt)=O, zEH,,+,.

Taking LJ = e,,+ 1 in (3.1) am!. z = r,,+, in (3.2) and adding yields the error equation

e n+ 1 - i,,

c At 9 e,,+l + (art,+,, rn+A

U n+ 1 - ii, = (x7 fn+A -c At = T, + T2

(3 1) .

(3 2) .

(3 3) . .

Next, we estimate (3.3) term by term. Applying the inequality a(a - 6) >, $(a’ - b2) and Lemma 3.1 to the left-hand side, we see that

(

e ,I+ 1 - ;,,

c At T e,,+, I

+ Car,, -t 1 7 rn + 1)

3 & Ik,,., 11~-~~~,1~1:] +&’ lIr,,+, II2 [

>- 2’,t ” [ e,,., Ilf-#,,~~f] ---~~~e”,~~~+~1lr,,+, 112,

where S = l/K,.

(3 4) .

D.Q. Yang / Tke characteristic mired FEM

To bound T,, we follow [9,28] and get

(3 5) .

For the estimation of T2, we cannot follow the usual analysis as in [9,10,28] since the numerical solution belongs only to L2(Jt) and does not belong to H*(O). Using the inequality ab < $( ea2

)withe=l-&wehave

IT,/’ 2At L[(l -S)IIe,+J,2+ (1 -W’llp,,, -iiIl$ RE (% lb

& (3.3)-(3&j we obtain the error inequality

~Ile,,,Ilf-Il&Jlf+6 AW-n+I II2

GK At(lle~+i:r,‘+l!~~jif)+A~2 II a2u 2 s L2~, L2~+(1-~)-111~~+l-~~l12 II - “’

where g E (4 1).

(3 6) .

9 (3.7)

Then we will find the relationship between II & II c and II e, II c= From (2.8) we have

(( c &-en), 0) = @(fin-pn), u), UE Vn+I-

Choosing v = & and applying the inequality ab < t<ea’ + b2/e) with E = 1 - 5, we see that

5(11%x - II en IIf) G 21 - oIlczll: + 2(1 l [) Ilii -Prlll:~ -

ThUS

#$Jf- IIqJ~~ (1 -S)-‘II& -pnllz, WE (0, 1).

Combining (3.7) and (3.8) we obtain

55lle,+Jf-- IIe,ll~+iX~~IIr,+, II2

[ At( Ile,,,

a2u 2 dK II 2 + II e, II ‘) + At2 s II II L2c, .L2, “*

(3 8) .

+P -WIlPn+1 -&Ii’+ (I- r)-‘llin -Pn112 1

l (3 9) .

Eote t!xf formula (3.9) is the error inequality in the case of implementing different grids and basis functions at different times. When the finite-element space remains unchanged at t = t, and f=fn+l, namely when Vn x H, = Vn + 1 x H, + 1, then the error relation is

5 II en+, llf - II e, llf + 6 At II m+l II 2

dK At( Ile,,, i

112+ Ild2) +k2 II L2)+ (I- W’II Pn+l -~nl12 : . 1 (3.10)

D.Q. Yang / The characteristic mixed FEM 349

Let %l+1 =l when i(IpIf~#~+,X&+, and q’,+] = 1 otherwise. Thus formulas (3.9) and (3.10) can be combined into

&7,+1)le,+J,2- lle,ll,2+h+1 AtIIr,+l II*

+(I -Wll&+1 -&II’+ (1 - 17n+J111& -&II2 1 l

(3.11)

The last term on the right-hand side can be taken to be zero when q, + l = 1. Note that 0 < 5 < 1, 0 < rli < 1, 1~ i < N, multiplying (3.11) by (“II:= lqi leads to

n+l

tn+l ~~~ll~n+*Il~~~n~~~ll~nll~+~Cnn~?~ AtIIrn+* II* i=l i=l i = 1

At( Ilen+

a*u * II * + II% II ') + At2(a7, L’(,_L’) I II I

n,

+(I-6)-*lIPn+l _Pll’+ (1 -rln+l)-‘llbn -PfZll* 1 l

(3.12)

Summing (3.12) from n = 0 to n = m - 1, 1~ m < IV, yields

1 Smfi~~~lle,Il~+~m~l AtIIr,+, II* i=l n=O

At 2 lIe,J2+m~* Al* p c~,.~~~+(1-5)-111pn+l (

a*u *

II II -E* II2

n=O n-0 "2

+(1-~)-111$n-~nI(2 l (3.13)

Note that 5” 2 tN, and ~~ Ir7i 2 lN. Substituting 6 = 6 = N/(1 + N) into (3.13), we have m- l

IIeml12+ C AtIIrn+l II* n=O

r

At 2 II en II * n=O

+mfl +N II&+1 -&II2 +N II& -&II2 n=O I + II e. II * 1 9

(3.14)

where we used EdA” =(I +N?k2.72, 6% -&‘<2.72(1 +N). An application of the discrete Gronwall’s lemma to (3.14) yields

111 - I

IIe,ll* + c At lb-,,, II*

m - 1

dC c At” #I=0 ( /I

ow we are ready to demonstrate our main results.

Theorem 3.2. Let (u, w} be the solution to (2.1)-(2.3) and {U, W) the solution to scheme

+ K II et, II ‘- (3.15)

(2.8)-(2.10X s uppose that hypotheses (1.7) and (2.6) hold. Then, for At sufficiently small,

E AtllW,I-w,$ I/2

tZ=l

11 u II Lz(,,~+~) + At , m=O,i ,..., N. f-3 LZ 1

(3.16)

proof, Note that

m-l

c N(II P n+l n=O

-iil12+IlB,,-PnIlt)

and N = T/At_ Then formula (3.16) follows immediately from the combination of (3.15), (2.7), Lemma 3.1 and the triangular inequality. •I

From Theorem 3.2 we know that, if the space and time discretizations satisfy the relation h = O(At), our approximation scheme has optimal rate of convergence for the gradient when k 2 1. The convergence order for the solution itself is one order lower than the conventional result [19,31], but is almost as good as the result of the dynamic finite-element method [32]. The reason for the slight influence on the accuracy is that the approximate solution U belongs only to L’(R), but does not belong to H’(R). Despite this, the convergence orders in (3.16) are good enough to meet the requirement of practical problems. It should be noted that our error estimates are not optimal in case k = 0. However, we still can get convergence in this case if we require h = o( At ). This requirement is not very restrictive, for our method allows large time steps.

Since the error estimates in (3.16) depend on second derivatives in the characteristic direction but not on second derivatives in J, time-+*--=- ...lXG rl~~lcdtion error is reduced compared with %ndard mcilhods, and large time steps should be possible. Numerical results reported in

D.Q. Yang / TIte characteristic mixed FEM 351

[14,27] have confirmed the ability to apply long time steps and other advantages, such as the avoidance of numerical diffusion and oscillations.

It is worthwhile to point out that, using our method, we can obtain the gradient simultane- ously with the solution. This is very important in many practical problems. For example, in petroleum reservoir simulation problems, the gradient of pressure represents velocity. Both quantities are important in the oil recovery process.

Numerical results of the standard finite-element method with local grid refinements for parabolic problems have been reported in [ 1,5,11- 13,16,18,20-221. They primarily demonstrate the capability of our dynamic characteristic mixed method for accurately solving time-depen- dent problems. In [5,12], a self-adaptive finite-element simulator (SAFES) for linear parabolic partial differential equations was described. In SAFES, the data structure supports the placement or removal of local refinement, and the grid analysis - based on the L*-norm of the gradient of the solution - directs the placement or removal of local refinement to represent the changing location of large gradient. In our method, the gradient is obtained simultaneously with the solution. Thus we can more conveniently predict large gradient areas if the gradient is used as a local refinement indicator. Furthermore, we can make basis function adjustments using our method.

Since the solution u changes slowly in the characteristic direction, we may want to move local refinements along characteristics for several time steps instead of calculating large gradient areas at every time step. This is advantageous from the computational point of view.

Thus, on the one hand, our method can accurately resolve localized and transient phenom- ena typical of convection-dominated diffusion problems; on the other hand, it can use large time steps and avoid or sharply reduce numerical difficulties.

Finally, we point out that our method can be extended to more general convection diffusion equations such as the case d = (x, u), f = f(x, t, u) in (1. l), and can also be extended to the three-dimensional case.

Acknowledgements

The author would like to thank Professor Jim Douglas Jr and Professor Richard E. Ewing for their encouragement and help. Thanks also go to Mike Dougherty for carefully reading the manuscript.

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