ll/ll*.,:o = E ll¿>7l|l' - Arizona State Universitymilner/PDF/Milner Math Comp 1985.pdf · april 1985. paces 303-320 Mixed Finite Element Methods for Quasilinear Second-Order

mathematics of computationvolume 44. number 170april 1985. paces 303-320

Mixed Finite Element Methods for Quasilinear

Second-Order Elliptic Problems*

By F. A. Milner

Abstract. A mixed finite element method is developed to approximate the solution of a

quasilinear second-order elliptic partial differential equation. The existence and uniqueness of

the approximation are demonstrated and optimal rate error estimates are derived.

1. Introduction. Let ß c c R2 be a domain with C2 boundary 9ß. We shall assume

that for some e, 0 < e < 1, and for each pair of functions (/, g) in He(ß) X

H3/2+e(dü) there exists a unique solution p e i/2+£(ß) of the quasilinear Dirichlet

problem

(a) Lp = -V ■ia(p)Vp + bip)) + cip)=f inß,

(b) p = -g on 9ß,

where Vw denotes the gradient of a scalar function w, and V • v denotes the

divergence of a vector function v. Note that then p belongs to Wx-Xiü), which will

be needed throughout the paper.

We shall also assume that the coefficients a: ß XR -» R, b: ß xR -» R2, and c:

ß X R -» R are twice continuously differentiable with bounded derivatives through

second order; moreover, assume that aix, q) > ax > 0. The variable x will normally

be omitted in this notation below.

For 1 < s < oo and k any nonnegative integer let

W*-'(Q) = {/e Z/(ß)|Z)a/e Z/(ß) if |a| < A:}

denote the Sobolev space endowed with the norm

ll/ll*.,:o = E ll¿>7l|l'<n>

(the subscript ß will always be omitted unless necessary to avoid ambiguity). Let

Hk(Q) = W*'2(ß) with norm || • \\k = \\ ■ \\k2 (the notation || • ||0 will mean || • ||L2(i2)

or || • ILw). For 0 < r < oo let Wrs(ü), Wrs(dü), Hr(tt), and Hr(dQ) denote the

fractional order Sobolev spaces with norms || • \\rs.a, || • ||rs;3ß, || • ||r.a, and || ■ ||r.aa.

Received August 3, 1983; revised April 10, 1984.

1980 Mathematics Subject Classification. Primary 65N15, 65N30; Secondary 41A10,41A25.*This paper represents work done by the author for his Ph. D. thesis at the University of Chicago, and

was partly supported by the Aileen S. Andrew Foundation.

©1985 American Mathematical Society

0025-5718/85 $1.00 + $.25 per page

303

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

304 F. A. MILNER

We shall denote by ( , ) the inner product in either L2(ß) or L2(ß)2, that is,

(tj,0) = ( t]8dx or ( t\ Qdx.

Let ( , ) be the L2-inner product on the boundary of ß:

(A, 77) = / Xtrds.

We shall use the same notations to indicate the dualities between Ws(ti) and

W*s(ti)' and Hs(dQ) and H~'(dQ), respectively.

Let

V = H(div; ß) = { v g L2(ß)2|divv e L2(ß)},

normed by

l|v||v=IK||o+l|divv||0,

and let

H^div; ß) = { v g L2(ß)2|divv g //s(ß)},

normed by

IKI|H»(div;a) = l|v||o + l|divv||J.

Let

W = L2(ß).

If

(1.2) u = -(a(p)vp + b(p)), a = \/a, ß = ab,

then (u, p) g V X Wis a solution of the following weak formulation of (1.1):

il3, (a) (a(/Ou,v)-(divv,/0+(ß(/>),v)=(g,v-v), v g V,

(b) (divu,w)+(c(p),w) = (f,w), W&W,

where v denotes the unit outward normal vector to 3ß. Since v • v e H~1/2(dil) (see

[12], [16]), the duality (g, v ■ v) is well-defined.

Mixed finite element methods for (1.1) are discrete versions of (1.3) and have been

treated for linear operators L by several authors [2], [5]-[7], [9], [12]-[16].

Let J4 be a quasi-regular polygonalization of ß (by triangles, rectangles, or

possibly parallelograms), with boundary polygons allowed to have one curved side,

of characteristic parameter h g (0,1), and let

\hX WhcY X W

be the associated Raviart-Thomas-Nedelec space of index k > 0, [11], [12]. To be

more explicit, for E c R2 let Pk(E) denote the restrictions of polynomials of total

degree k to the set E and let Qk(E) denote the restriction of Pk(R) ® Pk(K) to E.

Then, let Rk(E) = Pk(E) if E is a triangle (interior or boundary) and Rk(E) =

Qk(E) if E is a rectangle (interior or boundary), and let Rk(E) = Rk(E)2. For any

E g g-k let

V(£) = RkiE) e Span{xÄA(£)}, W(£) = /?,(£).


FINITE ELEMENT METHODS FOR SECOND-ORDER ELLIPTIC PROBLEMS 305

Set

\h = \(k,r„)= {VGV|V|£GV(£),£G^}

= ( v g fi V(£)|v|£j • v, + v|£ • v, = 0 on £,. n £,),I Eefh J

where v,, / = i,_/, is the outer normal to dE, on £, n £•; also, let

W„ = ^(fc,.TA) = (wG W|w|£G H^(£),£G^,}.

Let 7Th: V -> VA be the Raviart-Thomas projection, [6], [12], which satisfies (see

[13] for q + 2 below)

(1.4) (div[wAv - v],w) = 0, veV.welr'j,

(1.5) Ihr - 4>., < ßlMI,.,A', l/g<K¿ + l,ifveVn I*-«(ß)2,

(1.6) ||div(77Av - v)||0< ßlldivvll./j*, 0<5<fc + l,ifveVnHs(div;Ö).

Let i*A: W -» W^ be the orthogonal L2-projection into WA defined by

(1.7) (Phw-w,x) = 0, w<=W,XeWh,

which satisfies

(1.8) ||P*h> - w||o,, < ßlML*'. 0<j<it + l,ifiverr'nr''(8),

(1.9) ||/»Aw - w\\_r < ß|MI**''+,> 0<r,s<fc + l,ifweJ7'(Q),

(1.10) (divv, w - £Aw) = 0, w g W, v g VA.

We can now formulate the mixed finite element method to approximate the

solution of (1.1):

Find (uA, ph) g \h x WA such that

(in) (a) (a(/7ju,,v)-(divv,jpJ+(ß(/;A),v)=(g,vv), v G V„,

(b) idivuh,w)+iciph),w) = if,w), w<=Wh.

We shall demonstrate in Section 2 the existence of a solution (uA, /?A) g Va X Wh

of the nonlinear algebraic system (1.11) through an adaptation of the method used

by Douglas in [4]. In Section 4, we shall establish the uniqueness of that solution

inside a certain ball. Furthermore, we shall show that (uA, ph) converges to (u, p) in

L2(ß)2 X L2(ß) at an optimal rate as h |0 (Section 3) and also in (i/s(ß)2)' X

//s(ß)', 0 < s «s k + 1, provided that the boundary of ß, the coefficients a, b, and c,

and the solution/? of (1.1) are smooth enough (Section 6). In Section 5, we establish

the convergence of ph top in £?(ß), 2 < q < oo, at an optimal rate as h -* 0.

2. Solvability of the Discrete Problem. For pe Wh, we shall write

(2.1) a(p) -a(p)= -àp(p)(p - p) = -ap(p)(p - p) + àpp(p)(p - pf,

where

àp(p)= flap(p + t[p- p])dt,

àpp(p)= f(l - t)app(p + t[p-p])dt

are bounded functions in ß. Similarly, we can write

(2.2) p(p) - p(p) = -%(p)(p - p) + Vpp(p)(p - p)2 = 4p(p)(p - P),


306 F. A. MILNER

(2.3) c(p) -c(p)= -cp(p)(p - p) + cpp(p)(p - p)2 = -cp(p)(p - p),

where ß (p), cpip), ßpp(p), and cpp(p) are bounded functions in ß.

If we now subtract (1.11) from (1.3), we obtain the error equations

(a) (a(p)[u - uA],v) -(divv, p - ph) + [flp(p)[p - p„],v)

= ([cx(ph) - a(p)]uh + MPh) - HP) + fy(p)[p - PhU),

(2.4) v G VA,

(b) (div[u - uA],w) +(cp(p)[p - ph],w)

= {c(Ph)-c(p) + cP(p)[P -Ph]>w), wG Wh.

Substituting (2.1), (2.2), and (2.3) into (2.4), we see that (with p = ph)

(a) (aip)[u - uA],v) -(divv, p - ph) + ([ap(p)u + Vpip)]ip - ph),v)

(2 5) = ([&pp(Pû + $pp(P^](P -Ph? + apiph)ip-ph)iu- uj,v),

veVA,

(b) (div[u - uh],w)+(cp(p)[p-Ph],w) = (cpp(ph)[p-ph]2,w),

WG Wh.

Set T = ap(p)u + $pip) g Cg(ß) and y = cpip) g C¿(fi). Let us now replace u

by 7TAu and p by Php on the left-hand side of (2.5) to obtain (using (1.4) and (1.10))

the relations

(a) (a(/>)Ku - uA],v) -(divv, Php - ph) +(T[Php - ph],v)

= (aip)[irhu - u] + T[Php-p]+[âpp(ph)u + $ppiph)](p ~ p„f

(2.6) +àpiph)ip-Ph)iu-uh),\),

veVA,

(b) (div[wAu - uA],w) +iy[Php -ph],w)

= (y[Php -p] + cPPiph)ip -ph)\w), wG wh.

Now let M: i/2(ß) -» L2(ß) be the operator

Mw — -V -(a(/>)Vw + aip)Tw) + yw,

and let M* be its formal adjoint; that is,

(2.7) M*x = -V -(a(/>)Vx) + aip)T ■ Vx + YX-

We shall assume that the restrictions of M and M* to H2(ü) n //¿(ß) have

bounded inverses; that is, for any ip g L2(ß) there exists a unique $ G i/2(ß) n

//¿(ß) such that M<t> = rf, (respectively, Af*<f> = i//) and ||<>||2 < Q\W\0. This would

be guaranteed by assuming c > 0 (see, for example, [8]). Let

<D:VAX Wh^\hx Wh



be given by $(("., p)) = (y, z), (y, z) being the (unique) solution of the system

(a) (a(p)[trhu - y],v) -(divv, Php - z) +iT[Php - z],v)

(2 8) - («(/>)Ku - u] + T[PhP -p]+ [&ppip)u + %pip)]ip - p)2

+ *PÍP)ÍP- PH»" rl)>v),

veVA,

(b) (div[7rAu-y],w)+(Y[PA/>-z],w)

= (y[Php-p) + cpp(p)(p- p)2,w), we Wh,

the existence of which follows for small h from [5], since the left-hand side of (2.8)

corresponds to the mixed method for the operator M. Thus, (y, z) is the solution of a

linear algebraic system of the form

(a(/>H,v) -(divv, (p) +(r<p,v) = F(\), v g Va,

(div\|/,w) +(ytp,w) = G(w), w G Wh,

which for h sufficiently small has a unique solution (\|>, ) g Va X Wh for any

£ g V, G g W. (Existence follows from uniqueness. Thus it suffices to prove that if

£ = 0 and G = 0, then (\|/, ) = (0,0). This is done in [5] by an argument entirely

analogous to that of our Lemma 2.1.) We are taking in (2.8)

£(v) = -(/>,divv) +(a(p)u + Tp- [&pp(p)u + Ppp(P)](p - Pf

+ «p(p)(p- p)iu- u),v),

G(w) = (divu + yp - cppip)ip - pf,w).

The existence of a solution (uA, ph) g Va X Wh of (1.11) is equivalent to that of

the following problem.

Problem 2.1. The map 0 has a fixed point.

The solvability of Problem 2.1 will follow from the Brouwer fixed point theorem if

we can prove that 0 maps a ball of \h X Wh into itself.

In order to do that, we shall use the following technical result.

Lemma 2.1. Let 2 < 8 < oo. Let u G V, q G L2(ß)2, and r G £2(ß). // t g Wh

satisfies

(2 9) í («(/>)",¥)-(divv, t)+(Tt,v) = (q,v), v g Va,

\ (divw, w) +(yr, w) = (r, w), w g Wh,

then, there exists a constant C = C(8, a, T, y, ß) such that

(2.10) ||t||0.í < CfAÎMIo + ä1 -KV»Xi-»«^lldivu||o + llqllo + IM|o]for h sufficiently small. Also, if » G W°-e(di\; ß) = { v G Lö(ß)2|divv g Le(ü)},

qG £9(ß)2, andre L\Ü),

Ikllo.« < c[A||w||o.í + /!2"s°*||divu||0,» + A||q||0.9 + A2~S|»|kl|o.e + ||q||-i.» + lk||-2.«] •

Here, and throughout the paper, 8tJ will denote the Kronecker symbol.

Proof. Let 6' = 8/(8 - 1) be the conjugate exponent of 8. Since

Il II (T»*)lmlo.0= sup ——,

vi-^0


308 F. A. MILNER

we wish to bound (t, \p) for \p g Ls'(ü). Let <í> g W2-e'(Q,) be the (unique) solution

of M*<¡> = xp in ß, <p = 0 on 9ß. Then [1],

Hke- < kU\W-Observe that the Sobolev imbedding theorem [17] implies that W2~(-2/9)-6'(ü) c

H\ü) and W1-(2/e)-e'(ti) c L2(ß); so, ^-»'(Q) c H(2/e\ü) with

(2.11) ||X||2/S < K||A||M,.

Next, note that for any £ g 9~h

a(p)vê Wx'e\E)2âip)v<?lE& Wx-ê'^(dE)2,

and therefore the degrees of freedom defining trha(p)V on £ are well-defined, that

is, 7rAa(/?)v is well-defined.

Using (1.4), (1.10), (2.9), and integration by parts, we see that

(t,xP) = (t, M*(j>) = (t,-V -(a(p)v<?) + a(p)T ■ V</> + Y</>)

= (t,-V ■(Âa(/7)v<p))+(rT,7rAa(/7)v<í>)

+ (Tr,a(p)v<? - -rrha(p)v<?) +(yr,<?)

= (q, mha(p)v<¡>) -(a(p)u,7rha(p)v4>)

+ (Tr,a(p)v<¡> - irha(p)v<¡>) +(yr,4>)

= (q, a(/>)v<f>) +(q, irha(p)v<? ~ a(p)v<?)

+ (a(p)u + Tr,a(p)v<? - trha(p)v<?) + (divu,<p) +(yr,4>)

= (l,aip)V<f>) +(q,trha(p)v<? - a(p)v<?)

+ (a(p)u + IV, a(p)v<t> - irha(p)v<?) +(divto + yr, )

+ (r,<p)+(r,Ph4>-<¡>).

First, observe that

(q, a(p)v<p) < Al|q||o|| V4>||o « KhllolMki'-Furthermore, since V(£) z> P0(E)2, an Lp-version of the Bramble-Hubert Lemma

[3] implies that (using (2.11))

\\a(p)v<t> - vha(p)v<t>\\0 < *A2/'l|V*||2/i < *A2/1v*||i,,< < Kh2/"Uhe;

(a bound which cannot be obtained from (1.5) if 2/8 < 1/2) and therefore

(q - a(p)u,Ttha(p)v<? - a(p)v<t>) < A"(||q||0 + ||w||0)Ka(/>)V<|> - a(/>)v<p||0

<^(||q||o+Ho)A2/SW2,fl'.

Also, we see from (1.5) that

(IV, a(p)v<t> - TTha(p)v<?) ^ K\\T\\0Ja(p)v<t> ~ *haip)v<?\\0,e-

< ä-HtIIoâIIvÎI,,,. < a:||t||o,íA||*I|2,»'-Finally, from (1.8) and the Sobolev imbedding theorem we see that

(divw, </> - Ph<?) < /c||divu||0||<f> - Ph4>\\0

<^||divW||0A1^><1-^)||<i,||1+(2/(/)(1_Soi)

^K\\diw^\\0hï+^m-s^\\<t,\\2t$,

(vt, <#» - Ph4>) < tf|M|0,J* - p,fi>\\0j. < A:||T||0,,A2^»'|lll2,r,

(r,<j>)+(r, Ph<j> - <t>) < K\\r\\0\\<p\\o < J^|k||0tl4>llx-<2/*>.«- < tf|Mloll*lks-



It then follows that

(t, *) « K [a||t||m + A2/"||w||0 + A1+<2/"><i-^)||divw||o + ||q||Q + ||r||o] Wo>r>

and thus, if A is sufficiently small, it suffices to take C > (1 - Kh)~x in (2.10). The

second bound for ||t||0i# follows easily from the equation for (t, \\i) following (2.11),

(1.5), and (1.8). Q.E.D.

Let now irh = VA with the stronger norm ||v||y = ||v||02+e + ||divv||0, and let

Wh = Wh with the stronger norm \\w\\n = IMIo,(4+2E)a-

We are now ready to demonstrate the existence of a solution of Problem 2.1.

Theorem 2.1. For 8 > 0 sufficiently small (dependent on A), $ maps a ball of radius

8of-ThX Wh into itself.

Proof. Let 8 = (4 + 2e)/e so that (I/o) + (1/(2 + e)) = \. Let

Ik« - Úrh < s and ||£A/7 - p\\^ < Ä < 1.

Interpret q and r in (2.10) as

q = TiPkp - p) + a(p)(irâ - u)

+ [&ppip)u + \3pp(p)](p - p)2 + äpip)ip - p)(u - ,i),

r = y(Php - p) + cppip)ip - p)2,

and apply Lemma 2.1 to (2.8).

Note that, since e < 1 implies that 8 > 4 and that 2 + e < 4/(2 - 2s), #E(ß)2 c

L2 + E(ß)2, //1 + E(ß)2 c Wx>2+e(Q)2, and tf2+E(ß)2 c W2-2 + \ü)2. Thus,

Hx+^2\ü)2 c W12+E(ß)2 and ||A||12+e < ßE||A||1+(E/2). Then, (1.5) and (1.8) imply

the inequality

\\PhP - *llo.i < K[h2/%h» - yllo + A^^Xi-*«)!^^ _ y)||o

+ llqllo + Ikllo]

< K [A2/*hu - y||0 + A^/W-^^diviirrfi - y)||0

+ h\\p\\x + A||u||i + \\p - p\\0A + \\p - pIIo.Ju - Ji||o,2 + EJ

(2-12) < ^[A2/e|kAu - y||0 + A^/^-^'UdivKu - y)||0

+ Hp\\i + \\P - PU + {\\P - P„p\\o.e + \\PnP - P\\o,t)

(ll«-Â»llo,2+e + lk*U- rl|lo,2 + E)j

< ^[a^Hû - y||0 + Äi-K2/*)(i-*ot)jjdiv(OT.AU _ y)||o

+ A||p||2 + á2+(A||p||1,fl + 5)(A||u||1,2 + e + 5)]

< k[a2/«Ku - y||0 + hW^WM** - y)llo +(a + «2)W2+J.


310 F. A. MILNER

If we now take the last term on the left side of each equation in (2.8) over to the

right side, the left side becomes exactly the mixed method equations for the operator

-V • (a(p)V). It follows from [2] that then

(2 13) "77*U " y"V * K^PhP ~ Z"° + "q"° + "r"°l

<K,2

\\PhP-40.e+ih + S2)\\p\\;+e

If we now substitute (2.13) into (1.12), we see that, for h sufficiently small,

(2.14) \\P„p - z\\0J < Kx[h + 82],

with Kx depending on ||^||2 + E linearly. Putting (2.14) back into (2.13), we obtain

(WxthK2 = K(Kx + \\p\\\+t))

(2.15) ||divKu-y)||0<A-2[A + S2],

(2.16) \Wn-y\\0^K2[h + 82}.

It follows from (2.16), using the quasi-regularity of 3Th, that

IItt U - vll < K fc(2/<2 + t))-l|| _ II/„ 17x Whu vllo,2 + e ^ äe" IFAu y Mo

K Kth-u/(2+t))K2(h + 82) < AT3[A2/(2+£> + A-<E/(2+E»S2].

We now see that (2.15) and (2.17) imply that

(2.18) Ku - y||n < 2/C3[A2/(2+E) + A"«/<2+e»S2].

Now let A < (4a:3)-<4+2£)/(2-£) and take 5 = 4K3h2A2+c). Observe that in order that

2K3h2A2+e) < 8/2 and 2K3h-(eA2+e))82 < 8/2, we must have 5 G [4ä:3A2/(2+e),

(4A^3)_1AE/(2+E)] * 0, which is satisfied for A and 8 as chosen.

The theorem is now proved, since (2.14) and (2.18) imply that \\Php - z||^- < 8

and ||ttau - y||^ < 8; that is, 0 maps the balls of radius 8 = 0(A2/(2+E)), centered

at ( trhu, Ph p ) into itself. Q.E.D.

3. L2-Error Estimates. Note that Theorem 2.1 in fact shows that as A -» 0 we

obtain a sequence {(uA, ph)}hl0 which converges to(u, p) in V n L2+E(ß)2 X L*(ß)

and furthermore, that there is a constant C = 4K¿ + ßßE||u||1+E such that

(3.1) max{ ||u - u J0,2 + E; \\p - /> JM} < CA2/<2+E>,

since

II" - U*llo.2 + « < II" - n«llo,2 + e + Ik" - "aIIo,2 + e

< 0*Mu+. + S < ßAße||u||1+E + 4tf3A2/<2+E\

and

\\P - PhWo.e < \\P - PhP\\o.e + \\PhP - PhWo.e

< Qh\\p\\i,9 + 8 < ßAßE||p||2 + 4tf3A2/<2+E>.

Let us now rewrite (2.4) as

(a) (a(p)[u-uh],\)-(divv,p - ph)

(3.2) +([&PÍPhK + PipiPk)]ip-Pi,),1<) = 0, veVA>

(b) (div[u-uh],w)+(cp(Ph)[p-ph],w) = 0, weWh,

where ß„(/>A) and cp(ph) are bounded functions in ß defined in (2.2) and (2.3).



Observe that (3.2) corresponds to the mixed method for the operator N: H2(Q,) n

//01(ß)^£2(ß) given by

Nw= -V -(a(p)vw + a(p)[ap(ph)uh + $p(ph)]w) + cp(ph)w.

Its formal adjoint N*: H2(Q) n HX(U) -* L2(ß) is

(3.3) N*X = -V -(a(p)Vx) + a(p)[&piph)u„ + $p(ph)] • VX + cp(ph)x-

Before we turn to the rate of convergence of (uA, ph) to (u, p), we need the

following technical result.

Lemma 3.1. There exists an h0> 0 such that, if h < h0, N* has a bounded inverse

mapping L2(ß) onto H2(Q) n ¿/¿(ß).

Proof. Since M*"1: L2(ß) - H2(ti) n #¿(8) is bounded and N*-1 =

(M*~XN*)'XM*'X, it suffices to show that M*'XN* has a bounded inverse on

H2(ß) n H^ti). For a linear differential operator D: X -» Y, let |||D||| be its norm

as a linear functional; e.g.

|||M*||| = ||M*||^(W2(a)n//i(í2);L2(í2)).

Then, all that is needed is to prove that |||Af*_1(M* - JV*)||| is less than one, since

this will imply that / - M*~X(M* - N*) = M*~XN* has a bounded inverse. Thus, it

is sufficient to show that |||M* - N*\\\ is smaller than (UlAf*-1!!!)-1.

We have, by (2.7) and (3.3),

(M* - N*)x = a(p)[ap(p)u - àp(ph)uh + ßp(p) - $p(ph)\ ■ VX

+ {cp(p)-cp(Ph))x

(3"4) = a(p)[(apip) - äpiPh))U + &piph)iu - uA)

+(ß,(p)-t,(p*))]-vx

+ {cpip)-~c(Ph))x-

Observe that i = p - ph,

*p(p) - àp(ph) = fl[<*P(p) - oLp(ph + /€)] dtJo

(3.5) = ij\\ - t)Capp(p„ + tè + s(l - /)«) dsdt

= âppi,

where app is a bounded function. Similarly, we obtain

(3.6) %(p)-%(Ph) = %pí, cp(p)-cp(ph) = cppi,

where ßpp and c are bounded functions. Substituting (3.5) and (3.6) into (3.4), we

see that

(M* - N*)x = a(p)[[âppu + y (/> - p„) + àp(Ph)(u - uA)} • VX

+ cpp(p - ph)x,


312 F. A. MILNER

and thus, using (3.1),

||(M* - JV*)x||o < ff4[Hlo.JI/> -/»*llo.(4+2.)/.IIVxlloj+.

+ II" - «Allo,2 + ell Vxllo,(4+2e)/e + \\P ~ PJollxllo.oo]

</i5y|2 + f[Ä2/(2 + 1|Vxlli + A2/(2 + E)||x||1 + E]

< a-6a2/<2+e>||x||2,

since H\Q)2 c Lf(ß)2 for any finite r and //1+E(ß) c L°°(ß). Now, take A0 small

enough that K6h20A2+e) < (HIM*"1!!!)"1. Q.E.D.

We can now obtain a rate of convergence of (uA, ph) to (u, p) as A -» 0.

Theorem 3.1. £Aere is a positive constant C independent of A, depending on \\p\\2+e

quadratically such that

C\ II II *rlh^> '^ = 0'(i) \\p - ph\L < C<

\hs\\p\\s, 2<s < k + \,ifpeHsandk > 0,

(ii) l|o-uJ0<a'||/>|Ui. l^s^k + l,ifpeHs+l(Q),

(iii) ||div(u - uA)||0 < Ch*\\p\\s+2, 0^s<k + l,ifPeH*+2(Q).

Proof. Let £ = u - uA, i; = p - ph, o = 7rAu - uA, and t = Php — ph. Rewrite

(3.2) in the form

(a(p)S, v) -(divv.r) +([äp(ph)uh + $p(ph)]r,v)

< =([äpiph)uh + pip(ph)][Php-p],Y), VGVA,

\(divt,w) +(cp(ph)r,w) = (cp(ph)[P„p - p],w), we Wh.

It follows from Lemma 3.1 of [5] and our Lemma 3.1 that

Ho < ^[a||Ç||o + A2-^||divi||0 + \\[äp(ph)uh + %iph)]iPhp-p)\l

+ \\cp(ph)(Php-p)\\0]-

If pe HS(Q), then p e HsA2A2 + i))-e(Q,) and ||p|L_(2/(2 + E)),fi < K\\p\\s. Thus, using

(1.8) and (3.1), the penultimate term in (3.7) can be bounded by

\\[äp(Ph)uh + $h(ph)](Php-p)\\0

, < AT [||u||0-00||£A/7 - p\\Q + ||U - uJ02 + f||£A/> - /4.<4 + 2f)/E]

< K\\p\\22+e[hs\\p\\s + A2^+'>A'^+'»|WI^/(2+.>>.#]

< /:||p||2+EAlblL.We now derive a preliminary bound for ||£||0. Substituting (1.8) and (3.8) into (3.7)

gives the bound

IHIo ^ ^[aWo + A2-sH|divS||o + a>umiL],

which in turn implies, using again (1.8), that

(39) ll£lloHI/>-/>Jo<ll/>-n/>llo + IHIo

<^[ll/'ll2 + EAs||p||J + A2-^||divS||0 + A||Ç||o].



The quasi-regularity of $~h implies that

(3.10) Hollo.» < Kh-W+*\\°\\oa+t-

The Sobolev imbedding theorem implies that

(3.11) /Y1 + E(ß)2 c W^2)'x(ü)2.

Using (1.5) with q = oo and s = e/2, we obtain from (3.1), (3.10), and (3.11) the

inequality

Kilo.» < HUl|o.oc + Iku - Ullo.oo + ll'llo.«

<jrfllnll -4- AE/2lliill + A-2/(2 + E)llall 1=5 A [ \\P 2 + E + " U t/2100 T n ll°llo,2 + EJ

(3.12) L J< K [||p||2+E + A-2/<2+E)(|Ku - u||0,2 + E + ||u - uJ02+E)]

< K[\\p\\2+t + A-2/<2+E'(ß^ll«lli+E + CA2/<2+E')] < K(\\p\\22+t + l).

If we now rewrite (2.4) as

(a(p)a,\) -(divv, t)= (a(p)[ir,fi- u] -[äp(ph)uh + ßA(/>A)J£,v),

veVA,

(divo,w) = (-cp(ph)£,w), weWk,

we see using [2] (just as we did to obtain (2.13)), (1.5), and (3.12), that for

\ < s *S k + 1 and/? e Hs+l(tt)

(3.13) ||o||v < K|HlL[lku - u||0 + Hillo] < /:||p||2+E[lHllo + h°\\p\\s+i] ■

From (3.13) we now obtain by (1.5) and (1.6) the bounds

||s1lo<||u- wAu||0 + ||o||o

< K(\\p\\l+e + l)[U\\o + Al/>|Ui]> Ks<k+1,

(3 lldivJUo <||div(u - 77Au)||0 + ||divo||0

< k(mI+. + i)[||£llo + Al^lUj, o < 5 < k + i

which, when substituted into (3.9), yield the estimate

(316) Kilo < *IHl2+e[A||É||o + *-«-bL + *1p|J

< K(\\p\\22+i + l)[A||í||0 + *'-a«||^||,], 2 < s < k + 1 + 80k.

But (3.16) now implies (i) holds if A is small enough. Applying (i) to (3.14) and (3.15)

shows that (ii) and (iii) also hold. Q.E.D.

Observe that Theorem 3.1 shows that {(uA, ph)}hl0 converges in V X W to (u, p)

both at an optimal rate (for any A) and with minimal smoothness requirements on

the solution of (1.1) (if k > 1).


314 F. A. MILNER

4. A Uniqueness Result for the Discrete Problem. We shall now prove a unique-

ness result.

Theorem 4.1. If f e H2+eA1~e)Sot(ti) and A is sufficiently small, then for any

K > 0 there is a unique solution o/(l.ll) in the intersection of the balls

|llu - «aII^» +Wp -PhWn « (^4ll»llo.oollM*"1|l^(¿2(a);//2(S2)n/yi(S2))) I

^{h-uh\\0,OB+\\p-Ph\\o,oa<K}=B,

where K4 is the constant of Lemma 3.1 appearing after (3.6).

Proof. First note that Theorem 3.1 in fact shows that any solution (uA, ph) e \h

X Wh of (1.11) lying in B will verify the bounds (i), (ii), and (iii) of Theorem 3.1.

Assume now that, for i = 1 and 2, (u^'1, ph']) e VA X Wh is a solution which satisfies

the above hypotheses. Let Ç, = u - ví¿\ £,=/>- ph'\ U = u(hl) - u(A2), P = p[X) - ph2\

and a, = irhu - u(A°. Theorem 5.1(b) below will then imply that

(4.1) ItoHo., < ßA£/2|b||2+e+(1_e)8ot) ¿-1,2.

Also, Theorem 3.1(b) implies

(4.2) ||y0 < Qh\\p\\2, i = 1,2.

It follows from (1.5), (3.10), and (4.2) that

IIS/ILo «II» - *A»llo.oo + Ikllo.oc < KIÎIuIUoc + A^lloJIo]

(4.3) < a:[a-/2||ii|Ii+. + a-HII-a!" - »Ho + IIUo)]

< tf[AE/2||p||2+E + A-^lbHj < K(p), i = 1,2,

where K(p) depends on ||/>||2 + E quadratically.

It follows from (1.3) and (1.11) that

(a(p)U,v) -(divv, P) = ([a(p<2>) - «(p^)K2»,v)

+ {[a(p)-a(phxx)]V + fi(ph»)-ß(ph»),v),

v^VA,

(div\5,w) = (c(p^) - c(p^),w), weWh.

Let

«(/»?>) - "(Ph2)) = *PiP)P, ct(p) - «U1») = âpUx)îx,

ßU1') - Krf») = M*)/. c(pí1)) - c(pí2)) = tp(p)p,_where ctp(P), äp(£x), $P(P), and cp(P) are bounded functions in ß, where P is some

convex combination of phl) mdph2). Then

I («(/>)U,v) -(divv, P) = -([ap(P)uf + $,(?)] P,v)

(4-4) +(5,(€i)€iU,t), vgVa,

( (di\V,w) = -(cp(P)P,w), wewh.

It follows from [2], (4.1), and (4.3) that

(4.5) ||U||0 < K(p)[\\P\\o + A||U||0], HdivUHo < K(p)[\\P\\0 + A||U||0].

For A sufficiently small, (4.5) implies

(4.6) ||U|U^(p)||P||o,



(4-7) ||divU||o«i:(p)||P||o.

Rewrite (4.4) in the form

(a(p)U,v) -(divv, P) +([âp(?H2» + ß,(P)]i»,v) = (ap(Q^,y), v G VA,

(divU,w) +(cp(P)P,w) = 0, w e Wh.

Then, it follows from [4], an obvious variation of Lemma 3.1, and (4.1) that

(4-8) ||P||0<K(p)[A||U||0 + A||divU||0].

If we now substitute (4.6) and (4.7) into (4.8), we have ||P||0 < Kh\\P\\0, which

implies that P = 0 for A sufficiently small. Then (4.6) implies that U = 0. Q.E.D.

5. L'-Error Estimates (2 < q < oo). We shall first obtain a negative norm estimate

for t.

Lemma 5.1. There exists a positive constant C < ^(||/>||2+E + 1)> independent of h,

such that, if 3ß and the coefficients a, b, and c o/(l.la) are sufficiently smooth, then for

0 < s < k,

IMU < c\h^+x\\p\\n+x+(s_k+xy+ A"+'-1/>||r2+1|b||/+fiJ,

1 < rx, r2 < k + 1, 2 — 80k < / < k + 1, wherep is the solution o/(l.l).

Proo/. Since

ll II (T'*)IMI-»= sup ———,ê/7s(ß) llrlli

>/<#0

we wish to bound (t, ip) for ^ g //^ß). Let <j> e /£+2(ß) be the (unique) solution

of M*<¡> = \p in ß, <t> = 0 on 3ß, the existence of which we shall assume. Assume also

that ||<f>||J+2 < ^'ll'r'llj- Note that (1.5) and the Sobolev imbedding theorem give the

bound

(5.1) ||ö(/,)v* - *Aa(/0v*||o.(2 + e)/e « ê/2|l«(/')v«í'||E/2,(2 + E)/E

< /:ae/2||v<í»||i < âe/2||<i»|U2.

Also, the Sobolev imbedding theorem implies that a(p)'V||o, « ^ll^lh; me quasi-regularity of 3~h implies that if x g VA and

ê Wh,

(5.2) ||xllo,2 + E < Â-E/<2+E>||x||o, IHIo,2 + E < Ä»-^-">|Hlo.

Let k = àppiph)u + §pp(ph), \ = äp(Ph), and p = cpp(Ph). We have X, p e L°°(ß)

and k g L°°(ß)2. Rewrite (2.5) as

(a) (a(p)Ç,v)-(divv,T)+(rT,v)

(5.3) =i\«S + mS + T[PnP-p],v), vgVa,

(b) (divS,w)+(YT,w) = (p^2 + Y[PAp-jp],w), weWh,

where £, £, and t are the same as in the proof of Theorem 3.1. It follows from (1.4),

(1.10), (5.3), and integration by parts (exactly as in Lemma 2.1) that

(t, *) = ([icé + XSU + T[PhP - p], aip)v<?)

(5.4) +i[*t + W]t + T[Pkp-p],*haip)V4>-aip)V4>)

+ ia(p)t + Tr,aip)v<l> - trhaip)v<?) + (div£ + yr, <f> - Ph<t>)

+ (p£2 + y[PhP -p],4>) + (p£2 + y[Php -p], Ph<> - </>)■


(5.5)

(5.6)

(5.7)

(5.8)

316 F. A. MILNER

First, note that (1.5), (1.8), (5.1), (5.2), and Theorem 3.1 imply

([k| + XSU, aip)V<?) < Ki\\t\\02 + t +||S||o,2 + E)ll¿llo,2 + E||V<í»||o,(2 + E,/E

< KiWP - PhPWo,2 + c + IM|o,2+e + ||u - vrAu||02 + E + ||o||0i2 + ï)

x(l|p-^Pllo.2 + E+llTllo.2 + E)llV<í»||i

< K"f Ar2~E/(2 + E)llr>ll + A-e/(2 + e)IItL^ JS-\n IIPII|.2-e/t2 + i),2 + £ T " llTl|0

+ AÊ/<2 + E'||u||r2_E/t2 + E),2 + E + A-E/<2 + E»||a||o)

x(a'-e/<2+e»||p||/_e/(2+e,2+e + A-e/<2+e'||t||o)||<í.||2

< Kh-2°"2+<x{hr>\\p\\r2+i + \\PhP - Pilo + lltllo + Iku - »Ho + HSllo)

x(*Wll + l|i>*/'-/'llo+llillo)ll+IU2^Kh-2</<2 + <Xh'Mr2 + lh%p\\,+sJ<p\\s + 2

= Kh^-2^^M,+ 8jp\\r2 + lUi + 2,

2 - 80k < / < k + 1,1 < r2 < k + 1,

([k£ + AÇ]I, v(/>)V<í> - a(p)V)

< Â-2E/<2+E>A^||Jp|L2 + 1A/||p||/+ÄoJ|^fl(p)v<i. - a(p)v<p\\0,(2+t)/t

<^/+ri-2£/(2+1lp||/+SJ|pl2 + 1A£/2||<i»IU2,

2 - 80k < / < k + 1,1 < r2 < k + 1,

(p¿2,^)<^||o,2 + EWo.(2 + E)A

<ifA-2e/(2+e>A'||/>||/+8MA^|bL2+SoJI*ll2

< A'A/ + r^2E/(2 + F)llr)ll llnll IIaII^ A" IIPII/+S0tIIPIIr2 + lll<f)IU + 2.

2 - V < / < k + 1, 1 < r2 < A: + 1,

(p£2, PA -<},)< ^|||||o,2 + £||PA<i» - |lo.(2 + E,/t

*Kh'+'*-2"*+»M,+*Jph+iht/2Uh

<Kh,+r>-2eA2+np\\,+SJp\l2+Ms+2,

2 - 80k ) +iy[P„p-p],4>)

(5-9) <^l|i>A^-/'ll-J-l(l|V*||I+1+Wi+1)

<KA'.+i+1||/>y|<f>|U2, OK^Kk+1.

Also, (1.5) and (1.8) give the estimates

(T[PA/> - p], nha(p)v4> - a(p)v<?) +(y[PhP - p], Ph<> - <#>)

(5-10) < K\\Php - /4(lk«(p)V - a(p)v*||0 + \\Ph<? - *||0)

< Khr'+s + l\\p\\rUl + 2, 0<r1<fc + l,



(IV, a(p)v<? - iThaÍP)V4>) +iyr,4> - Ph4>)

(5-11) < A-||T||0(||a(p)v<i» - TThaÍP)V4>Wo + H ~ Ph<t>Wo)

< KAí+1||t||0||<í>|U2.

Finally, (1.5), (1.8), and Theorem 3.1 imply that

(a(p)S,a(p)V - trhaip)v<?)

(5.12) < K\\t\\oMp)V4> - irAfl(p)V*||0 < »*+'+1||p|k + i|WU2,

1 < rx < k + 1,

(divS, <? - P„<?) < ̂ Idivtlloll* - PA<í>|lo

(5.13) íll/>L + ilk»l|í+2, l<r1<A: + 2,if5<*-l,<£A"+J+1

(ll/'IU + 2+,-J*IUi+*-i. 0<r1<Ar + l,ifi>fe-l.

If we combine now (5.4)-(5.13), we see that

(5.14) ML < K[h°+X\\T\\« + A'+'-1b||/+ioJ|/;||ri+1 + h^x\\p\\rx + x+u_k + xy]

for 2 - 80k < / < k + 1, 1 < rx, r2 < k + 1, where K contains a multiple of ||p||2 + E.

If we now take s = 0 in (5.14), we obtain the estimate

Ho < *[aIMIo + a^'-eM|/+sJ/42+1 + A'i+1|H|ri+1+8J,

which, for sufficiently small A, implies that

(5.15) Wo<ii:[A'+'-«|H|/+a0JI/'m+i + Ar, + 1IHrI + i+J.

Substituting (5.15) into (5.14) completes the proof of the lemma. Q.E.D.

We can now demonstrate the convergence of ph to p to be at an optimal rate in

L1(Q), 2 < q < oo (for k > 0 if q > 2/e).

Theorem 5.1. PAere are positive constants C and C (both containing a multiple of

\\p\\\ + t), independent of h, such that, if p g W'"(ü) n Hr+1+¿<"-+'^'r\Q), 1 < r

^ k + 1 - (e - 2/q)+80k, where t(q, r) = -(2/q) + (1 + (2/q) - r)+, then

(a) \\P-Ph\\0,q< Cqhr\\p\\r+x+Sok + ,(qr), q<cc,

(b) IIP-PAllo,oo<CAr[||^||r>00 + ||/,||r+1+8oJ.

Proof. Using Lemma 5.1 (with s = 0, / = 1 + e(l — S0A:), /"j + 2/fl = r2 + (2/fl)

- e50/t = r), (1.8), and the quasi-regularity of S~h, we obtain the bounds

(a) Hp-pJo.^llp-PfcPlk + IHIo,,,

<^[ÎIpIL,? + ä-('-2)/1t||0]

<C,[Al|/;||ri? + Alb|U1+iot+/(?ir)+],

(b) Up - pjlo.oo < Up - âpIIo.oo + IMIo,«

^[aihu + a-ItIIo]

<K[h'\\pl,ao + h-1+'+1\\p\\r+1+Sm]. Q.E.D.


318 F. A. MILNER

6. Error Estimates in Hs(ü)' and (PP(ß)2)'. We can also derive negative norm

error estimates from Lemma 5.1.

Theorem 6.1. There exists a constant C > 0 (containing a multiple of \\p\\2 + e),

independent of h, such that, ifp is sufficiently smooth, then for O^s^k + lwe have

the following estimates:

(0 \\p-Ph\L^C\h^^r¡Ms_k,xy+h^-Ml+sJpWr2 + l},

2 - 80k < / < k + 1,1 < rx, r2 < k + 1,

(ii) ||u - uju < c[Ar.-||p||ri+1+(J_,)++ A^+^y/+E+s>||r2+1+EÄoJ,

2 - 80k < / < k + 1,1 < rlt r2 < k + 1,

(m) ||div(u - uh)\\_s < c[a—|b||ri+2 + A^'-E||plLE+ä>ll2+1 + ESJ,

2 - 80k < / < k + 1,0 < rx,r2 < k + 1.

Proof, (i) follows directly from Lemma 5.1 and (1.9), since

£ = T +(PA/> - p), (||É||-*-i < ||t||_a + ||P„/> - p|U_0.

Rewrite (5.3) as

i(a(pH,v) = (divv,T)-(n,v)+([K| + X5]l,v), VGVA,

\(divC,w) = (-Y¿,w)+(p¿2,w), weWh.

Note that (iii) for s = 0 is just a consequence of (iii) of Theorem 3.1. Let then

tf g PP(ß), 1 < s < jfc + 1. Then, ^ g £fl/2(ß) with ||^||9/2 < K\\4>\\s. The second

equation of (6.1) and (1.8) imply that

(div£, +) = (divÇ, Phrp) +(divÇ, V - P^)

= -(7^,PA^)+(p|2,P^)+(divL^-P^)

= -iyt,*) + (y£> * - Pht) +{pt2,4>) +(pt2, Ph* - *)

+ (<Kv5, *-/»**)

< K|hfl|,[ll«IL + *'||€||o + Ml*+. + h'UÙ + MldivSHo]

since [1/(2 + s)] + [1/(2 + e)] + [1/(0/2)] = 1. Therefore, (iii) follows from (i), (a)

of Theorem 5.1, (iii) of Theorem 3.1, and interpolation [10] for 0 < 5 < 1.

Finally, assume that the coefficient a is smooth enough that for x e iP_1(ß)

there is a unique ) = x in ß. <f> = 0 on <^, Wltn

\\<t>\\s + i < if llxll.-i. 0 < s < A + 1. Let i|/ e HS(Q)2; let <f> e tfJ+1(ß) be the solu-tion of v(a(p)V<#>) = div xj/ in ß, <¡> = 0 on 3ß, and let x = 4* _ a(p)V<f>. Then

divx = 0, llxll, < ffll+H,, and ||*||,+1 < K\W\\,. It follows from (1.4), (1.5), (1.3),



(1.8), (1.10), (5.1), integration by parts, Theorem 3.1 and Theorem 5.1 that

(a(p)t,^) = (a(p)t,\) +(S,V*)

= -(divg,<í>) +(a(p)t, trhx) + (a(p)s\X~ "aX)

= -(divÇ, Ph4>) + (div£, Ph<f> - <¡>) +(«(p)Ç, x - «*x) + («(p)£, "aX)

= (y{, P„<í>) -(p¿2, PA</>) +(divg, PA - <p)+(a(p)l x - Âx)

-(r{,irAx)+(["f + *S]*.»*x)

= (y*,*) + (y€, Pa* - *) -(p¿2,<#>) -(p¿2, PA<f> - 4>)

+ (dwi ph<? - <¡>) +(a(p)s,x - »*x) ~(r€, x)

-(r{, ttaX - x) +([«€ + K)l, x) +([«€ + XJ]i, "aX - x)

<ÎW,[||íIU-i + a,+1IIíIIo + II€|Im + A'+1IIíIIm

+ Aí+1||divS||0 + AS||o + ||¿||-í + Al¿||o

+ (l|£||o,2 + £ + l|í||o,2 + JI|l|lo.(2 + e)A + *' (H^llo + l|£||o)||l||o,=o]

1,-H+E«0JIPII/+E + S(U + Äri+1pllr1 + l+<,-A) +<^WjA^ + '-'b|L. + 1+^.b|L.+..+A^+'ll

from which (ii) follows immediately. Q.E.D.

Department of Mathematics

University of Chicago

Chicago, Illinois 60637

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Documents

ll/ll*.,:o = E ll¿>7l|l' - Arizona State Universitymilner/PDF/Milner Math Comp 1985.pdf · april 1985. paces 303-320 Mixed Finite Element Methods for Quasilinear Second-Order