Mixed Finite Element Methods for Nonlinear Elliptic Problems: The p-Version Miyoung Lee and Fabio A. Milner Department of Mathematics, Purdue University, W Lafayette, Indiana 47907- 1395

1. INTRODUCTION

The finite element method has been used for the numerical solution of elliptic and parabolic problems for quite some time. Mixed finite element methods have now been around for nearly two decades and much literature is available for the so called h-version of the method, based on obtaining error bounds for approximating polynomials of a fixed degree, asymptotic in the small parameter h , which denotes the typical size of an element [I]. In the p-version, the error estimates sought are for a fixed grid of the domain and asymptotic in the large parameter p, which represents the degree of the approximating polynomials. Very little work has been done on the p-version of mixed methods [2- 61. In spite of the split among supporters of one or the other form of the finite element method as being “better” than the other, evidence seems to indicate that for some problems the former may be better suited, while, for other problems, the latter. The p-version of the method is younger than the h-version and, frequently, numerical analysis of new problems using the finite element methods examines the h-version first and only later the p-version.

The application of mixed finite-element methods to strongly nonlinear second-order elliptic problems was studied quite recently for the h-version of the method [ 2 4 ] . In this article, we analyze the applicability of the p-version and derive error estimates for the approximation of both the scalar and vector variables found with the mixed method.

We shall consider the following nonlinear Dirichlet problem:

u(z) = -g(5), 5 E 80,

where s2 is a bounded domain in R’ with Lipschitz boundary 80. Assume that, for 0 < E << 1 and for each g E H“/2+‘(80), there exists a unique solution u E H 2 + € ( 0 ) of (1.1). The functions a, : fi x R x R2 ---$ R, 0 L i 5 2 are twice continuously differentiable with bounded derivatives through the second order. Also, assume that the quasilinear operator associated with (1.1) is elliptic in the following sense. Let X(z, u, z ) and A(z, u, z ) denote, respectively, the minimum and maximum eigenvalues of the matrix

Numerical Methods for Partial Differential Equations, 12, 729-741 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0749- 159W96I060729- I3

730 LEE AND MILNER

which we shall assume symmetric. Then, for all < E R’ - (0) and for all (x, u, z) E 0 x R x R2,

0 < v s , u, 4llE1I2 L E”AE I N x , u , z)llEl12 The variable x will normally be omitted in this notation below. For 1 5 q < 00 and k any nonnegative integer, let

IVk,q(i2) = {4 E L”(S2) : D(t4 E L(’(f2) if la1 5 k )

denote the standard Sobolev space endowed with its norm

The subscript f2 in the norm will be omitted, unless necessary to avoid ambiguity. The definition for q = cx) is obvious. Let H k ( , f 2 ) = W’,2(n) denote L2-based Sobolev spaces with the norm 1 1 . IIp = I( . Ilk,‘. In particular, the notation 1 1 . 1 1 ( ) will mean 1 1 l l l , ~ ( i ~ ) or 1 1 . I I L 2 ( 6 2 ) z . For 0 5 s < m, let 16’”,‘/(f2), W.Y(an), H”(f2), and H2(af2) denote the fractional order Sobolev spaces endowed with the norms 1 1 . ll,,,q:o, 1 1 . Il.\,q:i,(>, /I . I l s . < j r and I1 . Il,\i,r2. Next, let

V = H(div; 12) = { u E L’(fl)’, div E L2(f2)},

normed by

and let

The mixed finite-element method approximates at the same time the solution of (1. I ) , u , and the flux

y = 4 7 L , V7l) = - (n , (u , VU), 4 7 4 V u ) ) , (1.3)

which we assume to be in C”,’ ( 5 ? ) 2 . By the implicit function theorem and the ellipticity of A , ( 1.3) can be inverted to obtain Vu as a function of u and y, say

v u = -b( u, y). ( 1.4)

Wenowset in( l .1 )

f(., Y) = -ao(u, Vu) = -no(% - N U , Y)). ( 1 5)

Then, the mixed weak form of ( 1 . 1 ) we shall consider consists of finding (y, u ) E V x I+’ such that

(b (u , y ] , U ) - (div 71. u ) = (9, u . n) VV E V,

(div y, w) = (f(u, y), w) Vui E 1.2; ( 1.6)

where 71 denotes the unit outer normal vector to 30.

one grid element E = (I), and define Now we consider a fixed rectangular grid G on 0 with elements E (which may consist of just

’P7’,‘/(E) = {polynomials f(sl,x2) on E , of degree 5 p in 51 and degree 5 q in x.}.

NONLINEAR ELLIPTIC PROBLEMS.. . 731

Next, we define

VI’ (E) = p P + l . ? ’ ( ~ ) p?h~+l (E), W/’(E) = P7’J’(E),

where, for example, PlJf1.7’ denotes tensor products of polynomials of degree p + 1 in x1 and polynomials of degree p in 52. Finally, we let

V” x W’’ c v x w be the Raviart- Thomas -Nedelec space of index p 2 0, associated with this decomposition, which is defined as

V” x W” = (II,<,3V7’(E) x W7’(E)) f t (V x W),

where the inclusion in V x W is equivalent to the normal component of the vector part being continuous across inter-element boundaries [7,8]. This condition is trivial when consists ofjust one element E = 0. The reason we use rectangular elements is that the convergence proof we shall present depends on approximation estimates for projections into the finite element space, which are available for rectangular elements but not for triangular ones [4, 61. The estimates in those articles are explicitly found for rectangular Raviart Thomas Nedelec spaces, but, as indicated there, the same argument leads to the same estimate for other spaces such as BDM [9], as long as they are based on a rectangular grid.

The mixed finite element method we shall treat is a discrete form of (1.6) and consists of finding ( ~ 7 ’ ~ u7’) E V” x Wr’ such that

( b ( d ’ , y / ’ ) , u ) - (divu,u7’) = (9;u .n) V v E V”,

(div y7’, w ) = (f(uT’, yl’), tu) Vw E W7’. (1.7)

We shall use an L‘-projection onto W”, PJ’ : L2 + WJ’, given by

( P ’ U , - zu, x) = 0, x E W”, w E w, (1.8)

and having the property that, if w E H“‘(R),

l l ~ l l , r l , s 1 2, m 2 3/2 - 3/s. (1.9) IIpJJw - 2 v 1 1 0 , , 5 Qp-”’+:j/2-:j/”

We shall also use the Raviart--Thomas projection of V onto V”, XI’ : V + V ’ , having the following approximation property [4]: For v E W”.“(R)‘ 7 V,

ll7+’v - ~ I I O . , , I &pal r 2-7.-4/s 1 1 ~ 1 1 , - , s 2 1/2, T > KMX{ 1/2,3/2 - 3 / ~ } . (1.10)

We shall also need the following inverse-type inequalities [4]: For x E L”(R) n W7’ (or x E ~ ” ( 0 ) ~ n v?’),

IIxll0,s < - Q p 4 / T - 4 / S I l X l i O , ” , 1 I I s 500. (1 .11)

In Section I1 we shall linearize the system (1.6) about its exact solution by using second-order Taylor expansions; in Section 111 we will show that (1.7) is uniquely solvable by using a fixed point argument, and that its solution (yl’, uJ’) converges to (9, u) in V n L4(R)2 x L4((R). Then, in Section IV we prove a local uniqueness result for the approximation, and in Section V we derive optimal error estimates in L2 for the approximation. Finally, in Section VI, we provide a numerical example with a smooth (in fact, analytic) solution.

732 LEE AND MILNER

II. LINEARIZATION

Following [ 5 ] , for p E W p , p E VJ’, we shall use the following first- and second-order Taylor expansions:

f(P, .) - f(? Y) = - f l ‘ (K Y)(. - P ) - f Y ( % Y)(Y - P ) + Q,(u - p1 Y - P I

= -f;“P, P ) ( U - P ) - f , ,(P, P ) (Y - P ) . (2.1)

Here we have used the following notation:

Combining now (2.1)~-(2.5) with p = u7’ and ,LL = yp, we obtain the modified error equations that follow:

. NONLINEAR ELLIPTIC PROBLEMS.. . 733

Recall that [ 101

div o TI’ = P” o div : H’(R)2 + W P . (2.7)

Then, we can rewrite (2.6) in the following form, which we will need for our fixed point argument:

(B(u , Y ) [ T ~ ’ Y - $’I, V) - (div V , Ppu - u p ) + (rl [PPu - up] , V )

= (qU, Y)[x71 - Y] + rl [PU - U ] + Qh(u - U P , y - g)), V)

(div ( 7 ~ 7 ’ ~ - f ) , ~ ) - (r2[.rrpy - y ’ ] , ~ ) - (y[PPu - u p ] , w )

= (-r2[TpY - Y] - Y [ P P U - U ] - Or(. - U P , y - I/’), W )

vv E v”,

vw E ~ 7 ’ . (2.8)

Here we have set

B(u, Y) = b,(u, Y) = A-’(u, Y), rl = b,L(U, Y), r2 = fY(% Y), and Y = fu(% Y). (2.9)

Now define M : H 2 ( s l ) -+ L2(R) by

M W = -div (A(u , Y)VW + A(u, Y ) r l m ) + A(u, y ) r 2 x GW - (7 - rrA(U, y ) r l ) w

(2.10)

and its formal adjoint M’ by

M * x = -div (A(u , y)Vx + A(u , y)r2x) + A(u, Y ) ~ I x Vx - (Y - ri’A(u, ~)r l )x . (2.1 1)

From [ 101 we know that the restrictions of the operators M and IVf * to H2(R) n H i (9 ) have bounded inverses, provided that a R is C2. In our case it is only Lipschitz, but the result is still valid if we assume that (y, u ) can be extended to a pair (y, U) defined on a domain 0” with a C2-boundary, such that 0 c 0,) and meas(R0 - R) is arbitrarily small [ I I]. Then, for any d~ E L2(R) , there is a unique 4 E H 2 ( s l ) n HA(sl) such that Md = $ (respectively, M * 4 = $) and 114112 5 Gll$llo if we assume, for example, that the zero order term of M * is nonnegative:

Y 5 r;’A(U,Y)r* - div (A(u,Y)r2), (2.12)

where u E C0.’(f=2) and y E C0.’(f=2)2 [ 5 ] . In this article, we shall assume structure condition (2.12) to employ our duality arguments. Let @ : L’l) x Wp + V p x WP be the given by @ ( p , p ) = (s, q ) , where (s, q ) is the unique solution of the following system:

(B(u,y)[.rrPy-s],v) - ( d i v v , P p u - q ) + ( I ’ , [ P p u - q ] , v )

(div (7r7)y - s ) , ~ ) - (r2[.rrPy - s ] , w ) - ( y [ P p u - q ] , ~ )

= ( B ( u , ~ ) [ . ~ ~ P ~ - - ] + ~ ~ [ P P U - U ] + Q ~ ( U - P ! Y - ~ ) , V ) V V E v p ,

= (-r2[.rrpY - - Y[P% - 4 - arcu - p, - p ) , W ) vW E wp. (2.13)

Note that the left-hand side of (2.13) corresponds to the mixed method for the operator M

In the next section, we will show that this system is uniquely solvable, so the map @ is well given by (2.10).

defined.

734 LEE AND MILNER

111. ANALYSIS FOR THE LINEARIZED PROBLEM

We need to solve the following problem: Find ( s , q ) E VJ’ x 1C’” such that

( B ( u , y ) , s , U ) - (div (1.4) + ( T 1 q , U ) = ( 1 , U ) V U E V’, (div s, w) - ( r2s ,u t ) - ( y q , u: ) = (711, w ) Vul E bVJ’. (3.1)

We shall use the following duality result from [4].

Lemma 3.1. for suff iciently large p ,

Let s E V, 1 E Lz(12)’, and m E L‘(0). lf q E IZ;“ satisfies (3.1), then

IIqIIo 5 Q(p-”211sIIo + p-211(jiv .SII(, + I l l I Io + I I ~ ~ I I ~ ) ,

ifthe structure condition (2.12) holds. We shall also need the following bound for the vector function.

Lemma 3.2. lf s E Vi’ satisfies (3.1 ), then

II.4lo + lldiv SI I I I 5 C(llqll0 + I l ~ l l o + lIn41).

Proof. To bound llsllo, choose u = s, 21) = q in (3.1) and add the resulting equations. The

We can prove now that the operator CP is well defined. It suffices to demonstrate the following choice ui = div s in the second equation of (3.1) gives the bound for I(div yllo. m

result.

Lemma 3.3. There exists one and only one solution of the system (3.1) under the structure condition (2.12).

Proof. Existence follows from uniqueness, because the system is linear. Assume 1 = 0, m = 0; then Lemma 3.1 implies

Ilqllo 5 QF”’ II s II I ’

IISIIL, L Cll~Il(1~

11q11o L d 2 ~ ~ ” ~ l l s l l ~ ~ 5 Q P - ’ / ~ I I ~ ~ ~ o ~

where I Isl l r~ = IIsll,) + lldiv s / ( ( ~ . By Lemma 3.2, we have

This implies that

which gives q = 0 for large p . Finally, q = 0 yields s = 0.

IV. EXISTENCE AND UNIQUENESS

The solvability of ( 1.7) is equivalent to showing that CP has a fixed point.

Theorem 4.1. For p sufficiently large. CP has a f ixed point. To prove this, we shall need another duality lemma.

Lemma 4.2. Let w E V, 1 E L 2 ( 0 ) 2 and nx E L2(f2). l f ~ E “7’ satisfies the relations

(B(u , y ) ~ , 1 1 ) - (div U , 7) + (r,., U ) = (1 , U ) V U E V J ’ ,

(div w , ~ ) - ( r 2 w , u ) ) - ( ~ T , w ) = ( m , ~ ) VW E W1’, (4.1)

. NONLINEAR ELLIPTIC PROBLEMS.. . 735

then, for 2 I 0 < 00, there exists ( I constant C = C(0, y, u, rl , r2, y, 0, E ) > 0, independent of p, such that

The proof of this result follows from standard techniques and we shall omit it here. Now let V” = V7’ with the stronger norm llvllvl> = IIvllo,q + lldiv v[[o and let WP = W”

with the stronger norm llwllwp = \ \ ~ 1 1 ( ) , ~ . We shall prove the existence of a solution of (1.7). It follows from Brouwer’s fixed point theorem that Theorem 4.1 holds if we can show the following result.

Theorem 4.3. V7’ x W7’, cenlered at (7rpy, P7’u), into itself; provided the structure condition (2 .12 ) holds.

Proof. We apply Lemma 4.2 with the following functions: w = 7Py - s , T = P7’u - q,

For 6 > 0 sufficiently small (dependent on p), iP maps the ball of radius 6 of

i = B(U,Y) [T”Y - Y] + r , p U - U ] + &!,(u - p , y - p ) ,

and

nz = -r2[+, - y] + ~ [ P T ’ U - u] + Q,(u - p , y - p ) .

We have, for sufficiently large p,

[piu. - q [ j o . e I - sl[o +p-1-2/u lldiv (“”Y - s)llo + ll%l + Ilmllol 5 C[p1/2--”i@ I[7r1’y - s110 + p-’-2/elldiv (7r’”y - s)llo + 6* + p‘] , (4.2)

where t = max{ 1/2 - T , 3 - 2 ~ ) . and C = C( IIuII,., I[ yll,., r l , Q!,, Qr). Because we need t to be negative, we find a necessary regularity constraint for our argument to work T > 3/2.

Next, let us apply Lemma 3.2 to (4. I ) . It follows that

IW,Y - S I I v L C[IIP”u. - qII0 + 11111(1 + IImIIoj I Cb‘ + 6’1.

II7r’)y - SIjV’’ I Cpb‘ + s”],

I I P U - qllw. I C[p‘ + 6”.

(4.3)

Also, ( I . I 1 ) and (4.2) imply that

(4.4)

while (4.2) and (4.3) imply that

(4.5)

Therefore,

“ P P U - yJJw1. + II7rJ’y - SIIVI. I C1p[P + p’]. (4.6)

We want to choose p and 6 so that Clp’+‘ < 6/2 and Clp6* < 6/2. This requires that (Cfp’+‘) < i . Consequently, we need t < -2, that is T > 5/2.

Note that this is a severe regularity constraint, because we need u E H”(L?)*. This is alimitation of our proof, not really in the numerical method. This limitation appears because of our inverse estimates ( 1 . I 1 ) and, in fact, is needed to guarantee that the solution we find actually converges in L”.

Now let T 2 5/2 + ~ , t = 1/2 - T , and p 2 (2C,)’lE. Then the interval I = [2Clp1+‘, - ’ ac., p sJJvI. I 6 as needed. We are now free to choose 6 = 2ClpP1-‘ to have the best conver- gence rate.

- ‘ I is not empty. Moreover, for 6 E I , (4.6) implies that IjPl’u - q1Iwr, 5 6 and I[+’y -

736 LEE AND MILNER

Remark 4.4. Note that (4.6) already gives the convergence of the method

lI7-r"~ - silo + lldiv (+" - s)llo + IIPu - q1Io 5 C P - ' - ~ . (4.7)

Furthermore, combining (4.6) with ( 1.1 I ) , we also have convergence in L"

11x7'9 - + iidiv w y - S ) I I ~ . ~ + 11~71u - giio,m I W E . (4.8)

We shall prove a uniqueness result provided that the coefficients a,, i = 0, 1,2 of (1.1) are three times continuously differentiable.

Theorem 4.5. If p is sufficiently large, there is a unique solution of (1.7) near the solution (y, u ) of (1.6) under the structure condition (2 .12 ) .

Proof. Let (y f ' ,u : ) E V p x W",i = 1 ,2 , be solutions of (1.77, and let Y = y: - yg, U = uy - dj', C, = y - yf', E, = u - uI), z = 1,2. Next, it follows from (2.5) that

(b(u, y) - b(u;, y;), v ) - (div v , U ) = (b(u, y) - b(u7, y:), v ) Vv E V, (div Y, w) = (f(u7, yy) - f(uG, y:), w) Vw E W". (4.9)

Using (2.1) -(2.4), we may rewrite (4.9) as

(B(% Y)Y, v ) - (div u, U ) + Flu, v ) = ( Q b ( E 2 , ( 2 ) - Q b ( l l , C l ) , P ) )

(div Y,w) - F 2 Y > W ) - ( Y U , W ) = (Q , (E2 ,C2) - Q , ( E l , C 1 ) , 4

Vv E V" VW E W".

(4.10)

Then, combining (4.10) with Lemma 3.2 we see that

IIYllO + lldiv Yllo L C[ll~llO + I I Q b ( < l > C I ) - Qb(E2,<2)110 + l lQ , (< l>Cl ) - Q,(€2rC2)1101.

(4.1 1)

Also, by Lemma 3.1,

I lUl lo L c{P-''211yllo + p-"l(div YllO + IIQl>(Ell C l ) - Qb(E2?C2)1IO

+ I IQr(ElrC1) - Q f ( ~ 2 r C 2 ) l I O l . (4.12)

NONLINEAR ELLIPTIC PROBLEMS.. . 737

Similarly, we can see that

IIQb(<l?Cl) - Qb(<21C2)110 L CP-"I l~l lo + IlYllOl. (4.16)

Combining (4.1 1)-(4.16). the lemma follows for p sufficiently large. m

V. L2-ERROR ESTIMATES

We shall now establish our error estimates in L2

Since the left-hand side of (5.1) corresponds to a mixed finite element method for the linear operator M , it follows from [ 101 that

l l 4 + lldiv 410 5 C{11~110 + 11.11;.4 + 1 1 ~ 1 1 ~ , 4 + I17TPY - YI ;,4

+ IITPY - Yllo + IIPPU - 4 ; , 4 + IIPPU - 410)

+ P-2T-1/2 IIUIIT+l + P--I 1 1 ~ 1 1 T + 1 ~

+ P-r-lIIUllr+l(II~llr+l + 1)). (5.2)

I ~ { 1 1 ~ 1 1 0 + 11~110,4P11~110,2 + I l~ l lo ,4Pl l~ l lo + P3-2'11Yll~ + P1/2-TIIYIIT

L C{ll.llO +P-EO1l~l10 + P1~2-rllYllr~llYllT + 1)

Hence, for p sufficiently large, (5.2) yields

l l d o + lldiv 410 L C{llTllO +P1~2-rll~llT~llYllT + 1) +P-r- l l l~l lT+l~l l~l l r+l + 1)). (5.3)

Likewise, by Lemma 3.1,

11~110 I cLp-1/211~l10 +P-211div 410 + l l l l lo + Ilmllol L cb-1/211~110 + P-211div 4 1 0 + P-EIITIIO +P1~2-TllYllT~llYllT + 1)

+ P-7.-111~11T+1(11~11r+l + 111. (5.4)

738 LEE AND MILNER

By taking p sufficiently large, (5.4) leads to

IITllo F c [ P - F l l ~ l l \ + ~ ~ ~ ‘ ~ ’ l l ~ Y l l f ~ l l ! l l f ~ + 1) fp-’. ~ I I I ~ I / , + l ( / l U l I , , I + 1,). (5.5)

Substituting (5.5) into (5 .3) , we see that for p sufficiently large.

II4l. F ~ “ P l ~ ’ - - ’ l l Y l l ~ ~ ~ l l ! l l l + 1) +I)-‘- ~ l l ~ ~ l l f + l ~ 1 l ~ ~ l l , t l + 111. (5.6)

Finally, we substitute (5.6) into (5.5) to arrive at the relation

1 1 ~ 1 1 , 1 5 ~ ‘ [ ~ ~ ” 2 - ’ l l Y l l ~ ~ l l ! / l l f + 1) + ~ ~ ‘ - ~ l l ~ ~ l l f + , ~ l l ~ ~ l l , + . l + I ) ] . (5.7)

The theorem now follows by combining (5.6) and (5.7) with (1.9) and ( I . 10). m As indicated earlier, the proof we give imposes a severe regularity constraint on the solution

of the differential problem. In fact, we have assumed that u E H7/2t ‘(12). which entails that u E C’(C2)’. This allows the approximation estimates needed along the proof to hold. However, we also need the extension of u and ,q to a domain with C” boundary, which is possible if, for example, the second derivatives of and y are bounded, respectively. in 12 and 11’.

C‘’(c1) and

VI. EXPERIMENTAL RESULTS

We shall now apply the numerical methods described in Section I to approximate the solution of the minimal surface problem,

The approximation we compute in this example is based on the BDM (Brezzi Douglas Marini) spaces [ I ] , which are based on polynomials of some fixed total degree, rather than on tensor products. Consequently, the local dimension of these spaces is asymptotically much smaller than those of the corresponding Raviart Thomas Nedelec spaces and. as indicated earlier, the proofs we gave in the preceding sections work just as well for these spaces. We now give a succinct description of the BDM spaces. Let

QJ’( E ) = {polynomials of degree 5 p on E } .

For each element E E G. we base the space for the scalar part on QJ’ ~ ’ :

(6.2)

The space for the vector part is based on polynomials of total degree p augmented by a space of polynomials of degree p + 1 of dimension two. Let

\,v/’- I ( E ) = QJJ- 1 .

\’J’(E) = P”(E) (D Span (curl zJ’+ly , curl zyJ’+’) , (6.3)

where PI’(E) = &I’ x QJ’. Then, we define

V” = ( 1 1 E H(div ;R) : .tl,.; E V ” ( E ) , E E G}, w”-I = (711 : ‘UI(k; E wP-l(t;). E G}.

NONLINEAR ELLIPTIC PROBLEMS.. . 739

TABLE I. Error for the scalar part, mesh size = 1/2. Degree of polynomial, p u(z ,y) = h(cos(y - 0.5)/cos(z - 0.5))

llu" - UllcC lluP - ul10 2 0.3506E-01 0.1464E-0 1 3 0.1 1388-02 0.33378-03 4 0.14678-03 0.4926E-04 5 0.1024E-04 0.2459E-05 10 0.6405E- 10 0.1490E- 10 15 0.11698-32 0.2779E- 13

- 1 4 P ~- I 4 Mean convergence rate P

We now seek (yp, u7') E 1."' x W7'-' as the solution of (1.7) with V7' x W7' replaced by V7' x W"-'.

We consider f2 = [O, 112. The discretized weak form is

(b(u7 ' ,yP) ,2) ) - (d ivv ,d ' ) = (9,v.v) VIJ E IIp, (d ivv ,w) = 0 Vw E W7'. (6.4)

We use Newton's method to resolve the nonlinearity in the algebraic system using the following iterative procedure described in detail in [5] :

( B ( u " , 7 ~ ' ~ ) ( y " + ~ - y7'),21) - (div 21,u7'+-')

(div y''+l, w) = 0, = -(b(u",y"),w) + (gr2).vVr. ,) , 2) E V', (6.5)

w E Wp-',

where v, , denotes the outer unit normal to the edge e , of E , i = 1,2 ,3 ,4 . For this problem we have [3]

and

TABLE 11. Error for the vector part, mesh size = 1/2

Degree of polynomial, p u(z, y) = ln(cos(y - O.~)/COS(Z - 0.5))

2 3 4 5 10 15

Mean convergence rate

IIY" - Yll% 0.1982

0.5790 x 10- '' 0.1003E-02 0.5501 E-04 0. I659E-08 0.76648- 12

w - l I

IlY' - YIIO

0.5281E-01 0,12568-02 0.19068-03 0.1006E-04 0.192 1 E-09 0.1586E-I2

w - l I

740 LEE AND MILNER

Then we have

Now we present in Tables I and I1 the results from numerical simulations, respectively, for the scalar part and the vector part of the solution of problem (6.1) with the following boundary values:

(log(cos(y - 5 ) ) - log(cos(-.5))) (log(cos(y - 5 ) ) - log(cos(.5))) ((log(cos(.5)) - ~og(cos(z - . 5 ) ) ) )

i f0 < y < 1,z = 0, i f O < y < l , z = I , i f0 < z < I , y = I , i (log(cos(-.5)) - log(cos(z - 5 ) ) ) if 0 5 z 5 1,y = 0.

(6.7) g(zl y) =

The exact solution for this problem is

cos(y - 0.5) sm(z - 0.5) ' u ( 5 , y ) = log .

which belongs to C" (fl). From Tables I and 11, we observe that the convergence of the approximation to the true solution

is of the order of p - I " . This is due to the fact that the true solution actually belongs to H''(I1) for T as large as we want.

Finally, we present in Fig. 1 a picture of the numerical solution for the problem (6.1) with a different boundary condition g(z, y), given by

-0.05 sin(27ry) i f0 5 y 5 1,z = 0, 0.05 sin( 27ry) i f O < y < l , s = l ,

i f0 5 z 5 l , y = 1,

The picture shows the mixed method solution using p = 15. We see from the tables and the figure that the numerical method we described indeed produces

excellent approximations, when the domain is rectangular and the true solution smooth. This is, of course, consistent with our theorems.

In cases when our theorems do not apply, such as when the domain is smooth -and one necessarily must use elements with a curved boundary -or when the solution is not so smooth,

FIG. I . Approximate solution using boundary values (6.8).

. . . NONLINEAR ELLIPTIC PROBLEMS.. . 741

the method still seems to give good approximations. We have run many experiments with solutions that are not very smooth, but the approximations were still very good. However, they were all done on the unit square. Further experiments will be conducted on a circle, to see whether the approximations are still good even when using curved boundary elements.

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