JOURNAL OF
COMPUTATIONAL AND APPUED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 87 (1997) 3 9 ~ 0
Least-squares finite element methods for the elasticity problem 1
Suh-Yuh Yang, Jinn-Liang
Liu*Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan Received 21 January 1997; received in revised form 20 August 1997
Abstract
A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the HI-norm and in the if-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.
Keywords: Elasticity; Poisson ratios; Elliptic systems; Least squares; Finite elements; Convergence; Error estimates AMS classification: 65N12; 65N30; 73V05
1. Introduction
Over the past decade, increasing attention has been drawn to the use of least-squares principles in connection with finite element applications in the field of computational fluid dynamics (see, e.g., [4, 9, 10, 19, 22-26, 36], etc.). In this paper, we attempt to apply the methodology to develop two least- squares finite element methods for approximating the solution to the following two-dimensional linear
* Corresponding author. E-mail: jinnliu@math.nctu.edu.tw.
t This work was supported by NSC-Grant 85-2121-M-009-014, Taiwan. 0377-0427/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 7 - 0 4 2 7 ( 9 7 ) 0 0 1 7 4 - X
40 S.-Y. Yan9, J.-L. Liu/Journal of Computational and Applied Mathematics 87 (1997) 39-60 elasticity problem [13, 27]
-ItAu -
(2 +p)V'(!7, u) = f
u = 0 on Fo, 2 Z aiJ(u)nJ = gi j-1with the following notation:
in f2, (1.1)
(1.2)
on El, i = 1 , 2 (1.3)
• f2 C ~2 is a bounded domain representing the region occupied by an elastic body.
• ¢3f2 -- F0 tA F~ is the smooth boundary of f2 partitioned into two disjoint parts F0 and F~ with the measure o f F0 being strictly positive.
• #, 2 are the Lam6 coefficients where E
p - - - > 0 2(1 + v)
with v the Poisson ratio, 0 < v < 0.5, and E the Young modulus and
Ev
2 = >0.
(1 + v)(1 - 2v)
The upper limit o f the Poisson ratio, v ~ 0.5-, corresponds to an incompressible material. • u - - ( u l , u 2 ) T is the displacement vector field.
• f = (f~, f2)T is the density of a body force acting on the body. • g - - ( g l , g 2 ) T is the density of a surface force acting o n F 1. • n = (nl,nz) T is the outward unit normal vector to ¢3f2. • oij(u) are the stresses defined by
aij(U)=a:i(U)=2 ( ~ Skk(U)) rij +
l <~i, j<~2.
• e~i(U ) are the strains with
(Oju~ + Oeuj), 1 <.i, j<.2.
c , / ( u ) = c ~ i ( u ) =
• 6ij
is the Kronecker symbol so that 6ij = 0 ifi ¢ j ,
and,5ij
= 1 when i = j .In the analysis o f structural mechanics, the knowledge of the stresses
a,j
(strains cij) is often of greater interest than the knowledge of the displacementsui.
It is well known that the approximation o f the stresses can be recovered from the displacements by postprocessing in the standard finite element formulation for solving problem (1.1)-(1.3). From a numerical point o f view, however, their computation requires the derivatives o f the displacement field u which implies a loss o f precision. Thus, the most widely used approach for obtaining a better approximation of the stresses is based on the mixed finite element formulation which allows the stresses as new variables along with the primary variables (see [13] and many references therein). Consequently, the accurate stresses can be obtained directly from the discretized problem. Unfortunately, the approximation spaces in the mixed method must be required to satisfy the inf-sup condition which precludes the application o f many seemingly natural finite elements.S.-Y. Yang, J.-L, Liu l Journal of Computational and Applied Mathematics 87 (1997) 39~50 41 We provide herein an alternate way to avoid these difficulties by exploiting the least-squares prin- ciples on a new first-order system formulation of the elasticity problem. Introducing additional vari- ables of derivatives of the displacements, whose combinations represent the usual stresses (strains), the original system of second-order Eqs. (1.1) can be recast as an equivalent first-order square system in 6 equations with 6 unknowns, which is called the displacement-stress formulation here. The new formulation is very different from the standard one which is extensively studied in the mixed finite element method (see, e.g., [3, 13, 33, 34, 40, 41], etc.). We show that the first-order formulation is an elliptic system in the sense of Petrovski, and that, with the displacement boundary conditions, it satisfies the Lopatinski condition [43]. As a result, the problem can then be solved by using least-squares finite element methods (LSFEMs).
The least-squares approach represents a fairly general methodology that can produce a variety of algorithms. In this paper, we shall consider two LSFEMs. According to the boundary treatment, the first method is based on the minimization of a least-squares functional that involves only the sum of the squared L2-norms of the residuals in the differential equations. In this case, the trial and test functions are required to fulfill the boundary conditions. We refer it as the standard least-squares
finite element method (SLSFEM) (cf. [9, 14, 19, 20, 22, 24-26, 31, 32, 36, 39], etc.). The other is based on the minimization of a least-squares functional which consists of the sum of the squared L2-norms of the residuals both in the differential equations and the boundary conditions with the same weight h -1, where h is the mesh parameter. This method will be referred as the weighted
least-squares finite element method (WLSFEM) (cf. [4, 5, 11, 21, 43], etc.).
Recently, Cai, Manteuffel, and McCormick and their coworkers have developed a series of first- order systems least-squares (FOSLS) for the general second-order elliptic scalar equations [14, 16], the Stokes equations [15, 17], and the linear elasticity equations [15, 17, 18]. They have pointed out that one of the benefits of least-squares approach is the freedom to incorporate additional equations and impose additional boundary conditions as long as the system is consistent. Instead of applying Agmon-Douglis-Nirenberg (ADN) [1] theory, which is restricted to square systems, they use more direct tools of analysis for their overdetermined FOSLS (see also [31]). For example, the FOSLS of [17] for the elasticity equations with the pure displacement (homogeneous) boundary conditions, namely, (1.1) and (1.2) with /'1 = ~, consists of 11 equations and 7 unknowns. They prove that the FOSLS is uniformly coercive in the Poisson ratio in an Hi-norm appropriately scaled by the Lam6 constants. These FOSLS can be classified into the standard least-squares category mentioned
above since the least-squares functionals involve only the sum of the squared LZ-norms (or with the squared H-l-norms) of the residuals in the differential equations, and thus the trial and test functions are required to fulfill the boundary conditions. On the other hand, our formulation for (1.1)-(1.3) results in a 6 x 6 FOSLS in order to stay in the regime of the ADN theory. The advantages of the ADN-type FOSLS are that the system is smaller and that both SLSFEM and WLSFEM can be applied to the system. More specifically, the trial and test functions in the WLSFEM need not satisfy the boundary requirements, and thus, it is more convenient for treating nonhomogeneous boundary conditions. Convergence results of both approximations can be established in the natural norms associated with the least-squares bilinear forms. Furthermore, it is shown that, with displacement boundary conditions, both LSFEMs achieve optimal rates of convergence in the Hi-norm and in the L2-norm for all the unknowns. However, we do not obtain the uniform coercivity in the Poisson ratio under the standard H~-norm without any scaling, although numerical results given in Section 5 show the uniformity.
42 S.-Y. Yan 9, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 39~60
When compared with the classical mixed FEM formulation, the least-squares approach appears to require increased regularity and results in a larger system, i.e., with more equations and unknowns. Nevertheless, with a closer inspection, these shortcomings may be dispelled by the following im- portant features in practice:
• Since the approach is not subject to the Babu~ka-Brezzi condition, more flexible finite element spaces can be used. In fact, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns (cf. Section 5).
• The resulting linear algebraic systems are symmetric and positive definite and are highly vector- izable and parallelizable. The approach thus admits efficient solvers such as multigrid methods [16] or conjugate gradient methods [35].
• The solution of FOSLS can be accelerated by using two-stage algorithms [6, 19] that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired), see [18].
The layout of the remainder of the paper is as follows. In Section 2, we propose the displacement- stress formulation for (1.1)-(1.3). The LSFEMs are given in Section 3, as well as their fundamen- tal properties. A priori error estimates with the displacement boundary conditions are derived in Section 4. In Section 5, some numerical results are presented to demonstrate the approach. Finally, some concluding remarks are addressed in Section 6.
2. A new displacement-stress formulation
We first rewrite the system of Eqs. (1.1) as follows:
t3x (2 + z#)--~x + 2 fffy J - ~y # - ~ y + # Ox J = f~ inI2, (2.1)
(0Ul
& #ffyy + # ~ x J - f f f y 2~-x + ( 2 + 2 # ) O y J = f 2 inf2. (2.2)
Introducing the auxiliary variables ~3ul q91 = Ox ' (2.3) c3u2 q~2- ~3y' (2.4) ¢3ul q93 = Oy' (2.5) t3u2 ~04- OX' (2.6)
S.-Y. Yang, J.-L. L&/ Journal of Computational and Applied Mathematics 87 (1997) 39-60 43 defined on O and letting a = 2 + 2#, we can rewrite Eqs. ( 2 . 1 ) - ( 2 . 2 ) as
& 0
(/~P3 + #~o4) - 7 - (2~ol + ~o2) = f2 in f2.
& oy
(2.7) (2.8)
Note that a combination o f ~oi, i = 1,2, 3, 4, can represent the usual stresses rrij, i , j = 1,2. Also, by (2.3) with (2.5) and (2.4) with (2.6), we obtain the following two compatibility equations:
0qh &P3 0y & cq~o2 ~0 4 ~x 0y - 0 in O, ( 2 . 9 ) - 0 in O. ( 2 . 1 0 )
To recover the displacements, we have the equations ~ul ~ u 2 _
d---X- + ~ y ¢P~ - (P2 = 0 in f2, (2.11)
,~Ul ,3u2
Oy & ¢P3 + (P4 = 0 in Q. (2.12)
Eqs. ( 2 . 7 ) - ( 2 . 1 2 ) are the so-called displacement-stress formulation of (1.1) and may be written in the matrix form
5flU = AUx + B U y + D U = F in ~2, where A = D = - ~ - 2 0 0 0 0 0 - # -/~ 0 0 0 - 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 - 1 0 0 0 0 0 - 1 1 0 0 0 0 0 ' 0 - 1
0,/
o o - 2 1 B = 0 0 0/
q~2 U = ~o3 (/14 Ul U2 0 - / t -/2 0 - a 0 0 0 0 0 0 0 0 0 - 1 0 0 0 0 0 0 0 0 1/i/
A
and F = . (2.13a)i/
44 S.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 39-60
The system of differential Eqs. (2.13a) will be also supplemented with the boundary conditions (1.2)-(1.3) which may be written as
( ° 0 ° ° ° 1 0
0 0 0
one0
(~xnl)~nl /2n2 #n2 0 00) ( g l )
2n2 ~n2 /ml /znl 0 U = 92 on Fl. (2.15)
The boundary condition (2.14) implies that the tangential derivatives of ui, i - - 1,2, vanish n2q~l - nlq~3 = 0 on Fo,
-nl(p2 + n2q~4 = 0 on Fo, and also that
nlUl + n 2 u 2 = O on Fo. So, we have
(i
--hi 0 n2 0 0 U =/i/
0 0 0 n 1 n 2on F0. (2.16)
Conversely, we can show that (2.16) together with (2.3)-(2.6) implies (2.14) as well. The boundary condition (2.16) will play an important role in the later theoretical error analysis.
Rewrite (2.15) and (2.16) as the following operator form:
~ U = G on ~I2. (2.13b)
It is easily seen that Eqs. (1.1) and (2.13a) are equivalent for smooth solutions.
Theorem 2.1. u =
(ul,u2)TE[C2(~)] 2 satisfies
(1.1) /fand only
/f U ---- ((#1,(#2,(#3, q94, Ul,u2)TE[C1(-O)] 4 x [C2(~)] 2
satisfies
(2.13a).The existence, uniqueness, and smoothness of the solution of problem (1.1)-(1.3) are well known (cf. [30, 29]). Therefore, in the sequel, we shall always assume that problem (2.13a/b) has a unique solution U E [Hi(f2)] 6 with the given functions F E [L2(I2)] 6 and g E [L2(F1)] 2. Our LSFEMs will be performed over the first-order system (2.13a/b) to obtain approximations of the displacements and the stresses simultaneously.
3. Least-squares finite element methods
We shall require some function spaces defined on f2, F0, and F1 throughout this paper [27, 38]. The classical Sobolev spaces HS([2), s ~> 0 integer, L2(Fo), and
L2(F1)
with their associated inner products (', ")s,O, (', ")0,r0, (', ")0,r, and normsII'lls, , II'll0,Fo, II'll0,r,
are employed. As usual,L2(O)=n°(f2).
ForS.-Y. Yang, J.-L. Liul Journal of Computational and Applied Mathematics 87 (1997) 39~0 45 the Cartesian product spaces
[ms(~c~)] 6, [L2(/-'o)] 3,
and[L2(F1)] 2,
the corresponding inner products and norms are also denoted by (., ")+,a, (', ")0,to, (', ")0,r,, and I['[[,,a, [[']10,ro, [['[[o,r, when there is no chance for confusion.Let H~(~2) be the closure of ~(f2) for the norm [l'[[s,a, where ~(f2) denotes the linear space of infinitely differentiable functions on f2 with compact support. We denote by H - s ( O ) the dual space of H~(f2) normed by
lu(v)l
Ilull-s,
- - s u pSince the boundary 0f2 of the bounded domain f2 is smooth, there exists an operator 7o : H l ( f 2)
--'-~L2(~3(2),
linear and continuous, such that?or = restriction of v on ~3f2 for every v E C t ( O ) .
The space 70(HI(f2)) is not the whole space L2(c3f2), it is denoted by H1/2(c3f2) and define its norm by
ll~ll,/2,,~a = inf{llvll,,~;
vESa(a),
?0v-- ~o},which makes it a Hilbert space. Its dual is denoted by H-~/e(a~2) with the norm ll'll-~/2,aa. Also, the associated norms of the product spaces [H'/2(3~2)] 3 and [H-~/2(a~2)] 3 are still denoted by ]l'Ht/2,aa
and [l'll-1/2,aa, respectively.
We now introduce the standard and the weighted LSFEMs for solving problem (2.13a/b) in the following two subsections. For simplicity, we assume that G = 0 on 0g2, i.e., g = I} on /'1.
3.1. The standard least-squares finite element method
Let
3 ¢rs = { V E [ol(~-~)]6; ~ V = 0}, (3.1)
then define a standard least-squares energy functional gs : ~Us___~ N as
g s ( v ) = £ (5¢V - F ) - ( S e V - F). (3.2)
Obviously, the exact solution U E ~ s of problem (2.13a/b) is the unique zero minimizer of the functional g+ on ~ s , i.e.,
g s ( u ) = 0 = min{gS(V); VE Vs}. (3.3)
Applying the techniques of variations, we can find that (3.3) is equivalent to
£ ~U.£PV = £ F.~eV, VV E~t/~+. (3.4)
The SLSFEM for problem (2.13a/b) is therefore to determine U~ E ~ s such that
f ~U~.~Vh = f F.SfVh, VVhE ~t/'h+, (3.5)
46
S.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 39-60where the finite element space ~ s C y-s is assumed to satisfy the following approximation property. For any V E ~e's fq
[H p+l
(12)] 6, p ~> 0 integer, there exists Vh E ~/:h s such that[iV - Vhl10,12 +
hIlV - vhl11,12<~Chp+lllVl]p+~,12,
(3.6)with the positive constant C independent of V and h. Throughout this paper, in any estimate or inequality the quantity C will denote a generic positive constant and need not necessarily be the same constant in different places.
3.2. The weighted least-squares finite element method
Similar to the standard least-squares case, we define
y-w = [H,(12)16, (3.7)
and define a weighted least-squares energy functional gw : y-w__+ R as
g w ( v )
= l ( . t # V - F ) . ( S f V - F) + h -1 f ~lV. ~V,
(3.8),112 aa g2
where h is the mesh parameter. The exact solution U E ,//-w of problem (2.13a/b) is the unique zero minimizer of the weighted least-squares functional gw on ~ w , i.e.,
g w ( u ) = 0 = min{4"w(v); VE ~//'w}. (3.9)
Taking the first variation, we can find that (3.9) is equivalent to
VVE~//w. (3.10)
The WLSFEM for problem (2.13a/b) is then to determine U~'E ¢h w such that
~ U ~ . ~ V h
+ h - ' f ~ 1 2 J I U ~ . ~ V h = ~ F . ~ V h ,
V V h E ~ w, (3.11) where the finite element space ~ w C ~ w is also required to satisfy the following approximation property. For any VE .//-w N [HP+~(12)] 6, p>~0 integer, there exists Vh E ~ w such thatII
V -- V hll0,12 -Jr- hll
v - Vh [[ 1,12 ~ Chp+l IIFlip+l,12,
(3.12)
where C is a positive constant independent of V and h.
3.3. Some fundamental properties
In this subsection, we shall discuss the unique solvability of the numerical schemes (3.5), (3.11), and some of their fundamental properties. Before presenting these properties, it is of interest to note that the trial and test functions in the WLSFEM (3.11) need not satisfy the boundary conditions. In contrast, in the SLSFEM (3.5), both the trial and test functions are required to fulfill the boundary requirements. Moreover, since the original system o f second-order Eqs. (1.1) is transformed into the system of first-order Eqs. (2.13a), the same C o piecewise polynomials can be used to approximate all the unknown functions.
S.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 39-60
47 Denote the bilinear form and the linear form in (3.4) asas(v, w ) = fa&eV.~LPW,
fs(V) = ~ V.~V,
for all V, W E ~ s. Then (3.4) and (3.5) can be rewritten as
aS(U,
V ) - - fs(V), VVC~//~s, andaS(U~,Vh)=Ys(Vh),
VVhe~h ~,
respectively. Similarly, denote the bilinear form and the linear form in (3.10) as
aW(V, W ) = __~ £#V.SfW + h-1 _vf ~lv" ~lV,
( w ( v ) = faF.ZPV,
for all V, WE~/w. Then (3.10) and (3.11) can be rewritten as
aW(U,
V) = ~w(v), V V E ~ ~w, and (3.13) (3.14)(3.15)
(3.16) (3.17) (3.18) (3.19)Theorem 3.1.
Let
U E [ H I ( ~ r ~ ) ] 6be the exact solution of
(2.13a/b)with the given functions FC
[L2(f2)] 6
and G = O.
(i)
Problem
(3.16)has a unique solution U~ C ~h s which satisfies the following stability estimate
II u llas IlFIIo, .
(3.23)(ii)
The matrix of the linear system associated with problem
(3.16)is symmetric and positive
definite.
(iii)
The following orthogonality relation holds
aS(U - U~, Vh)=O, 'qVhE ~e"hs.
(3.24)aW(U~,Fh)=fW(Vh), VVhE~h w,
(3.20)respectively.
It is clear that aS( ., .) and aW( ., .) define inner products on ~e ~s × ~ s and ~e ~w × ~ w , respectively, since the positive-definiteness is implied by the fact that the problem (2.13a/b) possesses the unique solution U = 0 for F = 0 and G = 0. Denote the associated norms as
II wllo
={as(v,v)} 1/2, ~/vE3vs,
(3.21)[[VNaw = {aW(V,
V)} ~/z, VVE'U w.
(3.22)48 S.-Y. Yang, J.-L. Liu/ Journal of Computational and Applied Mathematics 87 (1997) 39-60
][U - U~[Ia~ ~ChPi[f[Ip+,,o.
(iv) The approximate solution U~ is a best approximation o f U in the II.[[~-norm,
I I f - u ~ l l ~ = inf I l u - V h l l ~ . (3.25)
(v) I f Uc[Hp+I((2)] 6, p>>-O integer, then there exists a positive constant C independent o f h such that
(3.26) Proof. To prove the unique solvability, it suffices to prove the uniqueness of solution since the finite dimensionality of ~h s. Let U~ be a solution of (3.16) then, by the Cauchy-Schwarz inequality,
u~ Ilas
-- a (u~, u~) = (F, ~q~U~)0,o~< IIFII0,~ll-~U~ll0,~
<. IIFIIo,~IIU~Ilos.
Thus, we obtain (3.23). Consequently, the solution U~ of (3.16) is unique.
Assertion (ii) follows from the fact that the bilinear form aS( ., -) is symmetric and positive definite. To prove (iii), subtracting Eq. (3.16) from Eq. (3.15), we get (3.24).
To prove (iv), by (3.24) and the Cauchy-Schwarz inequality,
II u - ush t2a, = aS( U - U2, U - U~)
= a s ( u - U L u -
zh),
V V h E ~ s <~ [ ] u - U~JJasllU-VhJJas.
Thus, we have (3.25).
Finally, assume that UE[HP+l(f2)] 6. Let VhE~h ~ such that (3.6) holds with V replaced by U. Then, by (3.25), we have
I I u - u~llos ~ IIU-
Vhlla~ <. Cll U - vhll~,o <. fhpllU[[p+~,o.In the second inequality above, we use the fact that ~q~ is a first-order differential operator with constant coefficients. []
Similarly, we have the following results for the WLSFEM (3.20).
Theorem 3.2. L e t UE[HI(~c~)] 6 be the exact solution o f (2.13a/b) with the 9iven functions F E
[L2(g2)] 6 and G = O.
(i) Problem (3.20) has a unique solution U~ E ~h w which satisfies the followin9 stability estimate:
{[ U~i{aw ~ ][F[{0,a. (3.27)
(ii) The matrix o f the linear system associated with problem (3.20) is symmetric and positive definite.
(iii) The followin9 orthogonality relation holds
S.- Y. Yang, J.-L. Liu/ Journal of Computational and Applied Mathematics 87 (1997) 39~0 49 (iv) The approximate solution U~ is a best approximation of U in the [[.[[aw-norm, that is,
IIU
-v;'llow
-- inf [ [ u - Vhll#.. (3.29)VhE ~
(v) I f
UE[HP+I(~-2)] 6,
p>>-O integer, then there exists a positive constant C independent of h such thatIIu - g2'll~ <~Ch"llvll~÷l,,~. (3.30)
I I u - v~llo, o~ ~< Thus,
Proof. The proofs for (i)-(iv) are similar to the standard least-squares case. For proving part (v), we need the following result whose proof can be found in [12]: there exists a positive constant C such that, for any V E [ H I ( ~ ) ] 6 and any 8>0,
"V"o,o~ <, C @[V[[l,~ + l [lV[[o,~) .
Taking e = h 1/2 and V replaced by U - Vh, where VhE~h w is chosen such that (3.12) holds with V
replaced by U, then we have
C(hl/ZllU - vhlll,~ + h-V=ll U - v~llo, o) Chp+l/2llUllp+l,w
U~ II~w ~ IIU
V~l[=aw
I I U - w 2
= I I ~ ( U - V ~ ) l l ~ o , ~ + h - l l l ~ ( U - Vh)[Io, o~
<~ C ( I I U - Vhll~.,~ + h - ~ l l U - v~ll0,0~) = <~ Ch=PlIUII%+I.,~.
This completes the proof. []
As a consequence of part (v) in the above theorems, the consistency of the approximations follows.
Corollary
3.3. Let U be the exact solution o f problem (2.13a/b) with the given functions F E [L2(~)] 6 and G = 0 . I fUE[Hp+I(~)] 6, p>/O
integer, then there exists a positive constant C independent o f h such that[ l ~ u x - FIIo,~ ~<Ch~llUIl~+l,~, (3.31)
II ~eu~ w - fll0,~ ~ Ch p II Nil ~+1,~, (3.32)
50 S.-Y. Yang, J.-L. Liul Journal of Computational and Applied Mathematics 87 (1997) 3940
4. Error analysis
The error estimates of the previous approximations in the H’- and L2-norm are primarily based on the theories of ADN and of Dikanskij [28]. Our approach in exploiting these theories follows principally that of Wendland [43, Section 3.1, ch. 81 for two-dimensional first-order elliptic systems in the sense of Petrovski. The application of the theories to our problem involves some unavoidable difficulties concerning the Lopatinski condition if the boundary condition (2.13b) is taken to be as that general. For simplicity, we only consider the displacement boundary conditions
0 -n1 0 0 0
&?U= 0 -?I] 0 n2 0 0
0 0 0 0 It1 n2
4.1. A priori estimates
on r0 = aa. (2.13b’)
We first show that _!Z is an elliptic operator in the sense of Petrovski, and that the boundary operator B in (2.13b’) satisfies the Lopatinski condition. So (2.13a/b’) is a regular elliptic boundary value problem and then (9,9) is a Fredholm operator with zero nullity. This enables us to get the coercive type a priori estimates (see Theorem 4.1).
For all (r, V)E R2 and (&r) # (0, O),
det(&4 + @) = -(I+ + 2p2)(12 + q2)3 # 0.
Thus, (2.13a) is an elliptic system in the sense of Petrovski. Obviously, by taking (5, ye) = (1, 0),
the matrix A is nonsingular and its inverse is
-l/a 0 0 -n/c? 0 0 0 00 100 A-’ r 0 0 -1 0 00 0 -l/p 1 0 0 0 0 00010 0 0 0 0 0 -1
1.
Then the original elliptic system (2.13a) can be transformed into the following form:
U,+SU,+&J=P in 0,
S.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 3940 51 0 0 0 00 0 0 0 00 0 0 0 00 -1 0 0 0 0 0 l-100 p = A-‘F= -f lb 0
-;2/r
0 0We now check the Lopatinski condition as follows. After elementary operations, we find that the eigenvalues of the matrix bT are the imaginary numbers i and -i both with multiplicities three. Consider the eigenvalue r+ = i in the upper half-plane, to which there exists a chain of linearly independent generalized eigenvectors p1 and p2 of BT defined by
BTp, - z+p1= 0,
BTP2 - 7+P2 =p1,
and a third eigenvector p3 is given by
BTp3 - z+p3 = 0, where p1 = l,l,-ei ei 0 0 T ( CI’X” > ’ 2P P(A-t3C1) P P2 = (jg0’ a(A + p) ) i(JJOS~ Then 9 = (~l,~l,P2,~2~P3>~3~T is nonsingular. Let
52 S.-Y. Yang, J.-L Liu/Journal of Computational and Applied Mathematics 87 (1997) 39-60
be the inverse matrix of ,#, then
2 = 0 0 - ( ( 2 + #)/4#)i ((2 + #)/4#)i 0 0 1/2 1/2 ((2 + #)/4#)i - ( ( 2 + #)/4#)i 0 0 ((2 + #)/4#)i - ( ( 2 + #)/4#)i (2 + #)/4# (2 + #)/4# 0 0 - ( ( 2 + 3#)/4#)i ((2 + 3#)/4#)i (2 + #)/4# (2 + #)/4# 0 0 0 0 0 0 1/2 1/2 0 0 0 0 (1/2)i - ( 1 / 2 ) i
Now, check the following determinant:
det 2 0 - n l
°-nl°°°o)
0 n2 0 (ql,q2,q3))
0 0 0 0 nl n2
= - (1/4#2)(2 + 3#)(2 + #)(hi + n2i) 3 ¢ 0 ,
since (nl,n2)yL(O,O). That is, the Lopatinski condition is satisfied for the boundary conditions (2.13b').
The following estimates then follow the standard results of [43].
Theorem 4.1. For the boundary value problem (2.13a/b'), (2.13a) is an elliptic system in the sense
o f Petrovski, and the boundary condition (2.13b') satisfies the Lopatinski condition. Thus, we have the a priori estimates: for each l>>-0 there is a constant C > 0 such that if VE[HI+I(O)] 6, then
(4.1) By an interpolation argument [28] (cf. [43, Lemma 8.2.1]), the estimate (4.1) can be extended to the case l / > - 1. Taking l = 1, l = 0, and l = - 1 in (4.1), we have
II vl12,~ ~ CIl~Vll
1,K2,II vll,,~ ~ CIl~ vll0,~,
II vll0,~ ~< CIl~e vll-~,o,
II vl12,~ ~< C(ll ~Vll ~,,~ + lt~vll3/z,a~),
VV E ~V's N[H2 (~"2)] 6,
(4.2) VV E ~V S, (4.3) V V e ' f s, (4.4) ~/VE 3 V'w f-) [H2(~2)] 6, (4.5) T w o FEM llVlll,~C(ll~Vllo,~ + l[~Vl11/2,0~), VVc ~'w, (4.6) IlVllo,~c(ll~vll-l,~ +II~vll-1/2,0~),
v v ~
w.
(4.7)sets of estimates (4.2)-(4.4) and (4.5)-(4.7) imply, respectively, the error estimates for SLS- and WLSFEM in the following two subsections.
S.-Y. Yang, J.-L. Liul Journal of Computational and Applied Mathematics 87 (1997) 39~50 53
4.2. Error estimates f o r the S L S F E M
For the standard least-squares case, by (4.3), we have
aS(V, V ) =
I[~evll~o,~>_.Cllv
I1,,~, v w c ~s, 2 (4.8)i.e., the bilinear form aS( • , .) is coercive on ~s. Thus, by using the standard argument, we have Theorem 4.2. Let U E~tFsM
[Hp+I(Q)] 6, UhSE ~
be the solutions o f (2.13a/b') and (3.16), respec- tively. Then1] U - Uh s [[ 1,o ~< Ch p [[ U[[p+I,Q. (4.9)
Proof. Utilizing (4.8) and (3.24), we have
IIV - u~ IIl,~ ~ C a S ( U - U~ s, U - U~ s) s 2
= C a S ( U - Uhs, U - Vh), VVhEU~
~< C l l U - U ; l l , , . t f u - V~lll,~.
Thus,
I I U -
u~ll,,o~CllU-
~lll,~, V V h ~ .Taking Vh E ~//-~ such that (3.6) holds with V replaced by U, we obtain (4.9). []
Theorem 4.2 shows that the SLSFEM (3.16) achieves optimal convergence in the Ht-norm. For deriving the optimal L2-estimates, we need the following regularity assumption: assume that, for any V E [H01((2)] 6 and Q E [Hm(Of2)] 3, the unique solution U* of the following problem
~EaU * = V in f2,
~ U * = Q on a~2, (4.10)
belongs to [H 2(Q)] 6, where ~ is the displacement boundary operator (cf. (2.13b')). This assumption is reasonable since ~ is a first-order differential operator.
Theorem 4.3. Let U E ~t/~s N [Hp+I(~'~)] 6, Uh s E ~//~ be the solutions o f (2.13a/b') and (3.16), respec- tively. I f the regularity assumption o f (4.10) holds with Q = 0 , then
II u - v,; IIo,~ ~ Chp+l N UIl,,+,.~. (4. l l)
Proof. For VE[H~(~2)] 6, let U * E [H2(O)] 6 be the solution of (4.10) with Q = 0 . Then,
I ( ~ ( u - u~s), V)o,~l
= I ( . ~ ( u - u~s), ~eu*)0,~l
54 S.-Y. Yang, J.-L. Liu/Journal of Computational and Applied Mathematics 87 (1997) 3940
< ~ C l l ~ ( u -
~)llo,~ll~(u* - ~)11o,~, v v ~
< ~ c I I u - ~=ll,,~llu * -V~ll,,~, v v ~
~<chllu- U,;lll,~llu*ll2,~,
(by (3.6))
~<chllu- ~ll,,~ll~v*ll,,~,
(by (4.2))
= ChlIU- U;II,,~II VlI,,~.
In addition, the L2-inner product (L~a(U- Uh s ), V)0, ~ defines a bounded linear functional on [H01 (f2)]
6,
sinceI ( ~ ( u -
u;), V)o,~l ~ I I ~ e ( u - U;)llo,~ll Vll,,~,
v v ~ [H~(a)] 6.
Therefore, by the definition of the [[. [[_,,a-norrn,
I l Z e ( e -
u~)ll_,,.<.Chlle-UhSl[1,12.
(4.12)The proof is completed by combining (4.12), (4.4), and (4.9). []
4.3. Error estimates for the W L S F E M
Following the techniques developed in [43, pp. 352-356], we shall first present the optimal L2-estimates and then the optimal HLestimates for the WLSFEM.
Similar to the proof of part (v) in Theorem 3.2, we note that, for any
WE[Hp+I(Q)] 6,
p>~Ointeger, there exists Wh e ~//'~' such that
II
z / -Whllaw ~<
ChPllWHp+I,O,
(4.13)where C is a positive constant independent of W and h.
Theorem 4.4. Let U E~I/'w
N
[Hp+I(Q)] 6, UhwEY/'~ v be the solutions o f (2.13a/b') and (3.20), re-spectively. Assume that the regularity assumption of (4.10) holds, then
II u - v~ w II0,~-<
Ch ~+~II u]lp+l,a.
(4.14)Proof. For V E [HI(K2)] 6, let U*E [H2(O)] 6 be the solution of (4.10) with Q = 0. Then,
I ( ~ ( u - u~w), v)0,~l
= I ( ~ ( v - v~w), ~U*)o,,~l
= laW(U - Uh w,
v*)l
= l a w ( U - Uhw, u * - Vh)l, V V h e ~ ; ' , ( b y ( 3 . 2 8 ) )
~< {aW(U- UhW, U - UhW)}'/2{aW(U* - Vh, U* -
Vh)} 1/2,
VVh e'~(/"h v < C h { a W ( U - U ~ v , u - uhw)}l/2llu*llz, a (by (4.13))S.-Y. Yang, J.-L. Liul Journal of Computational and Applied Mathematics 87 (1997) 39-60 55 <<.ChhPllUI]p+I,aII~LPU*III,a (by (3.30), (4.5))
= Ch p+~ 11 U]lp+l,al[ Vlll,a. Therefore, we have
II ~ e ( u - u~w)ll_,,~.< Ch '~+1 II UNp+I,(2" (4.15) On the other hand, take V = 0E[Hd(I2)] 6 in (4.10), then for any QE [H1/2(~3f2)] 3,
Ih-~(~(U - UhW), a)o,~al
_ - I h - ~ ( ~ ( u - UhW),~f*)o, ea[ --_ laW(U - Uh w, U*)I
=laW(U--Uhw, u * - - V h ) l , V V h E ~ , (by (3.28))
~<{aW(U - Uhw, u - UhW)}'/2{aW(U* - Vh, U * - Vh)} '/2, 'v'Vh E'U~'
~ C { a w ( U - Uhw, u - UhW)}l/211U*ll,,~ (by (4.13))
<-Ch"llfllp+,,,~ll~U*lll/2,a,~
(by (3.30),(4.6))= ChPllullp+~,~NOlll/2,a~.
Thus, for any QE [H1/2(OQ)]3 w e have
I ( ~ ( U - UhW), O)o, aal ~< Ch p+lll
ull,+,,oilall
1/2,~. Hence,I I ~ ( u - u~w)ll_ 1/2 ~ , o ~ C h p+I U I p+ ,a. 1 (4.16) The proof is completed by combining (4.7), (4.15), and (4.16). []
Note that in the proof of Theorem 4.4, we utilize the estimate (3.30) to circumvent the use of the optimal HI-estimates which is not yet established. In order to give the optimal HI-estimates, we need to define the following Gauss projection [43]:
~#. ~//-w ~ ~//-~v, f#W -z Wh w, (4.17)
where Wh w is the solution of the discretized problem (3.20) corresponding to problem (2.13a/b') with suitable data function F such that its unique exact solution is W. Since problem (3.20) is uniquely solvable, the Gauss mapping is well defined, and we have
~Vh = Vh, VVhE~//'~ v. (4.18)
Taking p = 0 in (4.14), then we get II~¢ello, a--IIU~Wllo,~
IIUllo,~ + IIu - u, hWllo,~ Ilfllo,~ + Ch[tUIl~,,~.
56 s.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 3940
Thus, we can conclude that, for any V E f-w,
IFcvH0,o ~< II vll0,o +
Chllvll,,~.
(4.19)We also need the following inverse assumption on the finite element space 7/~': there exists a constant C > 0 independent of h such that
IlVhll,,o~<Ch-lltVh/10,o, v v h c ~ v.
(4.20)
The inverse assumption is commonly used in many least-squares finite element analyses [4, 43]. More precisely, if the regular family {J-h} of triangulations of O associated with the finite element space Uh TM is quasi-uniform [27, 37], i.e., there exists a positive constant C independent of h such
that
h<.C diam(f2~), V g h E J h , ~--hE{J-h}, then (4.20) is satisfied.
The optimal order of convergence for the WLSFEM in the Hi-norm is thus concluded.
Theorem 4.5. Let UE~//~w n [Hp+I(Q)] 6, UhWE~/'~ v be the solutions o f (2.13a/b') and (3.20), re-
spectively. Suppose that the regularity assumption o f (4.10) and the inverse assumption (4.20) hold, then
IIv-
v~Wll,,o~Ch~llUll~+,,o.
(4.21)Proof. By (4.18), (4.19), and the approximation property (3.12), we have
I I u - U~Wll,,~
~< ~< I I u -VhIII,~-F
IIU~ w- ~11,,,~, v v ~ '
Ilu - ~11~,~
+II~(V
-V~)ll,,~, v v ~ '
I I v - V h l l l , ~ + C h - l l [ f # ( V -Vh)[Io,~,
V V h E ~ 'IIU - V~II,,~ + C h - ' { I I U - V~llo,~ + C h J J U - V~tll,~)
ChP[[U]lp+,,o. []
5. Numerical experiments
We shall give a simple example which will be solved by using the SLSFEM (3.16). Consider the displacement-stress elasticity Eq. (2.13a) supplemented with the homogeneous displacement boundary condition (2.13b'). Taking 12 = (0, 1 ) x (0, 1 ) and choosing
f l = (c¢ + #)re 2 sin(~x) sin(try) - (2 + #)rt 2 cos(rex) cosOty), f2 = (c¢ + p)rt 2 sin(Ttx) sin(lty) - (2 + p)rt 2 cos(rtx) cos(r~y),
S.-Y. Yang, J.-L. Liul Journal of Computational and Applied Mathematics 87 (1997) 3940
Table 1
The SLSFE approximations with E = 2.5 and v = 0.25
57
1/h Llelq0,~ RelErr Conv. rate Ilella~ RelErr Cony. rate
2 0 . 6 9 0 1 2 2.1431-10 -1 - - 1 1 . 5 0 1 8 3.68523.10 -1 - - 4 0 . 1 9 0 0 7 5.90235.10 -2 1.86 5 . 8 6 0 7 0 1,87780.10-1 0.97 8 0.05076 1.57625.10 -2 1.90 2 . 9 6 6 0 1 9,50326-10 -2 0.98 16 0.01304 4.05067.10 -3 1.96 1 . 4 8 8 6 8 4,76981.10 -2 0.99 32 0 . 0 0 3 3 0 1.02456.10 -3 1.98 0 . 7 4 5 1 1 2,38739.10 -2 1.00 Table 2
Rates of convergence in the aS-norm with E = 2.5
1/h v = 0.05 v = 0.15 v -- 0.35 v = 0.45 v = 0.49 v = 0.499 v = 0.4999 2 . . . . 4 0.98 0.98 0.96 0.93 0.90 0.90 0.89 8 0.99 0.99 0.98 0.96 0.95 0.95 0.95 16 1.00 1.00 0.99 0.99 0.98 0.98 0.98 32 1.00 1.00 1.00 1.00 0.99 0.99 0.99
the exact solution is then g i v e n b y
~ol = rt c o s ( n x ) s i n ( n y ) , q~3 = n s i n ( n x ) c o s ( n y ) , u 1 = s i n ( n x ) s i n ( n y ) , q~2 = n s i n ( n x ) c o s ( n y ) , (/)4 - - ll~ c o s ( u ) s i n ( x y ) , u2 = s i n ( n x ) s i n ( n y ) .
T o s i m p l i f y the n u m e r i c a l i m p l e m e n t a t i o n , w e shall a s s u m e that the square d o m a i n f2 is u n i f o r m l y partitioned into a set o f 1/h 2 square s u b d o m a i n s f2 h with side-length h. Piecewise bilinear finite
e l e m e n t s are u s e d to a p p r o x i m a t e all c o m p o n e n t s o f the exact solution. F o r the case o f P o i s s o n ' s ratio v = 0.25 a n d Y o u n g ' s m o d u l u s E = 2.5, the results are collected in T a b l e 1, w h e r e e denotes the exact error U - Uh s and R e l E r r denotes the relative error. Since the H i - n o r m is equivalent to the aS-norm for the standard least-squares case, T a b l e 1 exhibits that the S L S F E M achieves o p t i m a l c o n v e r g e n c e b o t h in the LZ-norm and in the H i - n o r m for all the c o m p o n e n t s .
T h e influence b y the P o i s s o n ratio v for the b e h a v i o r o f c o n v e r g e n c e is also e x a m i n e d . T a b l e s 2 and 3 s h o w that, except on v e r y coarse m e s h e s , the o p t i m a l c o n v e r g e n c e is still essentially insured for various P o i s s o n ratios e v e n for nearly i n c o m p r e s s i b l e elasticity. T a b l e 4 shows that the c o n v e r g e n c e in the aS-norm s e e m s to be u n i f o r m in the P o i s s o n ratio. It is not surprising since, r o u g h l y speaking, the aS-norm can be v i e w e d as the H i - n o r m w e i g h t e d a p p r o p r i a t e l y b y the L a m 6 coefficients 2 and #. H o w e v e r , w e find that the situation is quite different for the full L 2 - n o r m case since the values largely increase w h e n v a p p r o a c h e s to 0.5. Thus, one can c o n c l u d e that in order to get o p t i m a l c o n v e r g e n c e in s o m e S o b o l e v n o r m w h i c h is u n i f o r m in the P o i s s o n ratio, the least-squares functionals need to be w e i g h t e d a p p r o p r i a t e l y b y the L a m 6 coefficients (cf. [15, 17]).
58 S.- Y. Yan 9, J.-L. L i u / Journal o f Computational and Applied Mathematics 87 (1997) 3 9 4 0
Table 3
Rates of convergence in the L2-norm with E = 2.5
1/h v = 0.05 v = 0.15 v = 0.35 v = 0.45 v = 0.49 v = 0.499 v = 0.4999 2 . . . . 4 1.82 1.83 1.85 1.57 1.47 1.45 1.45 8 1.90 1.91 1.86 1.77 1.74 1.73 1.73 16 1.96 1.96 1.95 1.91 1.89 1.89 1.89 32 1.98 1.98 1.98 1.97 1.97 1.96 1.96 Table 4
The values of h -111ell~ II UII~ l
1/h v = 0.05 v = 0.15 v = 0.35 v = 0.45 v = 0.49 v = 0.499 v = 0.4999 2 0.758 0.748 0.735 0.763 0.789 0.797 0.797 4 0.767 0.759 0.756 0.803 0.844 0.856 0.856 8 0.774 0.767 0.768 0.824 0.874 0.889 0.890 16 0.777 0.769 0.772 0.832 0.885 0.900 0.902 32 0.777 0.770 0.773 0.834 0.888 0.904 0.905 6. Concluding remarks
In this paper, a new first-order displacement-stress formulation for the elasticity equations is introduced. Standard and weighted L S F E M s are proposed and analyzed. C o n v e r g e n c e results for both methods are established in the natural norms associated with the least-squares bilinear forms. Furthermore, with the displacement b o u n d a r y conditions, both the methods achieve optimal rates o f c o n v e r g e n c e in the H i - n o r m and in the L2-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical analysis.
Although it is interesting to note that the results o f Tables 2 and 3 do not deteriorate as the Poisson ratio v tends to 0.5, we cannot say that the least-squares methods for the elasticity p r o b l e m b y using the n e w first-order system formulation avoid the locking p h e n o m e n o n [2, 7, 8]. H o w e v e r , utilizing the techniques developed in the appendix in [42] and weighting the least-squares functionals b y suitable parameters as that in [15, 17], a theoretical verification about possible i m p r o v e m e n t in regard to the locking p r o b l e m based on the present first-order formulation appears to be promising. In this case, the auxiliary variables ( 2 . 3 ) - ( 2 . 6 ) m a y be replaced b y (~01 = ~ U l / ~ X , q)2 = ~ U l / ~ y , q)3 ~ ~ U 2 / ~ X , and
p = - ( v / 1 - 2 v ) V ' . u. This issue has b e c o m e the subject o f a separate investigation in progress.
While the basic c o n v e r g e n c e theory derived in Section 3 works well for the system (2.13a) with the displacement-stress b o u n d a r y conditions (2.15) and (2.16), the error estimates d e v e l o p e d in Section 4 m a y not c o v e r this type o f b o u n d a r y conditions since the boundary-value problem (2.13a/b) with m e a s u r e ( F 1 ) > 0 does not satisfy the Lopatinski condition. H o w e v e r , it is possible to d e c o m p o s e the system (2.13a) into two subsystems. One is the stress system ( 2 . 7 ) - ( 2 . 1 0 ) and the other is the displacement system (2.11 ) - ( 2 . 1 2 ) or ( 2 . 3 ) - ( 2 . 6 ) . The optimal c o n v e r g e n c e properties
S.-Y. Yang, J.-L. LiulJournal of Computational and Applied Mathematics 87 (1997) 3940 59 for both methods may then be retained for the more general boundary conditions by means of the two-stage techniques [6, 18, 19].
Acknowledgements
The authors would like to thank Professor C.L. Chang (Department of Mathematics, Cleveland State University, Cleveland, Ohio, USA) for many helpful discussions when he visited the Depart- ment of Applied Mathematics of National Chiao Tung University in September 1996. In addition, the authors would also like to thank Professor Steve McCormick and an anonymous referee for their valuable comments and suggestions on an earlier version of the manuscript.
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