A unified analysis of a weighted least squares method for first-order systems

(1)

~£THZ~{AT[CS Abl]~

,C(ON ~<,]TAV[OM ELSEVIER Applied Mathematics and Computation 92 (1998) 9-27

A unified analysis of a weighted least squares

method for first-order systems 1

**Suh-Yuh Yang *, Jinn-Liang Liu 2**

Department (~f Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan

Abstract

A unified analysis of a weighted least squares finite element method (WLSFEM) for approximating solutions of a large class of first-order differential systems is proposed. The method exhibits several advantageous features. For example, the trial and test functions are not required to satisfy the boundary conditions. Its discretization results in symmetric and positive definite algebraic systems with condition number O(h 2 + u~). And a single piecewise polynomial finite element space may be used for all test and trial functions. Asymptotic convergence of the least squares approximations with suitable weights is established in a natural norm without requiring extra smoothness of the solutions. If, instead, the solutions are sufficiently regular, a priori error estimates can be derived under two suitable assumptions which are related respectively to the symmetric positive systems of Friedrichs and first-order Agmon-Douglis-Nirenberg (ADN) elliptic systems. Numerous model problems fit into these two important systems. Some se- lective examples are examined and verified in the unified framework. © 1998 Published by Elsevier Science Inc. All rights reserved.

Keywor&v Boundary value problems; First-order systems; Friedrichs" systems: ADN elliptic systems: Least squares methods; Convergence; Error estimates

* Corresponding author. E-mail: syyang@math.nctu.edu.tw.

i This work was supported by NSC-grant 85-2121-M-009-014, Taiwan, ROC. 2 E-mail: jinnliu@math.nctu.edu.tw.

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10 S.-E Yang, Z-L. Liu / AppL Math. Comput. 92 (1998) ~ 2 7

1. Introduction

The purpose of this paper is to give a unified analysis of a weighted least squares finite element method (WESFEM) applied to a large class of first-order differential systems. The Friedrichs symmetric positive systems [24] and the first-order Agmon-Douglis-Nirenberg (ADN) elliptic systems [l] are of particular interest in this general framework.

Although there has been considerable attention to the use of least squares principles in connection with finite element applications during the last decade, the modern theory of least squares finite element methods (LSFEMs) for the approximate solution of elliptic boundary value problems dates back at least from the work of Bramble and Schatz [7,8] in 1970. In Refs. [7,8], the approximate solution is defined to be the minimizer of a least squares functional over a finite-dimensional approximating function (trial function) space. The functional consists of a weighted sum of the residuals occurring in the differential equation and the boundary condition. This method has the feature that the trial and test functions are not required to satisfy the boundary condition. On the other hand, it requires that the trial and test functions are smooth enough to lie in the domain of the elliptic operator. For example, they must be in the space H2"(Q) for a 2mth-order problem. Thus, many seemingly natural finite elements are never admissible. However, this difficulty may be circumvented by introducing the derivatives of the unknown function as new dependent variables (in general, the combinations of these new dependent variables present certain physical meanings such as, fluxes, vorticity, and stresses, etc.), then the original higher order problem can be reformulated as a system of differential equations of first-order with possibly additional compatibility equations. Applying the least squares principles on this extended first-order system, the smoothness requirement on the trial and test function spaces can then be re- laxed, which eliminates the main disadvantage of this approach.

The least squares approach to boundary value problems of first-order systems represents a fairly general methodology that can produce a variety of al- gorithms. Thus, various LSFEMs appeared in the literature. Roughly speaking, according to the boundary treatment, these methods can be classified into the following two categories.

• The least squares functional involves only the residuals in the differential equations. In this case, the trial and test functions are required to fulfill the homogeneous boundary conditions and thus more than L 2 regularity, say

H I/2,

for the given boundary functions is necessary in the nonhomogeneous cases. See, e.g., Refs. [5,11,13,14,17 20,22,23,26,35].

• The least squares functional consists both of the residuals in the differential equations and the boundary conditions. The trial and test functions need not satisfy the boundary conditions. Hence, only L 2 boundary data is required whenever the problem is well posed. See, e.g., Refs. [2,4,12,15,16,25,37].

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S . - E Yang, J.-L. Liu / Appl. Math. Comput, 92 (1998) 9 27 I I Both types of least squares functionals can be combined with the weighting techniques to enhance the stability and accuracy of the approximate solution, even allowing different equations and boundary conditions equipped with different weights [2,5].

Motivated by the W L S F E M of Aziz and Liu [4], we generalize the method (of the second category) in a unified framework for both Friedrichs' and A D N systems. More specifically, the method is applied to the boundary value problems of first-order systems in the general form:

0u

L f u : = ~ . . , A i ~ x i + A o u = f inf2, (1)

i~l

. ~ u : = B u = g on0~2,

(2)

where f2 c ~ d d ~> 2, is a bounded domain with a smooth boundary 0f2, and u -- (ul . . . . ,urn) t, f -- (fl . . . . ,fro) t, g = ( g l , . . . ,g,)¢. In the sequel, we shall always assume that the entries of m × m matrices Ai E [L~(E2)] m×m, 0 <<. i <<. d, and of n × m boundary matrix B E ILk(Of2)] "×m are regular enough on ~ and 0f2, respectively, such that problem (1), (2) has a unique solution u E [H 1 (~c2)]m with the given functions f c [L2(f2)] ", g E IL2(Of2)] ".

LSFEMs offer many attractive features in practice when applied to boundary value problems formulated in first-order systems, we refer to the references mentioned in the above. We summarize our results as follows.

• With a minimum regularity of the (known or unknown) functions as posed in Eqs. (1) and (2), asymptotic convergence of the approximate solutions ob- tained by the W L S F E M is given for the general problem (1) and (2). • Under suitable assumptions ((19) and (20) in Section 4), an analysis of a pri-

ori estimates for both Friedrichs' and A D N systems is presented. In particular, the recent works on LSFEMs for the Stokes equations by Bochev and Gunzburger [5], Chang et al. [18,19], and Jiang and Chang [26] may be extended by using the W L S F E M . Consequently, the regularity requirement on the boundary conditions can be lessened and the trial and test functions are not required to satisfy the boundary conditions. However, it is not clear that the estimates are sharp under the general assumptions. If, in addition, stronger conditions such as, e.g., that of Ref. [37] are met, optimal convergence can be expected for certain systems.

• The condition number of the resulting system of linear equations is O(h -2 + w2), where h denotes the mesh parameter and w the weighting parameter.

• The framework is independent of the type of differential systems, i.e., it is for elliptic, parabolic, hyperbolic, or mixed type problems.

The remainder of the paper is organized as follows. Some notation and pre- liminary results will be introduced in Section 2. The W L S F E M is presented in Section 3 with its fundamental properties and asymptotic convergence result.

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12 S.- E Yang, J.-L Liu / AppL Math. Comput. 92 (1998) 9 27

A priori estimates are derived in Section 4 under the two general assumptions. Two model problems, namely, the neutron transport equation and the Stokes equations cast in the framework of Eqs. (1) and (2) are examined in Section 5 to validate the assumptions. An estimate for the condition number of the resulting symmetric positive definite matrix is given in Section 6. Finally, some concluding remarks are drawn in Section 7.

2. Notation and preliminaries

Throughout this paper, we shall require some function spaces defined on f2 and 092 [33]. The classical Sobolev spaces H+(~2), s ~> 0 integer, and L2(0f2) with their associated inner products (., .)~..~, (., ")0,0e and norms ]].

[IL,'.Q'

11' II0.0e are employed. As usual,

L2(~):=H°(~).

For the Cartesian product spaces [H~(~)] m and [L2(0(2)] ", the corresponding inner products and norms are also denoted by (., ).,.a, ( ,

)o,00

and l l ]1.+.~, [1 ll0.o~2, respectively, when there is no chance for confusion,

By L ~(~2) and L ~(0~2) we denote the usual Banach spaces of measurable and essentially bounded real-valued functions defined on ~2 and 0 ~ with the norms II'

[l~.a

and 11.

I]~.0~2,

respectively,

Since the boundary 0 ~ of the bounded domain f2 is smooth, there exists an operator 70 : H~ (~2) ---+ L2(0f~), linear and continuous, such that

70v = restriction of v on 0• for every v c C 1 (~).

The space 70(H j (Q)) is not the whole space L2(0(2), it is denoted by

HI/2(0(2)

and define its norm by

II~IP~/2,oQ

= inf{llvll,,n; v ~ H ' ( n ) , 70v = ~},

which makes it a Hilbert space. Also, the associated norm of the product space

[HI/2(Of2)] "

is still denoted by PI " ]ll/2,0Q.

Define the following bilinear form and linear form: for any v, w c [H 1 ((2)] m,

a+(v, w) = ( d v , ~w)0~, + w(.~v, :~w)0 o+,, (3)

g,~(v) = (f, Sv)0.Q + w(g, .Sv)0 o,2 , (4)

where w is a positive weight maybe depending on the mesh parameter h which will be introduced later. It is easily seen that, for each weight w, a,,.(., .) defines an inner product on the space [H l (Q)}m × [H l (~2)lm, and the reduced norm shall be given by

Ilvll~. = a+(~, v) Vv ~ Et/' (~)]". (5)

Note that the homogeneous property of ]] - H~ is ensured by the fact that problem (1) and (2) possesses a unique solution in [Hl((~)] " for given functions f E [L2(g2)] ~, g E

[L2(On)] ".

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S.-Y. Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9 27 13 To approximate (1) and (2), we consider a regular family [21] o f triangulations {,Y-h : 0 < h < 1 } of O, where the parameter h measures the mesh size o f each discretization. F o r each h, define the finite element space ~F'h,p C [H ~ (~2)] ~, p ~> 1 integer, which is assumed to possess the following approximation property: for any v • [H p+I ((2)] m there exists Vh.p • ~°h.p such that

IIv - v,,,,~ll,,.~ ÷ hllv

-

v~.,,ll,.o ~ C,h'~+' Ilvll,,+,.~,

(6)

where C~ is a positive constant independent of v and h.

Using the following lemma, further approximation properties of the finite element space ~"h.p can be deduced•

Lemma 2.1. There is a positive constant C2 such that, ,['or any v • [m I (Q)]m and

a n y ~; ~ O,

(

1 )

I/vllo.,,~ ~< c2 ~llvll,.~ ÷-Ilvllo.~, _g;, • (7)

A p r o o f of Lemma 2.1 can be found in, for example, Refs. [9,33]• With (6) and (7), we immediately have:

Lemma 2.2. Let v • [Hp+I ( (2)] m, p ~ 1 integer. Then there exists Vh,p • :t h.p such that, Jor an)' e > O,

( l hp+l )

I1 v - "~,,,11,,.~ ~< C3 ~ h " + - Ilvll,,+,,~, (8)

where C3 is a positive constant independent o f v, ~, and h.

Lemma 2.3. Let v e [H p+I (~r~)]m P ~ 1 integer. Then there exists Vh,p • ~th.p such that

II v - Vh.,,II,,, ~< C4(hP+ whP+')ilvlkp.,,~ (9) Jbr some positive constant C4 independent o f v, w, and h.

Proof. Let Vh,p be the same as in (8) with e, = 1/xfw, by (5) and (3), IIv -

v~,,H.,, <~

I I S ( v - v ~ p ) l l 0 ~ + v ~ l l . ~ ( v - v~,,)ll0~,~

~< c5(11., - v,,.ll,.o + v%llv - v~.llo.,),P

C6(hP + (h p -I- wh"+'))llvll,,~,.,~, (lO)

where the second inequality is ensured by the fact that cj~ is a first-order differential operator and Ai • [L~(Y2)] m×m, O<~ i<~d, B • [L~(0(2)] ~×~. This completes the proof. []

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14 S . - Y . Yang, J.-L. L i u / Appl. M a t h . Comput. 92 ( 1 9 9 8 ) 9 - 2 7

3. Weighted least squares approximations

Define a weighted least squares functional J : [H i (f2)] m ~ ~ as

j ( v ) - - II ev -

fl[2o,~ +

wll v - gll0,o , 2 ( l l )

where w is the same parameter of (3) and (4). Evidently, the exact solution u E [H ~ (f2)] m of problem (1) and (2) minimizes the functional and vice versa, i.e.,

J ( u ) : min j ( v ) . (12)

v~[H~(Q)]"

Taking the first variation, the solution equivalently satisfies the equation

aw(U,V) = gw(V) Vv

• [Hl(~r~)] m,

(13)

where the bilinear form and linear form are given in (3) and (4), respectively. The W L S F E M for problem (1) and (2) is then to find UhW • ¢~h,p such that

a,.(u~'p, Vh,p) = gw(Vh,p) VVh,p • Uh,p. (14)

Note that the trial and test functions are not required to satisfy the boundary conditions in the approximation.

We first have the following results concerning existence, uniqueness, stability estimates, and some important properties of the approximate solution.

Theorem

3.1. Let u be the exact solution of problem ( 1 ) a n d (2)

f E [L2((2)] m, g E [L2(0(2)] ".

with

(i) Problem (14) has a unique solution u~,p E ~/~h,p for each given positive weight w, and the solution satisfies the following stability estimate:

Ilu~,llo,,

~< Ilfllo,Q

+ v~[Igllo,o~.

(15)

(ii) The matrix of the linear algebraic system associated with problem (14) is symmetric and positive definite.

(iii) The following orthogonality relation holds:

a w ( u - u w v ~ = 0 h,p ~ h,p ] VVh,pC~hp , . (16)

(iv) The approximate solution UhWp is a best approximation of u in the II " [la~,- norm, that is,

[lu-

u~"p[[ .... = inf I l u - Vh,pl[a,. (17)

Yh.pE I h.p

Proof.

To prove the unique solvability, it suffices to prove the uniqueness of

solution since the finite dimensionality of ~h,p. Let uhWp be a solution of

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S . - E Yang, J.-L. L i u / Appl. Math. Comput. 92 (1998) 9 27 15 w 2 a ( u w U w ~ Iluh.pL[,,+ = w~ h,p', h,pl = (f,L-~ w , _{uh,p)o. Q + w(g, ~Uh.p)O,O~}+,. f w ~ w f w ]Aft w ~< I1 II0.~llu~,plk, + v~llglt0,o~llu~.vll.+.

Thus, we obtain (15). Consequently, the solution u)ilp o f problem (14) is unique.

Part (ii) follows from the fact that the bilinear form uw(., .) is symmetric and positive definite.

Part (iii) follows easily from Eqs. (13) and (14), since ~' ),.v is a subspace o f

[H'

(~2)]".

Finally, to prove (iv), by (16) and the C a u c h y - S c h w a r z inequality, I1" - %,11 .... + , 2 - - "~,1 ( u I ~ h ,,,, u I + Uh ,p ) +,

&/w( u w U

--: __ Uh,p~ Vh,p)

~< II. - .LII.,, II. - vh+,ll .... for all vt,.v c ~ )w. This completes the proof. []

As a consequence o f part (iv) in T h e o r e m 3.1, we have the following asymptotic convergence.

T h e o r e m 3.2. Suppose that the positive weight w in the least squares

approximation Eq. (14) is a constant or a bounded mesh-dependent fzmction in

h E (0, 1). Then we have

liml]u - u L I < -- 0. (18)

h ~ 0

P r o o f . W i t h o u t loss o f generality, let constant C7 > 0 represent an upper b o u n d

o f x/~ on (0, 1). Let ~ ( Q ) denote the linear space o f infinitely differentiable functions on ~2 such that all the derivatives have continuous extensions to 0Q. Since [ ~ ( ~ ) ] " is dense in [H l ((2)] 'n with respect to the ]]. Ill.Q-norm+ for any c > 0, there exists u * E [~(P)I m independent o f h such that

I[u - u+ll~ +~ <

2C5(1 + 2C2C7) ' which implies (cf. (10) and (7))

C, Ilu - u+ll.~ ~< c 5 ( 1 + 2 c 2 c v ) l l u - u+ll,.~ < ~.

F o r this fixed s m o o t h function u* E [~y(~)]m by (6), we can find Hh,pU* E ~ h,v so that

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16 S.-K Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27

which implies, for sufficiently small h,

g,

Ilu* - n~,t,u*ll~,,, ~<c~(1 + 2c~_c7)11u* - r / j * H , . ~ < ~ .

By Eq. (17), we immediately obtain

0~< Ilu - u',i',,ll,,, ~< Ilu - r/,,~u*llo,, ~< Ilu - u ' L , , + Ilu* - r / ~ u * l l ~ , , < ~.

This completes the proof. []

4. E r r o r e s t i m a t e s

Theorem 3.2 indicates that, for example w = const, the approximate solution

u})'p

satisfies the differential equations and the boundary conditions asymp- totically in the I1" ]10.e -n°rm and the [1. I]0.oe-norm, respectively, without assuming additional regularity assumption on u, that is,

I I ~ u ~ - fll0,~ - ~ 0 as h - ~ 0, II/~u2,t, - gll0.e,.~ - ~ 0 as h ~ 0.

Of course, one may expect better convergence properties for the approximation provided that the exact solution is sufficiently regular and that the system (1) and (2) satisfies certain coercivity conditions. In fact, these conditions associated with some specific numerical methods are often circumstantial and hence somewhat restrictive to a wider class of problems. For example, the LSFEMs of the references cited in the second category in Section 1 are similar in princi- ple and yet quite different in terms of the coercivity conditions or some related approximation assumptions. On the other hand, an attempt to create a univer- sal conditions for the general system (1) and (2) in the context of LSFE approximation is very intractable if not impossible. Nevertheless, with W L S F E M (14), we classify the conditions for the Friedrichs and A D N systems by the following two respective assumptions.

(H1)

There exists a constant Cs > 0 such that:

Ilvllo,~ ~< Cs(ll~vllo.~ + II,~vllo.~) Vv ~ [HI(~r~)] m. (19) (H2) There exists a constant C9 > 0 such that:

Ilvll, ~ ~< C+(llSvll0~ + [I.~vll~/2,0~) Vv c

[H~(~)] ".

(20) Associated with the assumption (H2), we also need the following inverse assumption [21] on the finite element space ~'h,p: there exists a constant G0 > 0 such that

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S.-Y. Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9 27

17

11 ~vh,~

II ~/2o~ ~<

C~oh-'/2

II~vh,~/10.o4 Vvh,~

C ~h,p.

(21 )

This type of assumption is commonly used in W L S F E M s (see, e.g., Refs. [2,15,25,36,37]) and holds for a large class of finite element spaces Y'~h,p. More precisely, if the family {¢-h } of triangulations of O is quasi-uniform [21,27], i.e., there exists a positive constant v independent of h such that

h ~< vdiam(f2~) VO~ E-Y-h, .Y-h E {3-h}, (22)

then the inverse estimates (21) are satisfied.

We now state the main results for the approximate solution u w _{tl .p "}

Theorem 4.1. Suppose that the exact solution u o f problem (1) and (2) belongs to

[H p+I (f2)] m. Then there exists a constant C > 0 independent o f u, w, and h such that

U w I

Ilu - ,,~ ,,,, ~<

C{hP +

whP+'}llull,,~,.~,

C_~uW

_~,,

_{- fll0~ ~<}

_{C{hP + wh ~+' }}

_{Ilull~+, ~,}

r h , '

}

II.~uL, - glloo~ ~< c ~

+ v ~ h ~+~ Ilull~+, ~.

I f in addition, (HI) holds then

[[u-u)[pl,0,~ ~< C { (1 q - ~ w ) h P q - ( w +

V/w)h p+I },[u[[p+l.~.

~.

I f (H2) and (21) hold with 1/w = O(h), then

Ilu - u ~'h,~ I ,Q -.~-< C{ (1 +

v%)hP+w3/2hp+'}l[ullp+,. ~.

(23)

(24)

(25)

(26) (27) Proof. Let vh,p E ~t'h,p such that (9) hold with v replaced by u. Then, by (17), we get (23) immediately. By the definitions (5) and (3), we obtain

u w i/2 w

.

Uh,p)

IIo.Q + w l l ~ ( u

Uh~)

II;.o~" Ilu h,, ... I I ~ ( u w 2

Then estimates (24) and (25) follow easily from (23).

The estimate (26) is an immediate consequence of (HI), (24) and (25). To prove (27), assume that assumptions (H2) and (21) hold with

1/w = O(h). Then we obtain, for any Vh,p ¢ "l%.p C [H 1 (f2)] m,

Ilvh,,,ll[,~ ~< C~(ll~ev~,,llo, +

II~vhpll,/2.~,o

<~ c~(ll sv,,.,~llo.~ + h -~ ~

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18 S.-]~ Yang, J.-L Liu / Appl. Math. Comput. 92 (1998) 9-27

A p p l y i n g inequality (28) to u~i p -vh,p c ;tJ'h,p and using (16), we have

w - - V U w

a"' - v~,~lll.~ ~< cl3a~ h,p Uh, p h,p ~ h.p - - Vh,p

U w __ V h , p )

Cl3aw ( u - Vh.p~ h,p

V ~

~<

c.4{Jl. - ,,,,ll,~lt.,,p - v~ll, ~,

A p p l y i n g L e m m a 2.1 to Uu - Vh.pllo,o~2 a n d [lu~i¢, - Vh.pll0o~ with e = l / v @ and e = 1, respectively, we get

uW

h.t, - v~,t, ll..~ -< c,4

~

I

Ilu - v,,.,,ll,.~,llu~i,, - v~,,,ll,.~

+ c: ( liu

-

+ .,'J21tu-v .,ilo, )

+ H<,

Hence,

Ilu£1,,- v,,,~ll,.Q -<

C15{

Ilu - v,,,,,ll,..~ + v%llu - v~.,ll,.o + w3/~llu - v~.,,ll,,.o }. Using the approximation property (6), we can choose Vh.p E "~"h4, SO that

Ilu - v,,,,,ll0.Q + hllu - v~.,,ll,.~ ,< C,h~-' Itull,,+,.~.

Then, by the triangle inequality,

uW i w

Ilu - ~.,, ,.~ -< Ilu - vh.~ll,.~ + IIv~.p - u~.~lt,.:,

-< Gh"llull,,+ ,.~ +

c,s{ClhellUjjp~,.~+Civ@h'liull,,T,.a

-~-

CI W3/2hp÷l

Ilull~+l,~,}

~< C { ( I + x/w)h t' + u,'3/2hP+l

}llullp+l.O.

T h e p r o o f is complete. []

Corollary

4.2. Under the s a m e a s s u m p t i o n s as in T h e o r e m 4.1, /f we t a k e w = const or w = h I then the error e s t i m a t e s (23)-(26) b e c o m e respectively as

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S.-Y. Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27

Ilu - u~',,llo., ~< Ch"llult,,+,,o, ll~u,~'p - fllo.~ ~< ChPllul[,,+~,~,

w Ch p+k u II~uh,~ - gll0,o~ <~ ,, ,,~+,,~, Nu - u~',,l[o,~ ~<

Ch~llull~+~,~,

19 (23') (24') (25') (26') w h e r e k = 0 i f w = constant, a n d k = ½ i f w = h -I . M o r e o v e r , i f w = h i then ( 2 7 ) b e c o m e s as

Ilu - u~pll

1,Q ~ ChP-(1/2)Ilull~+l,~.

(27')

R e m a r k 4.3. Evidently, the error estimates (26) and (27) are not optimal. It is unclear if these estimates are sharp under the above assumptions. For certain systems, it is possible to achieve optimal convergence with stronger conditions (see, e.g., Section 8.4 in Ref. [37]).

5. First-order s y s t e m s

There are many important first-order problems such as the Friedrichs and A D N systems that satisfy (H1) or (H2).

5.1. Friedrichs' symmetric' positive s y s t e m s

In Ref. [24], Friedrichs introduced the notion of symmetric positive linear differential equations independent of type. The criterion of symmetric- positive- ness has many advantageous features. For example, suitable boundary conditions can always be determined and equations of different types are treated in a unified way. In particular, the Friedrichs theory has been shown to be a very useful tool in the theoretical analysis for mixed type PDEs such as the Tricomi equation and the forward-backward heat equation that are cast into equivalent first-order systems. For the details, we refer to Refs, [3,4,24,28-31].

Consider the following system of differential equations of first-order which is a special form of problem (1) and (2),

d OU

LPU : : ~ _ A i ~ x i + A0U = f in (2, (29)

i : 1

.~u := (/~ - ]7)u = 0 o n 01"2, (30)

d A

where/7 = ~i=l ni i, the n~, 1 ~< i ~< d, being the components of the unit outer normal vector n on 0(2, # is a given continuous m x m matrix defined along 0(2. The differential operator L/~ in (29) and the boundary conditions (30) are

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20 S.-E Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27

symmetric positive and admissible, respectively, in the following sense. The op- e r a t o r 50 is symmetric positive if

1. the m x m matrices Ai, 1 ~< i ~< d, are symmetric on (2,

2. M = Ao + A~ - ~ d OA~/Ox~ >~ c J inEL

where c~ is a positive constant and 1 denotes the m x m identity matrix. The b o u n d a r y conditions Eq. (30) is admissible if

3. tt + / -It ~ 0 . . . . on 0Q,

4. K e r ( p - fi) @ Ker(ll + fl) = ~ " on 0~2.

T o verify the assumption (H1), we use the well-known (second) identity o f Friedrichs [24]

2(v, Lfv)0,~ + (v,,Sv)0.a ~ = (v, Mv)0.~ + (v, l,v)0.0~2 Vv E [HI(Q)] m. (31) Assume further that tt +/~t ~> c2I on 0EL c2 > 0 constant (see also T h e o r e m 2.1

in Ref. [28]). Then, by the C a u c h y - S c h w a r z inequality and the basic inequality,

ab <<. (e2a2/2) + (b2/2~2), for all real numbers a, b, and ~, > 0, we get

c, Ilvllo.~

+

~c2llvll0.o~ ~< 21rvllo.,,llSvllo.,

+

Ilvllo.o~ll~vllo.o~

for any positive constants ~1, ~,2. Choosing ~1, e2 such that C18 := cl - 1 / ~ > 0 and 1 / 2 ~ = c2/2, we thus have

which illustrates (H 1).

Example 5.1 (The neutron transport equation). Let d = (1, 1) t c ~2. Consider

the following n e u t r o n t r a n s p o r t e q u a t i o n in plane that no neutrons are entering the system from outside,

V u . d + u = f i n ( 2 : = ( 0 , l) x (0,1), (32) u = 0 on 0~2 ,

where 0~2_ is the inflow b o u n d a r y defined by

o ~ = {x c o ~ : n ( x ) . d < 0}

= {(0,y)t: y E (0, 1)} U {(X,0)t : X E (0, 1)},

n(x) being the o u t w a r d unit n o r m a l vector to 0 ~ at the point x E Or2. Then p r o b l e m (32) is a simple symmetric positive system with m = 1, A1 = A 2 = A 0 = 1, f i = n ( x ) . d and t t = i f l [ = 1 > 0 . Thus, the assumption (H I) is fulfilled.

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S.-Y. Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27 21 5.2. First-order A D N elliptic systems

Another interesting class of differential systems are the first-order A D N elliptic systems. In Ref. [1], the ellipticity of the general system of partial differential equations is determined by three ordered sets of integral indices {Si} = (SI,... ,Sm) , s i l O , {tj} = ( h , . . . ,tm), tj >1 0, and {rk} = ( r l , . . . ,r,) cor-

responding respectively to differential equations, unknown functions, and boundary conditions. Based on the A D N theory, the operators Lf, ~ appear- ing in (1) and (2) must satisfy the so-called uniform ellipticity condition, sup- plementary condition and the complementing boundary condition in order to have the following coercive type estimates.

For each l/> 0, there exists a constant C20 > 0 such that if v = (Vl,..., Vm) t, /)j E Ht+t~(O), j = 1,... ,m, then

llvjll,+,,, c20

+

II( v)klt/_r

,

(33)

j=I k=l

where 5¢v = ((~/~¥)1,''', (~¥)m) t' '~¥ = ( ( ~ ¥ ) 1 ' " " ' ' (~¥)n) t"

We shall not state these conditions here. To fulfill these three conditions in turn leads to a rather complicated algebraic checking on the three ordered sets {s~}, {tj}, and {rk}.

Recently, Bochev and Gunzburger [5], Chang et al. [18,19], and Jiang and Chang [26] have successfully formulated the Stokes equations into first-order systems in two- or three-dimensional bounded regions and then proved that, under appropriate formulation (see the next two examples), (33) is satisfied with

{s,} = ( 0 , . . . , 0 ) ,

{tj} = (1 . . . . ,1), (34)

{rk}= ( - 1 , . . . , - 1 ) .

Consequently, we have (H2) by taking l = 0.

Example 5.2

(The Stokes equations in the velocity-vorticity-pressure formula- tion). Let Q C ~2 be a bounded domain with smooth boundary 0fL The Stokes equations for incompressible flow can be expressed as

- A u + g r a d p = f in (2,

d i v u = 0 in [2, (35)

where u = (ul, u2) t denotes the velocity, p the pressure, and f = 0q,f2) t the body force. By introducing the vorticity o3 := curlu = Ou2/Ox - Oul/Oy as an auxiliary variable and utilizing another two-dimensional curl operator curl~o = (e~y,-~ox) t, (35) can be transformed into the following first-order system in velocity-vorticity-pressure form

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22 S.-Y Yang, £-L. Liu I Appl. Math. Comput. 92 (1998) 9 ~ 7 c u r l ~ + g r a d p = f in (2,

- ~ o + c u r l u = O in (2,

div u = 0 in (2. (36)

If system (36) is supplemented with the following boundary conditions,

p = 0 on 0(2,

ulnl + u2n2 = 0 on 092, (37)

then Eqs. (36) and (37) is an A D N elliptic system and (33) holds with (34) (cf. Ref. [5]). Unfortunately, if system (36) is imposed by the homogeneous velocity boundary conditions,

u l = 0 on 0~2,

u 2 = 0 on 0(2 (38)

with (p, 1)0,a = 0, (34) will not hold, i.e., the assumption (H2) fails to hold for this problem. However, the following formulation works well for this type of boundary conditions which are more useful.

Example

5.3 (The Stokes equations in the velocity-stress-pressure formulation).

In Ref. [19], the velocity-stress-pressure formulation for the two-dimensional Stokes equations is proposed as follows:

Oq) 1 0(t9 2 Op Ox Oy

+ ~ = J ' "~

inf2, 0(/91 0@3 Op Oy OX + ~y = f2 in(2, --&-x +-b-f-Y&°J &°3 = 0 in (2, (39) &°i & ° 2 - 0 in (2, Oy Ox d i v u = 0 in (2, c u r l u - o 3 + ~ p 2 = 0 in (2

with (p, l)o,a = 0, where the auxiliary variables ¢Pl, q~2, and ~P3 are introduced as

Oul in ~, ~ol- Ox Oul in (2, (40) ~ 2 - Oy Ou2 in (2, ~°3-- Ox

and their combinations represent the usual stresses. If system (39) is supplemented with the boundary conditions

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X-Y. Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27 23 n l ( P 2 - - nzcPl = 0 o n OQ,

n l q ~ l + n 2 ¢ P 3 : 0 o n 0~2,

(41)

nlul + n2u2 : 0 o n 00,

which are equivalent to (38), then it is an A D N elliptic system and (33) holds with (34).

Many other boundary value problems can also be proved to have the estimates (33) with (34) by using the A D N theory. For the details, we refer to Refs. [1,2,13,14,16].

6. Condition number

In this section, we analyze the asymptotic conditioning of the linear system arising from problem (14). Let {u~,..., uK} be a set of basis functions for the finite element space ~%,p and we assume the basis is chosen so that the following two conditions hold [2,6,25].

There exist positive constants A~ and A 2 such that for all ~ . . . . , ~K E ~, A , h a Z ~ ~ <~ ~iUi, Z ~ j u j <~ A 2 h a Z ~ , (42)

i=l \ i=l ]=1 ,It 0,I2 i=1

¢iUi~ ~jUj ~ A2hd-2 g~. .2 (43)

"= IZ2 i=1

Note that the above inequalities hold for most finite element spaces ~/%,p if condition (22) is satisfied.

Theorem 6.1. Suppose that the basis conditions (42) and (43) are satisfied. I f (H1) hold.~ with w >~ 1, or (H2) and (21) hold with 1/w = O(h), then the condition number of the resulting linear system of (14) is O(h -2 + w2).

Proof. Since the matrix

M : : (Mij)Kx = (aw/ui,uj)) xK

is symmetric and positive definite, we find that 2~ax max p(E) condition number of M - 7 -

Zmin min p(E) ' (44)

where 2max and '~min are the largest and smallest eigenvalues of M, p(E) is the Rayleigh quotient,

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24 S.-E Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9 ~ 7

D ( E ) : = Eta, -- Et, ~,

for any E = ( ~ l , . . . , ~ x ) t C N~, E # 0.

Let Vh,p = ~ixl ~iu~. Suppose (H1) holds with w ~> 1, then, by (42),

Ajhd Z 2

i=1 j=l 0,Q .<

c~ (ir~evh~llo~+ll~v~pfloo~) 2

Or 2 ~- C21 (H~Vh,p]]:.(~ ~-]]~¥h,p]]o,oQ)

= Czlaw(Vh,p,

vh,p).

If (H2) and (21) hold with

1/w

= O(h), by (42) and (28), we have

Athd Z 2

~i <~

~iu,,

~juj

= [Ivh,~ll~,~ ~< IIv~,pll2~.~

i=1 0,Q

C13aw(V,~.p, Vh,p).

On the other hand, by (3), we obtain (cf. (10))

aw(Vh,p, ¥h,p) = (~Q-Q~V,~,p, ~v,~,p)0,~ + W(~Vh,p, ~Vh,p)o,O~ 2

By Lemma 2.1, we get

IIv~,AIo,o~ ~< 2c= ~ £11v~,~ll~,~ + 1

j IIv~,pllo,~

Taking e 2 = 1/w in (49), then (48) becomes

aw(Vh,p,

Vh,p)~< C23 (]]Vh,pH~,~ + w ( 1 ,]Vh,pl,~,e +

w][Vh,p]]:,e))

2 2 2

4Grv plll o + iiv lt0o)

K 2 d 2

<.

C=4A:(h ~ ~ + w h )

~

i=1

The proof is completed by (44)-(47), and (50). []

(45) (46) (47) (48) (49)

(5o)

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S.-E Yang, J.-L. Liu / Appl. Math. Comput. 92 (1998) 9-27 25

7. Concluding remarks

A unified analysis of a weighted least squares finite element method applied to a general class of first-order differential systems is presented. The method is based on the minimization of a least squares functional that is a sum of the residuals in the differential equations and the residuals with the same weight in the boundary conditions. Compared with other LSFEMs, the most significant feature of the method is that the trial and test functions need not satisfy the boundary conditions. Consequently, it applies to a broad scope of problems with only L 2 regularity required on the boundary data.

Asymptotic convergence is established in a natural norm without any extra regularity conditions on the exact solution. Many mathematical model problems fit into this general framework. In particular, we present two types of assumptions which are respectively suitable for Friedrichs' symmetric positive systems and for first-order Agmon-Douglis-Nirenberg elliptic systems. Under these assumptions, more specific convergence properties can be analyzed. The resulting linear system is symmetric positive definite with condition number O ( h - 2 +

W2).

Three examples, namely, the neutron transport equation and two first-order formulations for the Stokes equations with various boundary conditions are examined.

It is evident that the least squares approximation involves more degrees of freedom in the solution procedure since there are more unknowns to be determined at each nodal point and more equations to be approximated under the reduced first-order system. Nevertheless, with its advantageous properties such as symmetric positive definiteness and uniform finite element spectral order, this drawback may be alleviated via effective and efficient adaptive process [32,34] and/or parallel implementation.

Acknowledgements

We would like to thank Professor C.L. Chang (Department of Mathematics, Cleveland State University, Cleveland, Ohio) for his helpful comments and suggestions on the paper when he was visiting our department in September 1996. The second author would like to express his gratitude to the Department of Mathematics, Texas A & M University for a stimulating and enjoyable visit during which part of this work was undertaken.

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