QUARTERI,Y OF APPLIED MATHEMATICS OCTOllER, 1975 MIXED FINITE-ELEMENT APPROXIMATIONS OF LINEAR BOUNDARY-VALUE PROBLEMS* By J. N. REDDY·· AND J. T. ODEN University of Texa.~, A 11stin 255 Abstract. A theory of mixed finite-elementjGalerkin approximations of a class of linear boundary-value problems of the type T*Tu + ku + f = 0 is presented, in which appropriate notions of consistency, stability, and convergence are derived. Some error estimates are given and the results of a number of numerical experiments are discussed. 1. Introduction. A substantial majority of the literature on finite-element approxi- mations concerns the so-called primal or "displacement" approach in which a single (possibly vector-valued) variable is approximated which minimizes a certain quadratic functional (e.g. the total potential energy in an elastic body). A shortcoming of such approximations is that they often lead to very poor approximations of various derivatives of the dependent variable (e.g. strains and stresses). The dual model, also referred to as the "equilibrium" model, employs a maximum principle (complementary energy), and can lead to better approximations of derivatives, but it leads to difficulties in com- puting the values of the function itself for irregular domains. The alternative is to use so-called mixed or hybrid approximations in which two or more quantities are approxi- mated independently (e.g. displacements and struins are treated independently). Numerical experiments indicate that this alternative can lead to improved accuracies for derivatives in certain cases, but the extremum character of the associated variational statements of the problem is lost in the process. This means that most of the techniques used to establish the convergence of thc finite element method in the dual and primal formulations are not valid for the mixed case. In thc mid-l!l60s, use of mixed finite-element models for plate bending were proposed, independently, by Herrmann [1]and Hellan [2].These involved the simultaneous approx- imation of two dependent. variables, the bending moments and the transverse deflection of thin elastic plates, and were based on stationary rather than extremum variational principles. Prager [3], Visser [4], and Dunham and Pister [5] employed the idea of Herrmann to construct mixed finite-element models from a form of the Hellinger- Reissner principle for plate bending problems with very good results. Backlund [5} used the mixed plate-bending elemcnts dcveloped by Herrmann and Hellan for the analysis of elastic and elasto-plastic plates in bending, and Wunderlich [7}used the idea of mixed models in a finite-elcment analysis of nonlinear shell behavior. Parallel to the work on mi.-xcdmodels was thc development of the closely related hybrid models by • Received November 2il, 1973; revised version received March 7, 19i4. .. Current address: University of Oklahoma, Norman.
Transcript
QUARTERI,Y OF APPLIED MATHEMATICS
OCTOllER, 1975
MIXED FINITE-ELEMENT APPROXIMATIONS OF LINEARBOUNDARY-VALUE PROBLEMS*
By
J. N. REDDY·· AND J. T. ODEN
University of Texa.~, A 11stin
255
Abstract. A theory of mixed finite-elementjGalerkin approximations of a classof linear boundary-value problems of the type T*Tu + ku + f = 0 is presented, inwhich appropriate notions of consistency, stability, and convergence are derived. Someerror estimates are given and the results of a number of numerical experiments arediscussed.
1. Introduction. A substantial majority of the literature on finite-element approxi-mations concerns the so-called primal or "displacement" approach in which a single(possibly vector-valued) variable is approximated which minimizes a certain quadraticfunctional (e.g. the total potential energy in an elastic body). A shortcoming of suchapproximations is that they often lead to very poor approximations of various derivativesof the dependent variable (e.g. strains and stresses). The dual model, also referred toas the "equilibrium" model, employs a maximum principle (complementary energy),and can lead to better approximations of derivatives, but it leads to difficulties in com-puting the values of the function itself for irregular domains. The alternative is to useso-called mixed or hybrid approximations in which two or more quantities are approxi-mated independently (e.g. displacements and struins are treated independently).Numerical experiments indicate that this alternative can lead to improved accuraciesfor derivatives in certain cases, but the extremum character of the associated variationalstatements of the problem is lost in the process. This means that most of the techniquesused to establish the convergence of thc finite element method in the dual and primalformulations are not valid for the mixed case.
In thc mid-l!l60s, use of mixed finite-element models for plate bending were proposed,independently, by Herrmann [1]and Hellan [2].These involved the simultaneous approx-imation of two dependent. variables, the bending moments and the transverse deflectionof thin elastic plates, and were based on stationary rather than extremum variationalprinciples. Prager [3], Visser [4], and Dunham and Pister [5] employed the idea ofHerrmann to construct mixed finite-element models from a form of the Hellinger-Reissner principle for plate bending problems with very good results. Backlund [5}used the mixed plate-bending elemcnts dcveloped by Herrmann and Hellan for theanalysis of elastic and elasto-plastic plates in bending, and Wunderlich [7}used the ideaof mixed models in a finite-elcment analysis of nonlinear shell behavior. Parallel to thework on mi.-xcdmodels was thc development of the closely related hybrid models by
• Received November 2il, 1973; revised version received March 7, 19i4... Current address: University of Oklahoma, Norman.
256 J. N. REDDY AND J. T. ODEN
Pian and his associates (e.g. [8,9,10)). Reddy [11], Johnson [12], and Kikll<:hi llnd Ando(13] obtained some error estimates for mixed models of the biharmonic equlltion; however,their approach is not general and the biharmonic equation has the speciul feature thatit decomposes into uncoupled systems of canonical equations which arc themselveselliptic. In all of these studies, results of numerical experiments suggest that mixedmodels can be developed which not only converge very rapidly but also ma)' yield higheraccuracies for stresses than the corresponding displacement-type model. 1'Iore import-antly, the stationary conditions of the mixed formulation are a set of canonical equationsinvolving lower-order derivatives than those encountered in the governing equations.This makes it possible to relax continuity requirements on the trial. functions in mixedfinite-element models.
It is the purpose of the present paper to describe properties of a broad class of mixedfinite-element approximations and to present fairly general procedurcs for establishingthe converJl;ence of the method and, in certain cases, to derive error estimates. Prcliminaryinvestigations of the type reported herein were given in [14] and centered around notionsof consistency and stability of mixed approximations. The present study utilizes a similarbut more general approach, and we are able to obtain the conclusions of [141 as well asthose of previous investigators (e.g. [13)) as special cases.
2. A class of linear boundary-value problems. W l' arc concerned with 1L class ofboundary-value problems of the type
'j'*Tu + kit + f = ()in n, N('l'It) - (/2 = 0 on c'm~. (2.1)
Here l' is II linear operator from a Hilbert space 'U into a IIilbcrt 8pace '0, '1'* is theadjoint of '1' and its domain D7'* is in '0, the dependent variable u(x) is an element of'lLand is a function of points x = (XI , x~ , ... , x,,) ill an open bounded c10mnin nCR".The boundary c'mof n is divided into two portions, ani v rJ12, = an on whidl the imagpsof u and 1'1l under the boundary operators ill and N are prescribed, as indicated. If[UI , u~l and [VI' v2] denote the inner products associated with spuces 'U and '0,respectively, then T and T* are assumed to satisfy a generalized Grecn's formula ofthe type
where I. , .IaD. and [. , . ]aD, arc associated bilinear forms obtained using the extensionsof u and v and Mu and .Vv to the indicated portions of the boundary. Clearly, the formsof M and N depend upon T and the definition of the ilmer products (for a completepicture see [15)).
The boundary-value problem (2.1) can be split into a canonical pair of problemsequivalent to (2.1) of the form
T*v + ku = - f in nTu == v in n llfu - gl = 0 on an" (2.3)
Our mission is to study finite-element-Galerkin approximations of this pair.
3. Mixed Galerkin projections. 'Vc 1I0W iuentify finite, lincnrly-indepl'ndellt setsof functions 1'l>o(x)lo_,G E 'lL and Iwd(x)ld_lll E '0, which, respectively Mpan the
:MIXED FINITE-ELEMENT APPROXBU TrONS 257
finite-dimensional Subspllces :JTl:/ and mJ/. ?{ow if u( x) and v( x) arc arbitrary elementsin 'U and 1), respectively, their projections into :J1Ir/ and ml/' arc of the form (see [16])
G
IlA(u) = U(x) = L: a".p,,(x, 11);,,-I
II
P,(v) = V(x) = L: b.1w.1(x, 1).fJ-)
(3.1)
Here aa and bA are constants, uniquely determined by u, .pa , v, fiud wA• It must be noted
hcre that there is no relation between the spaces 'U and 1), and thc biorthogonal basesin mIG A and mil I are completely independent of each other.
Consider the case in which :JlIGA C :DT and mil' C :D T *. In general, T(;}lIl/) is not asubspace of mill, find T*(mll') is not a subspace of mIG A. The operntors 7' and T* can beapproximated by projecting T(;}lI/) into mH' and 'l'*(mH
I) iuto :J1I/. This projection
process leads to a Ilumber of rectangulfir matrices of which the following are encounteredIlaturally:
(3.2)
where (.pa 1 and (WA I are the biorthogonal bases, and
Analogously, the boundary operators M and .v can be approximated using projectionsto yield
P,(M<l>,,) = L: M:Aw,J.A
where
(3.4)
(3.5)
In view of Green's formula 7'a'A can be also written in terms of 7'a *.1:
Ta'A = Ta*A + Ma'A + NaA. (3.6)
.Mixed projections. Primal-dual projection. The primal-dual projection, together withdual-primal projection to be discussed subsequently, give mi.xed approximation ofboundary-value problem (2.1). In primal-dual projection, approximate solut.ions U* =L: aa<l>" and V* = L: bA(.,A of (2.3) nre sought simultaneously by requiring
This yields
I1.(T*V* + kU* + f) = 0 in n, (3.7)
II 0
L: (Ta'/i - M"'/i)b,, + kL: a/lG"fJ + fa = O. (3.8)A fJ
Since, in general, mIG A and m/ are of different dimensions, (T" A - M " ") is a rectangularmatrix; and since (3.8) involves (G + If) unknowns with only G equations, no uniquesolution to (3.8) exists. The remaining H equations are provided by the dual-primalprojection.
Dual-primal projection. Here thc approximate solutions U* and V* are obtainedby requiring
PI(1'U* - V*) = 0 in n, (3.9)
258 J. N. REDDY AND J. T. ODEN
which upon simplification lead toG
L (T,,*'" + N,,'''')Ta"11
L brHf'" + j'" = 0
r(3.10)
where f'" = [gl , w"']ao •. Note that (3.10) involves H equations in (G + H) unknowns.Eqs. (3.8) and (3.10) combined lead to a determinate system for the approximate solutionsU* and V*. Solving (3.10) for br , we obtain
where If",r (Hlif)-l. Substitution of (3.11) into (3.8) leads to
L [(~"a" + P{I = 0"
(3.12)
where
(3.13)
Since T{I'''' - Mil'''' = TII*'" + N~·t:., clearly [(II" is symmetric. Eq. (3.12) determines thecoefficients aU, and hence leads to the approximate solution U*. The local form of (3.12)can be generated using uSlIlLi finite-clement approximutions (see [16]); techniques forconnecting elemcnts together to obtain the global model are well known (see [17]).
(4.1)
(4.2){N v - {!2 , 11I = 0 on a112 •{T*v + leu + I, ·al = 0 in 11,
4. Some basic properties of mixed finite-element approximations. The proof ofconvergence and the establishment of errol' estima.tes for conventional primal and dualfinite-element approximations follow easily from extrcmum properties of the associatedvariational principles, and concrete results are available for a number of different approx-ima.tions of this type (see, for example, [18-24]). \Vhile a great deal of numerical evidencehas accumulated on the utility of mixed models, rigorous studies of their advantages ordisadvantages as compared to traditional formulations have not heretofore been made.Indeed, the true utility of mixed models can only be determined when answers to anumber of basic questions concerning their intriIlSic propert.ies are resolved. The mainobjective of this section is to examine some of these questions for linear boundary-valueproblems of the type (2.3).
Let U and V denote the typical elements of ~JR{Jhand m.1I1 respectively, and U*
and V* denote the mixed finite-element (or Galerkin) approximations of the weaksolutions u* and v* of the boundary-value problem (2.1):
[Tu - v, v] = 0 in 11, [Mu - gl , v] = 0 on a111 ,
By construction of the mixed models (3.7) and (3.9), U* and V* satisfy the followingorthogonality relations:
[TU* - V*, VI = 0 in 11, [MU* - gl , V] = 0 on 0111 , (4.3)
{T*V* + kV* + I, Vj = 0 in 11, {NV* - 02' Vj = 0 on a112 . (4.4)
MIXED FINITE-ELEMENT APPROXIMATIONS 259
These relations can be expressed in a more general form [see (3.7) and (3.9)] by employingthe projection operators IThand PI of (3.1):
PI'l'U* - V* = 0 in n; P,MU* - P,gl = 0 on ani (4.5)
llh(T*V* + f) + kU* = 0 in n; llh(NV* - Y2) = 0 on an2 (4.6)
where advantage is taken of the fact PI V* = V* and IlhU* = U*.
THEOREM 4.1. Let (u*, v*) be the weak solution of (4.1) and (4.2) and let U*and V* be the corresponding mixed finite-element solutions satisfying (4.5) and (4.6).Then the following relations hold:
(P,'l'IlhT* + kI)V* + P,TIlhf = 0,
(lIhT*P,T + kI)U* + IIht = 0,
(4.7)
(4.8)
where I and I are identity operators.Proof: The relation (4.7) is obtained from (4.6) by eliminating U* and (4.8) is
obtained from (4.5) by eliminating V*. Indeed, operating with P,T on (4.6) and sub-stituting for P,TU* from (4.5) yields (4.7). Similarly, operating with [hT* on (4.5)and substituting for IlhT*V* from (4.7) lead to (4.8).
At first glance at (4.7) and (4.8), it may seem that the approximate solutions U*and V* are required to satisfy a greater degree of differentiability, equal to that ofexact solutions u* and v*. However, no extra smoothness of U* and V* is requiredsince projections of T*V* and 'I' Ih 'l'*V* are always continllolls, even if T*V* is piecewisecontinuous. Now define
(P,TIIS* + kI),
(IThT*P,T + kI).
Note that Qhl and Rih are mixed discrete approximations of
(4.9)
(4.10)
Q = (T*T + kI), R = (TT* + leI), (4.11)
respectively. For the sake of simplicity, define
e" = u* - U* = approximation error in u*,
ev = v* - V* = approximation error in v*,
Bv = u - Ilhu = interpolation error in u,
Hv = v - P,y = interpolation error in v,
Eu * = u* - IIhu* = interpolation error in u*,
Bv * = v* - Plv* = interpolation error in v*.
(4.12)
The following theorem establishes some fundamental properties of the approxinlationerrors eu and e.. in terms of the interpolation errors E" * and Ev *.
THEOREM 4.2. Let the conditions of Theorem 4.1 hold. Then the approximationerrors eu and e.. satisfy
(4.13)
260 J. N. REDDY AND J. T. ODEN
II,,7'*e. + ke. - kE,,* = 0,
Il"(T*e,, + ke,,) = 0,
P,(Te" - e,,) = 0,
PlMeu) lao. = 0,
IIl(Ne,,)laa, = O.
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
Proof: Proof of this theorelll is straightforward und can be found in (14].By using the relations (4.13) and (4.14), equations of the type in (4.7) and (4.8)
can be obtained for e" and ev .
COROl,(,ARY4.1. The approximation errors c" and ey satisfy the relations
(4.19)
5. Consistency of mixed variational approximations. The notion of consistencyof approximation of a. differential ('quation is fundamental to conventional methodsof numerical analysis. It is a measure of how weIl the problem is discretized Ilnd whetherthe discretizcd operators Qu and R,,, approach the exact operators Q and R, respectively,as the mesh parameters Ii and I approach zero. Consistency of a discrete model assuresthat the discretization errol' goes to zero as the associated mesh parameters approachzero. For primal and duul problems, the notion of consistency of variational approxima-tion is studies by Aubin [22], and diffcrs from the consistency of diffcrence approxima-tions defined in Isaacson and Keller 125]. III the prescnt analysis, notions of consistencywhich are appropriat.e for the problems considered here are introduced.
Suppose that it is required to obtain approximate solutions of (2.3). Variationalmethods of approximation involve seeking solutions to the weak problem (4.1) and(4.2). The discrete approximations of (4.1) and (4.2) are obtained by replacing II byII"u and v by P,v:
[T(II"u) - P,v, V]a; [J/(Il"u) - g, , VJaa.
(T*(P,v) + k(Ihu), Ula; [.\'(P,v) - y" Ulaa,
(5.1)
(5.2)
for every U E guo" and V E f1tIl'.
lVeakl!/ consistent approximatiolls. The discrete s~'stcm (5.1) and (5.2) shall berefcfI'f'd to as weakly cOllsistent with the variationlll equations (4.1) and (4.2) if
lim [T(II"u) - PlY' PlY'] = [Tu - v, VI]'".1-0
(5.3)
lim IT*(P,v) + kIIhu, II"u,1".1-0
lim [M(II"u), P,vtlaa.".'-0
(5.4)
(.5.5)
lim [N(P,v), II"uJ\all. = [Nv, u1lau •.".1-0
(5.6)
The qUllntities
A~I(tl, v) = II T*(P,v) + kIIAu, IIhu.1 - IT*v + ku, u.1\, (5.i)
From (5.13)-(5.16), it is clear that the lack-of-consistency Ahl(U, v) and B'h(U, v)depend on the interpolation errors. This fact is emphasized in Theorem 5.1.
THEOREM 5.1. Let the interpolation errors E" and E. be such that E" , E.. , TE" ,and T*E. approach 7.ero with hand t. Then the approximations (4.3) and (4.4) areweakly consistent with (4.1) and (4.2).
Proof: It is sufficient to show that the lack-of-consistNlc~' (5.7)-(5.10) are boundedby the interpolation errors. Indecd, by replacing UI by II/,u, and VI by PlY' in (5.7)and (5.8) (i.e. Bv = 0, B. = 0), we obtain
Ahl(U, v) = IIT*E. + kE,., UII, U E mta\
BIA(U, v) = I[TE" - E. , YJI, V E m.H'.
Now, by using the Schwarz inequality, (5.22) and (5.23) become
By comparing the approximate problems (4.7) and (4.8) with the strongly-posedboundary-value problem (2.1) and its adjoint, an alternate definition of consistencycan be given.
Strongly consistent approximation. The discrete system (5.1) and (5.2) shall bereferred to as strongly consistent with (2.3) if
lim Ehl(u) = lim IIIQh,(Ilhu) - IlhQu111 = 0, u E 'U, (5.30)A.I .....O ".1-0
lim F'h(V) = lim IIR,h(P,v) - PIRvl1 = 0, v E 'U. (5.31)h.I-O h,I-O
Here QhI , Q, RIA , and R are the operators defined in (4.0)-(4.11) and Eh,(u) and F'h(V)are lack-of-con8iste11cy functions. Here it must be pointed out that the discrete operatorsQ{" and R'h nrc associated with the mixed problem (2.3), and quite difTerent from thediscrete operators associated with the primal and dual problems.
It is convenient to define
Q = T*T, R = TT*.
(5.32)
(5.33)
TlIF~OHE~l5.2. Suppose that the interpolation errors Eu = u - Ilhu, and E. =v - PI V, and the operators Qah' and R'h are such that
lim T*ETII = 0,h-O
lim QhIE .. = 0,A,I_O
lim TET •• = 0,1-0
lim RuE. = 0,h.l-a
(5.34)
(5.35)
uniformly, where ETu = 7'u - P,Tu, and Ej". = T*v - Ilh(T*v). Then the approxima-tions (4.3) and (4.4) are strongly consistent with (4.1) and (4.2).
Now, by hypothesis, the right-hand sides of (5.36) and (5.37) vanish as hand l approachzero, imploying (5.10) and (5.31).
6. Stability, existence and uniqueness of mixed approximations. The growth ofround-off errors in the numerical solution of (4.3) and (4.4) is related to the notion ofstability. For arbitrary choices of the mesh parameters hand l, it may not be possibleto bound the round-off errors. This suggests that there be some criteria to select themesh parameters It and l so that the numerical scheme is stable. In this section theconcept of stability as applied to mixed approximation is discussed.
Guided by the form of the approximate equations (4.7) and (4.8), thc followingdcfinitions of stability are introduced:
Weak stability. The mixed approximation scheme in (4.7) and (4.8) shall bereferred to a weakly stable if positive constants 'Yl and µl exist such that
IIIQAt(II AU) III ~ 'Yl I I III AUIIIIIR/h(Plv)11 ~ µ1 liP/vii
U E 'U,
v E 'U,
(6.1)
(6.2)
where QA' and Ru are given by (4.9) and (4.10).The approximate scheme (4.5) and (4.6) suggest another definition of stability.Strong stability. The mixed approximation scheme (4.5) and (4.6) shaH be referred
to strongly stable if there exist positive constants 'Y. and µ. such that
Define
IIIIIAT*P,vlll ~ 'Y2 IIP/vllI IPI Til AUI I ~ µ. IIIIlAul1i
v E 'U,
u E 'U.
(6.3)
(6.4)
TAI* = IIA'I'·PI , 'I"A = P,TTIh
Now suppose that P,v = L~b~WA, and II~u = L" a" 'P" . Then
ITAT*P,v = 1:bA L !T*wA''P"I'P'' = L LT,,*~'P"bA6 0' a f1
and
IlIIIAT*p/vIW = L L bAbrT" *AG"fJTfJ*r,R.fJ A. r
where T R·A is given by (3.3). Also,
IIP/TIIAuW = L L a"afJTR·AHAfTfJ·r".fJ A. r
where T R • A, HAf and G"fJ arc defined in (3.3). Similarly,
QuIhu = IIAT*P,TIhu + kIl1u
1: 1: a"[T'P" ,WA] IT*wA, 'PfJIl + k La" 'P"
".fJ A
(6.5)
(6.6)
(6.7)
(&;8)
(6.9)
264
and
J. N. REDDY AND J. T. aDEN
R,AP,V = L (L Ta*!l.(;"PTIJ·r + kllAr)bo1wr .01. r ".P
(6.10)
Thus the stability conditions (6.1) and (6.2) are equivalent to
and the stahility conditions (6.3) and (6.4) are equivalent. to
"" ("" ']' *tJ.G"/lT *1' _ 2HH)b b > 0£..... £..... " T fJ "Y2 01 r _ ,01. r ".fJ
It, is convenient to dpfine the following matrices:
H = [BAr], G = [Ga/l], M = [Ta·4], N = [Ta*tJ.].
Then (6.11)-(6.14) imply the following fundamental result.THEOREM6.1. Let the following matrices be positive definito:
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
MTH-1N + (k - "YI)G,
NTG-1M + (Ie - µI)H,
k > 0,
k > O.
(6.16)
(6.17)
Then the mixed approximations (4.7) and (4.8) are weakly stable. :\'[oreover, if thematrices
N7'G -IN - "Y2HMTH-1M - µ22G
(H. IS)(6.19)
are positive definite, the mixed approximations (4.7) and (4.8) arc strongly stable.Since G and H are the fundamental (Gram) matrices, they are always positive
definite. Consequently, from (6.16) and (6.17) it is clear that k has the stabilizing effecton the s~'stcm.
Existence and uniqueness of solutiolls. The stability conditions (6.1)-(6.4) can beused to establish the existence and uniqueness proofs for approximate solutions of (4.5)and (4.6). We shall prove hero the existence and uniqueness in the ('llseof weak stability.
THEoRf:~16.2. Let the mixed approximation (4.7) and (4.8) be weakly stable inthe sense of (6.1) and (6.2). Then the approximate scheme (5.8) is uniquely solvable.Moreover, if the operator TIA = P,TTIh is bounded above
c = constant (6.20)
then the approximate scheme (4.7) is uniquely solvable.Proof: From (4.8) and the assumed stabilit.y condition (6.1),
Thus, Qhl is bounded and hence (see Naylor and Sell [26, p. 244]) invertible. This impliesthat (4.8) has at least one solution. Note from (6.21) that (4.8) has only the trivialsolution if f is identically zero. This proves unique solvability of (4.8).
MLXED FINITE-ELEMENT APPROXIl\IATIONS
To prove unique solvability of (4.7) note from (6.2) and (6.20) that
11II111~ c IlPlTfhfl1 = c IIRIAV*II ~ ell I IIV*IIThis completes proof of the theorem.
265
(6.22)
(7.1)
7. Convergence of mixed finite-element solutions. Thus far the notions of con-sistency and stability of mixed approximations are discussed. Now the more importantissue of convergence is to be resolved based on the knowledge of previous sections.Convergence proof based on the assumption of stability will be given. A more directproof of cOIlvergencc, without using thl! stability concept, is given in [27].
TIIEOUE~1 7.1 (Convergence Theorcm I). The mixed finite-element approximations(4.3) and (4.4) are convergl'nt; that is, IlIe.1I1 and 1Ie.1I approach zero as " and l tendto zero in some manner, if the interpolation errors E,,*, b'y*, TE.* and T*Ey* vanishas II and I approach zero and the following sets of conditions hold:
Case k = O. The approximate scheme is strongly stable in the scnse of (6.~) and(6.4).
Case k > O. The approximate scheme is weakly stable in the sense of (6.1) and(6.2) and the operators QloI and Ru of (5.32) arc continuous in the t.opologies inducedby the norms 111·111 and 11·11 respectively.
Proof: Case k = O. Using the triangular inequality,
II Ie" III = Illu* - IlAu* + lIAu* - U* III,~ IIIE.*III + IIITI.u* - U*III,
and
lIeyli ~ IIE.*II + IlPlv* - V*II·In view of stability conditions (7.3) and (7.4),
Eqs. (7.6) and (7.7) prove convergence of e, and e" .COROI,LARY7.1. Let the assumptions of Theorem 7.1 hold. Then the mixed
approximations (4.3) ancl (4.4) are weakly consistent if Ie = 0, and strongly consistentfor k > O.
The proof of this corollary follows directly from Theorem 5.1. In conventionalmethods of numerical analysis (for example, finite-differences), for consistent schemesstability implies convergence. 'With the particular definitions of consistenc~' and stabilitygiven here for mixed finite-element schemes, it seems such conclusions cannot be drawn.However, for consistent mixed finite-element schemes, stability implies the followinginequalities:
THEOHEl\17.2. Let the mixed approximiltion scheme (4.3) and (4.4) be weaklyconsistent for k = 0 ancl strongly consistent for k > O. Then strong stability impliesconvergence of IIlculll alld lIevll for Ie = 0, and weak stability implies convergence ofIlIe ..111for k > O.
Proof: Consider the case k = O. From the strong stability condition (6.3) and (6.4),
Then the mixed approximations (4.3) and (4.4) ~re convergent for all k ~ 0, providedthe interpolation errors Eu*, E.*, TEv*, and T*Ey* vanish as h and I approach zero.
Proof: Since V* and V* satisfy (4.3) and (4.4), relations (4.13)-(4.18) hold fork ~ O. Now suppose that k = O. Then (4.15) becomes
No\\' suppose that k > O. From (4.13) and (4.13) and (4.1.'5), it can be shown that[P,Te" , Te.] + kle" I e,,1 = kle" , E ..*l + [P,Te. , TE,,*] + [E ..*, TB.*] - [E ..*, Te..]and
+ E IITe"W + L (1IEv*1I + IITE.*1!)2 + IIE..*II·IITE.*II
whcro 0 and E fire arbitrary positive constants. Choose 0 = i, and E = -y/2.
Let
Then
c. = min (-y, 1); Dz = 1 + 1/-y. (7.16)
117'e.1I+ k IlIe,,11I~ «2D.jC2)r/2(k IIIE.*III + IIEv*" + IITE.*ID. (7.17)
Similarly, from (4.14) and (4.16), the following result can be obtained:
IIIT*e.1I1+ Ie Ile..1I ~ «2Da/Ca»1/2(k IIEy*1I+ k IIIE,,*111+ IIIT*E.*11D (7.18)
where
Ca = min (µ, 1); D = 1 + 1/µ (7.19)
Thus, IIle.lIl, IITe"ll, 1Ie..1I,and IIIT*e.111approach zero as It and l tend to zero. Thiscompletes the proof of the theorem.
It must be ohserved that Theorem 7.3 assures convergence of not only e. and e..but also of Te. and T*e.. . This indicates that Te. and T*e.. converge at thc same rateas TE.* and TE..*, respectively. Intuitively, the errors e. and e.. may approach zeroat the rate of E.* and b'v *, respectively. In that case, faster convcrgence of e. and e. isl'stablished by Theorem 7.3.
8. Some error estimates. Consider the casc in which
'U = W~k(n), 'l) = w."(n)
where W 2k(n) is the Hilbcrt space of order k, and Ict
(8.1)
MIXED FINITE-ELEMENT APPROXIMATIONS 269
<p. = the space of polynomial of degree 8 on 11C En; h, p = finite-elementmesh parnmeters (see [28]) of approximations U*(x) E ;JUGA of u*(x)j (8.2)l, a = finitc-element mesh parameters of approximations V*(x) E m./of v*(x).
Let the inner products in 'U = JV2k(l1) and'U = W2 ·(11) be defined by
lUi ,u21 = (UI ,U-2)w,.( 0) = f 1: DOu.D·U2 dx, (8.3)o loiS.
Assume that
s < k, r < q. (8.5)
In most of the applications the operators T and T* are differential operators of the form
T(M)* = :E (-l)IOID"(aa(x»101 S ..
(8.6)
where m < 8, 1'.
Now assume that there exist interpolants 0 = ll.u and V = PlY
IIIT.u - ullll'.~lo) :$ (: lulw.>+'(I)W+1/p"),
IIPlv - vllw.M(O) :$ jj Ivlw".'(O)(l'+1/a"'),
(8.7)
(8.8)
where ITAand PI are projection operators such that ITAu= u and Plv = v for all u E <P,and v E <P r , and C and jj are constants independent of the mesh parameters. Inter-polation formulas of the type (8.7) and (8.8) are derived by Ciarlet and Raviant [28, 29].If the coefficients a satisfy the condition
then
(8.9)
m < k (8.10)
for every u E W2·(I1). :\Ioreover, if the estimate (8.7) holds, then
h••1(..) - I IliT KIIL.(o) < C -.. U w.···CO)- P
holds for any u E JV~"I(11), 111 < S + 1.Similar results call be obtained for T*:
It can bc shown, in a similar way, that ,"IIlTc")Evl\w ••( 0) :$ C* ; .... lullV ..•. ( 0) ,
l'+1IlT",*E"llw •• (O) :$ D* ak'" Ivlw.,.·cO)
(8.11)
(8.12)
(8.13)
(8.14)
270 J. N. REDDY AND .J. T. ODEN
(8.16)
(8.15)
Now error estimates for mixed finite-element solutions can be derived using (8.11)-(8.14).THEolmM 8.1. Consider a mixed finite-element approximation based on poly-
nomial bases for which the relations (8.7) and (8.8) hold for the spaces defined in (8.5).Then the following error estimates hold, if the conditions of Theorem 7.1 arc satisfied.
Case Ie = o.
lleullw,.(O) S (Cp-k + ~~p-m-·)h·+1 lu*lw"+'(Il)
+ ] (1-) -. + D* -m-t) I *1- a - a v II' " + 0( 11) ,µ, 1',
Ilevllll',.+, (Ill S (.Da-. + ~,*a-"-k)r+ 1 Iv "'I IV " +. (Ill •
Case k > O.
(CJI) h·+J D* 1'+1
Ilevllw"(1l) S 1 + - C-" lu*llI',d,(1l1 + - hm Iv*IIl"'+'(1l1 I'YI p 'YI a
(8.17)
(8.18)
Proof: The proof is straightforward. These estimates can be derived directlyfrom (7.3), (7.4), (7.6) and (7.7) with interpolation enol'S (8.7), (8.8), (8.13), and (8.14).
It is clear from above estimates that the errors depend on both sets of meshparameters.
COROI,LARY8.1. Let the condition.c; of Theorem 8.1 hold, and let p = v1h anda = V2l. Then
Case k = O.
Ilevllw,'IO) S vl(Ch-k + C::I h-"-·)hHI lu*lw"+'(Il)
The rates of convergence from (8.25) and (8.21i) for Ttl and 7'*v are, respectively(for k > 0),
t1 = min (r + 1 - q, s + 1 - q - m, S + 1 - k),
E = min (r + 1 - q, r + 1 - k - 111, S + 1 - k).(8.29)
The convergence rates for Tev and T*e. seem to be of the same order as compared tothose of eu and e. in (8.24). Thus, the error estimates obtained from Theorem 7.3 aresharper.
9. Numerical results. There exists ample literature on numcrical analysis of mixedfinite-element models. For examplc, Herrmann [1, 30] and lIellan [2] have developedmixed plate bending elements, and later Backlund [6] (sec also Conner [31] and Visser[4]) used these elements in the analysis of elastic and elastoplastic plates in bending.Dunham and Pister [5] employed the Hellinger-Reissner (mixed) variational principleto construct mi."ed finite-clement models of linear elastic problems. It was observedthat the mixed models are particularly effective in capturing steep stress or displacementgradients that can occur near singularities in boundary-value problems. In recent timesthere has appeared a vast literature on the closely related idea of the hybrid finite-element method [8, 9, 10] applied to stress concentration problems. In all these works,numerical examples have been presented \\ith extremely good results; however, thesedo not contain any information 011 the behavior of the error (in energy).
The primary purpose of the examples presented here is to demonstrate, numerically,that thc mixed models yield higher accuracies for certain quantities (e.g., stresses),
272 J. N. REDDY AND J. T. ODEN
and to give precise rates of convergence for the mixed finite-element solutions. Inparticular, the error in energy norm (or Lz-norm) is computed in each exumple to finc!the rates of convergence. For simplicity, only one-dimensional, second- and fomth-orc!erequations are considered here.
1. Second-order differential equation. Consider the boundary-value problmn
o < .r < 1 (9.1)
subjected to the boundary conditions u(O) = u'(l) = O. Clearly, (9.1) is of the generalform (2.1) with T and T* given by
T = -T* = d/dx, J[ = N = 1, U, = (/2 = 0 (9.2)
and the inner products I· , .I and [. , .] are defined
(9.3)
The following sets of basis functions are selected for the problem at hand:
~I(X) = 1 - (x/h),
~a(x) = (x/h) - (a - 2),
= a - (x/h),
(a - 2)h 5 X 5 (a - l)h (a = 2, ... ,N. - 1),
(a - l)h 5 x 5 ah
~.v(x) = ~ - (N, - 2), (N, - 2)h 5 x 5 (N. - 1)11, (9.4)
and similar expressions for wA, A = J, .... M, are taken with h rr.placcd by I. Here
N, is the number of nodes (or (N, - 1) is the number of clements) in approximationby tpS and ill. is the the number of nodes in approximation by ws; i.e., It = l/(N. - 1)and I = l/(M. - 1). Let the ratio of the meshes be denoted by II = h/l. The fundamentalmatrices aa~and J-lH are given by
where
[G ,,~] = h[G], [HAl'] = I[G] (9.5)
1[G] = {3
2 1
"'''1 4 1
"'''''''1 4: 1~
o
141
"'''''''o 1 4 1
"''''1 2.
MIXED FINITE-ELEMENT APPROXIMATIONS
Also, f" and r/" fire given by
1
273
6
12
6(p - 1)
6(N, - 2)
3N, -4
2l I
g.1 =?"',
2
:2
1
(9.6)
Thc matrices (1',,':1. - M,,·J.) and (T,,*J. + N,,·oJ.) of (3.8) and (3.10) arc given inTable I. Notl' that (1',,':1. - M" '.1) = (T" *!. + .V" ':1.). Here it is assumcd that II is aninteger. WIIl~nll is not fin integer, it is not possible to compute the matrices (7'" ..1 - ill" '.1)and ('I", J. * + iV" ':1.) for arbitrary X, and ill, . l\'iixed tinite-element 50lutions 1l* and v*,for ditTefl'nt vahws of II, arc obtained and compared with the exact :,;olutions (SllC Figs.
TABLE I. The rnlltrix (7'".J. - M,,·J.) ror any inte~er vllllle or v.
(1',,·J. _ M .. 'J.)(.\1 ,XN,)
a=1I a=II+1 211 211+1 :311 311+1I I I I I II I I I I II I I I I I
It Ii h It0 0 0 02l
.. . .. . ...l l 2l
-It -h -Ii 0 It It It 0 02ll"'l l... - 0 ...
l 2l
0 0 0-It -It -h It h.. . -- ... -0 1... 2l 021 I l I
-h It h h 0- 0 l...
l '2ll
0-h -h -It 0 It2l l .... l l
0 0 0 0 -It -h.... --2l l
N,
o
o
o
o
It It
l 2l
-It 11l 21
9.1 find 1).2). It must be noticed that the mixed solutions are less stable and more in-accUI'ate as tIw mesh ratio II = lilt increases. This can bc explained in vicw of the stabilityconditions (li.l)-(uA). A close examination of the matt'ix in Table I reveals that as
274 J. N. REDDY AND J. T. ODEN
v increases the sum of the off-diagonal terms inel'enses, and consequently the matrix]( ,,{J of (3.13) becomes ill-conditioned. }loreover, the error on the boundary is moresensitive to the mesh mtio, with the error increasing with the mpsh ratio. For v = 1,the solutions U* and V* are plotted against the primal and exact solutions in Fig. 10.3and against the dual and exact solutions in Fig. !)A. For v = 1 the matrix in Table Itakes the simple form
1 -1 0
'" '"1 0 -1 0
'" '" '"010
o.",o -1
'" '" '"1 0-1
'" '"o 1 1
Thl' broken linc ill Fig. 9.3 is the solution V = dU* /dx obtained by differentiating U*;the brokcn curve ill Fig. 9.4 is the solution U = 101 V* elx obtained by integrating V*.Thus, the solution V, in the casc of the primnl problem, is discontinuous and the solu-tion U, in the case of the dual problem, is in error.
0.0
0.1
0.2
0.3
exacto 0" "'.1, N '" 6
-'- "'" 2, N = 6A-.--b, " '" 3, N = 6__ \I '" 2, N = 10.4--~ " '" 3, N '" 12
0.4 0.5 0.6 0.7 0.8 0.9 1.0
FIG. 9.1. :\lixed finite-element solutions ufoI' valious values of the mesh ratio ...
MIXED FINITE-ELEMENT APPROXIMATIONS 275
exact
0.90.80.70.60.50.4
6
6
6
6
0.3
4, N
1, N = 6
3, N
2, N
5, N
v =
0.2
o V_.~o---tJ v
0.10.0
0.2
0.1
0.3
v
FIG. 9.2. :\Iixed finitc-elemcnt solutions v for various values of the mesh mtios v.
o mixed finite element solution
• • primal solution
exact solution
x1.00.90.80.7
o
0.60.50.40.30.20.1
0.1
0.3
0.2
FlO. 9.3. Comparison of mixed and primal solutiol1.'l with exact solutions.
276 J. N. REDDY AND J. T. aDEN
• • dual solution
o mixed finite element solution
exact solution
x1.00.90.7
o
0.60.50.4
u
v
0.30.20.1
0.3
0.1
0.2
FII;. 9.4. Comparison of mixed and dU1l1solutions with exact sohlt,iolUl.
The error ill energy is computed for v = 1 case and plotted llgainst the mesh size h.In this case, where same basis (or trial) functions (linear polynomials) arc cmplo~'edto approximate u and v, the rntes of eonvergcnCl)for U* us well as for V* is 2. In Fig. 9. I,the value of k is ]. The Sllllll' problem is solved with k = 0, and same rates of convergenceare obtained in this case also (with the Sllmebasic functions).
2. Fourth-order differential equatio/l. Consider the fourth-order equation
(cl'u/d./) + .r2 = 0, o ~,L' ~ I, (9.7)
u(O) = u(1) = 0; (d2u/d:r2)(O) = «(flt/d.l)(l) = o.In this case the basis functions are cubic polynomials:
"'1° = 1 - 3(~r + 2(~r 0 ~ x ~ 11,l/'aD = 3[~ - (a - 2)r - 2[~(a - 2)T, (a - 2)h ~ X ~ (a - l)h
= 1- 3[~- (a - 1)r + 2[~- (a - l)T, (a - l)h ~ :r ~ all
(a = 2,3, ... ,N. - 1)
(N, - 2)h ~ x ~ (N. - 1)It ,
1
"'a
[ ( )2 ()3J1 X X X"'1 = It h - 2 T, + h ' 0 ~ X ~ h (9.8a)
= h[{~ - (a - 1)}- 2{~- (a - 1)r + {~- (a - 1)r] , (a - l)h ~ ;1' ~ all
h[ {X - (a - 2)Y - {~ - (a - 2)YJ, (a - 2)h ~ X ~ (a - 1)11
MIXED FIXITE-ELEJ\1ENT APPROXIl\lA nONS 277
'PNI = h[{f, - (N, - 2)f - {~ - (N, - 2)2}] ,
(N, - 2)h ::; :r ::; (N, - l)h. (9.8b)
The fundllnH'lllal matrices G"fj and HH in tills case (for J.I = 1) are given b~rIt
[Ga8] = 420
156 54 2'2h -13h~ ~ ~ ~
54 312 54 13h 0 -13h~ ~
54
54~
2'2h 13h~ ~
-13h 0 131t~
-131t
312
~54
54
156
13h
13h~
4~ -3~~ ~
-3~ ~2 -3~
~-~
o -131t~
13h -22h
-1311 0 13h
~ ~-1311 -22h
-3h2 8h2 -:Ht2
~ ~-3112 4h2
= [IlH).The matricl's (To*:' + lV"a':'), and fa are computed to be
(T *:. + N 'A) _ -.La 0 - 30h2
3611 -36h 33112 3h2
~ ~ ~ ~-36h 72h -3611 -3112 0 3h2
~ . ~-36h -3112
(9.9)
- 3611 72h - 3611~ ~
-36h 36h
- :3h2 0 3h2
~ ~-3h2 -33h2
3h2 -3lt2
33h? -3lt2
~ ~3lt2 0 -3lt2
~
3h2
~
~o -3h2
~3h2 -33h2
4h' -h'~ ~
_II' 8h' -h'
~_ha /3-I
~_It' 8h' -h'
~ ~_II' 4h3
(9.10)
278 J. N. REDDY AND J. T. ODEN
and
(9.11)
whereh"
f~~ 15 'h
3( 2 )30 30{j - 60{j + 34 ,
~~ (15N,2 - 39N, + 26),
{j = 1,
{j = 2,3, ... ,N, - 1,
{j = N, ,
f' h4~ = 60 '
~: (-15{j2 + 62{j - (2),
:~ (15N,2 - 39N, + 25),
{j = 1,
{j = 2,3, ... ,N, - 1,
{j = N, .
The mixed finite-element solutions U* and y* arc plotted against the exact solu-tions in Fig. 9.5. The rates of convergence in this case, where the same basis (cubic)
- eXAct solut.ion
1.0 x0.80.6
• mixed finite clementsolution
0.40.2
0.4
0.2
0.3
0.1
- (!xact solution
omixed finite clementsolution
l.0
2.0
3.0
4.0
0.2 0.4 0.6 0.8 1.0 x
FIG. 9.5. Comparison of mixed fillite-element solut.ions with exact. solutions.
MLXED FINITE-ELEMENT APPROXIl\lATIONS 279
functions arc (~mployed, are 4. It is also noted that the first dcrivatives of U* and V*are approximated vcry closely to the exact derivatives.
Acknowledgement. The support of this work by the U. S. Air Forcc Office ofScientific Hescarch under Contract F44620-69-C-0124 to the University of Alabama inHuntsvillc is gratefully acknowledged. We arc also grateful for support of t.hc Engineering~Iechanics Division of the ASE/EM Department of the University of Tcxas at Austin.
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