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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 8 (1976) 1-16© NORTH·HOLLAND PUULlSHING COMPANY
A THEORY OF DISCONTINUOUS FINITE ELEMENT GALERKIN APPROXIMATIONSOF SHOCK WAVES IN NONLINEAR ELASTIC SOLIDS
PART I: VARIATIONAL THEORY
L.c. WELLFORD. Jr. and J.T. ODENTexas Inslitllte for Compl/tational Mechallics, The University of Texas at AI/still. AlIStill, Texas
Received 25 February 1975
COll/ellts:
I. Introduction2. Some notation3. Mechanical preliminaries4. Regularity theory
I. Introduction
I237
5. Galerkin models for waves willi multiple shocks6. Some fundamental approximation theory lemmasReferences
II1216
In the early 1950's von Neumann [ II conjectured that the problem of shock propagationthrough a given domain might be attacked by considering it as a free boundary value problem forthe partial differential equations involvcd - that is. a nonlinear boundary valuc problem in whichthe boundary varied with time. and its location at any time was an unknown. Then the shockpropagation problem could be characterized by a system of partial differential equations definedon certain shockless domains, together with a collection of boundary conditions (jump conditions)valid at the shock surfaces which form the interfaces of these domains.
For several reasons. not the least of which was a lack of a complete theory of free boundaryvalue problems. thc mathematical theory of shock waves has developed along quite different lines.The thrust of basic work in this area has been toward a global theory that would accomodateshocks, and this has necessarily led to viewing the shock solution in a distributional setting. Thatis. one seeks from the outset generalized solutions of certain nonlinear hyperbolic equations. For
! a recent account of the state of the subject. see Lax [2].Our interest here in these interpretations is from the viewpoint of approximation: i.e. what
specific types of approximation schemes do these contrasting philosophies suggest? The answer isfairly clear. The distributional theory forms the backbone of virtually all variational methods ofapproximation. In particular. it is well known that the Galerkin concepts. when used in conj unc-tion with finite element interpolants. represent one of the most powerful methods of approxima-tion of elliptic and parabolic equations. Among their attractive features arc their accuracy. condi-tioning. and the case with which they can be applied to very irregular domains all which mixedboundary conditions are prescribed. Unfortunately, Galerkin schemes have not been consideredseriously for shock wave calculations. Increased continuity requirements are implied in the
2 L. C. Wellford. Jr. 01/(1J. T Odell. A theory of diSCOlI/illllOUS fillite element Galerkill approximatiolls - Part I
Galerkin formulation. and. consequently. they endow the shock wave approximation with an un-realistic degree of smoothness. In other words, they tend to smooth solutions which are actuallydiscontinuous, and in transient problems this often leads to an unacceptable amount of artificialdissipation. On thc othcr hancl. the free boundary conjectures of von Neumann have been thebasis of the shock-fitting schemes that are popular in gas dynamics calculations. There, however.questions of irregular domains and complicated boundary conditions are rarely important, and theprincipal aim is to dcpict accurately the strength of the shock. It would seem that a scheme thatadopted the advantages of both of these classes of methods would be the most appropriate forgeneral shock problems. It is toward the development of a theory for schemes of this latter typethat the present paper is directed.
In this paper wc cxtend the Galerkin method to include shape functions with built-in discon-tinuities. We accomplish this by using two types of finite element interpolating functions. Oneconsists of the conventional piecewise polynomial approximations for representations overshockless domains: the second involves a set of local trial functions with prescribed discontinuitiesthat are introduced in a collection of elements forming a boundary layer around the shock surface.By using this device, we prove conclusively that we are able to retain the desirable properties ofconventional Galerkin techniques (that is accuracy. geometrical independence. stability) and atthe same time modcl cffectively the strength of the shock. its propagation and decay. For sim-plicity. we confine our attention to shock propagation in one-dimensional domains.
Following this introduction. we present preliminary remarks on notation and pose the physicalproblem to be analyzed. Wc introduce certainmcchanical constitutive assumptions involving thefirst Piola-Kirchhoff strcss tensor for a class of isotropic hyperelastic materials, and we discussthe implications of the assumptions as far as the regularity of the exact solution is concerned. Wethen dcvelop Galerkin models for wave propagation using variational equations valid for multipleshocks. We ncxt dcvclop certain approximation theory results consistent with the nonlinearmatcrial characterizations. We then discuss the accuracy and convergence of the semidiscrete ap-proximation. Finally. we develop a priori estimates and present certain stability criteria for a fullydiscretized schemc.
2. Some notation
Let I be a bounded open subset of R. We denote by W;' (I) the Sobolev space of functions withgeneralized derivatives of order E;; 111 in Lp(l). 11';(1) coincides with the space obtained by com-pleting the ("'I (I) functions in the Sobolev norms
{ak }IIP
lIullwm = ~ II~IIPP (I) k .;; //I ax k L p(I)
IE;;p<oo. (2.1 )
The completion of the space C~I(I) of m-times differentiable functions with compact support in Iin this norm is denoted ~tJ~'(I). We also use the notation
w~'(I) = Hili (I) . (2.2)
It is well-known that thc spaccs 11';'(1) are complete and that the spaces HI/l(l) are Hilbert spaces
I..C. Wellford, Jr. and J. T. Oden, A theory ofdiscontinllol/s finite elemelll Galerkin approximations - Part I 3
with inncr products
(( ' •• ))/'2(1) is the L2 inner product.
(2.3a)
(2.3b)
The temporal behavior of a function u(X. t) may also be in a space Lq(O. n: therefore we write
(2.4)
Thus. L2(0. T: 11'"(I» denotes the spacc of bivariate functions which are measurable and sq uare-integrable in the tcmporal variable ovcr (0. ncR and are in H"'(l) for each t E (0. n. Likewise.we use the notation
(2.5)
Since we shall usually be concerned with solutions over somc specific time intcrval (0, 1'), we shallsometimes omit thc interval designation and simply write Liw~n (I)) for Lp(O. T: W;'(I».
Now suppose Z is an arbitrary point of I and K, LeI are open sets bounding the point Z. Then.as special notations. we let (·,.)z denote the scalar product of two functions evaluated at Z, andwe define a "boundary norm" associated with Z by
1IIIIIIIZ{K,L) = sup III(X)!.XEKU/.
Other notation will be defined where it first appears in subsequent sections.
3. Mechanical preliminaries
(2.6)
We wish to consider wave propagation in a class of materials in which dissipativc effects areinherently small and for which a fairly fine resolution of specific physical charactcristics is possible.To keep kinematical considerations simple. we confine our attention to bodies which are essen-tially one-dimensional but in which smooth initial data may. in time. lend to thc formation andpropagation of shocks.
In particular wc wish to consider two specific classes of elasto-dynamic problcms:I. For compressible materials, the motion of a finite slab of hyperelastic material
(3.1 )
in plane strain. on which initial displacements. velocities, and body fort.:es have been prescribed
4 L.c. WeI/ford. Jr. and J. T. Odefl. A theory of disCOllliflllOIlS finite element Galerkin approximaTiofls - Part 1
which arc uniform in Y and Z. but vary with X. We includc in this class the half-space
1 = {X : X = (X, Y. Z) E R3. X> O} (3.2)
wherein uniform initial velocities and displacements may be prcscribed over the plane X = 0 att = O.
2. For incompressible materials. the plane axisymmetric longitudinal motion of a thin cylin-drical rod of length Lo composed of a hypcrelastic material in which the principal stress compo-nents normal to the axis of the rod are taken to be zero. In this case
1= {X: XER, O~ X~ Lo}' (3.3)
We now consider construction of the elastodynamics problem of class I or 2 for the case inwhich a finite number N - I of shock waves may exist at any time I;;;' 0 in the material. We con-tinue to denotc by 1 a possibly unbounded sct of matcrial particles equivalent to an open subsetof R. Let {Yk}~=O = Q denote a set of N + I real valued functions from [0, T] c R into R suchthat for each IE [0. T]. Q is a partition of I: i.e. if 0 = inf {f}. Lo = sup {f} ~ 00, then
At IE [0, T] wc denote by Ji(f) thc open sets
JiU) = (Yi-1(t). Yi(t)). I~i ~ N.
and
(3.4)
N
I/Q = U Jii= ]
YIEIO.T].
In particular Q subdivides J into a number of disjoint open sets of particles JI at each t E [0, T].These sets shall correspond to shockless subdomains of J at time t. and the functions Yj(t), whichtogether with Ui Ji describe the closure of U, Ji, represent particles at which there may existshocks - i.e. surfaces of discontinuity in thc displacement gradients ux(X, t) and the velocityli(X. t).
With these conventions in mind. the basic physical conservation laws take on the following form:(iJ balance of linear momentum
N N-I
E f[i.. (pli) - PI] dX + E P ViRilD y. = a(Lo_ t) - a(O. t) .i= I 01 i=t IJ.
I
(iiJ conserJlation of energy
(3.5)
(3.6)
L. C. Wellford. Jr. and J. T. Odell, A theory of discontinl/ous fillite e1emellt Galerkill approximations - Part J 5
{iii} Ille Clal/sills- Dllllelll il/£'ql/alilY
..
N N-I
L; f~dX+ ~ Vi[~h.~ {qIO}I~:~Oi=1 J. at 1=1 I
1
Here Vi is the intrinsic speed of the ith wave
Vj(t) = d Y;U)dt
n '" h· denotes thc jump in any field quantity'" at the surface YiI
(3.7)
(3.8)
(3.9)
II = /leX. t). a(X. t) and 1·I(X.I) arc the displacement. Piola-Kirchhoff stress and velocity at X attime I, f is the body force per unit mass. e is the internal energy per unit of referencc volume,q(X. r) is the heat flux. ~ is the entropy, and 0 is the absolute temperature. We also use the ob-vious notation
(3.10)
In classical formulations of wave problems in mechanics. it is argucd that within each open setJi the integrands in (3.5). (3.6). and (3.7) are sufficiently smooth to give meaning to a pointwisestatement of the wal'e problem: that is. together with ccrtain bounclary- and initial-conditions, it
, is asserted that the set {It, a, e, q, to} is such that
pii - ax = pf
v (X. 1) E JI(t) X (0. T] . I~ i~ N:
(3.lla)
(3.11 b)
(3.Ilc)
(3. 12a)
on Yi(t) X (0, n. I ~ i ~ N -I . (3.12b)
When the initial data arc smooth. it is possible to develop a priori estimates of "breakdowntimes" in which a smooth signal develops into a shock. Such estimates have bcen describcd by Laxand othcrs. In more gencral situations. it is necessary to calculate the charactcristic surfaccs of thcsolution to dcterminc when and whcrc they coalesce: the coalescence of characteristic surfaces.of coursc. indicates the shock front. In practical calculations, less sophisticated methods can beused. For example. one can estimate the magnitude of the gradients within each finite element by
6 L.C. Wellford, Jr. alld J. T. Odell. A theory of discominllol/S finite element Galerkill approximatiolls - Part I
simple differencing. When these gradients exceed a large preassigned number, it can be assumedthat a shock has developed.
The character of the solutions to (3.11 )-(3.12) (and indeed the existence of solutions to thisproblem) depends upon the data and thc properties of the function a. Initially we discuss thecompressible plane strain case (class I). In this case the only nonzero displacement gradientoccurs in the X direction: thus a = a(ux)' We restrict ourselves to the class of materials in whicha(u x) corresponds to an isotropic. hyperclastic material. Then a is derivable from a potentialfunction W(IlX) which represents the strain energy per unit volume Vo of the body when itoccupies the refercnce configuration. Then we write
or a(X) = a W(X). ax . (3.13)
where
X(X.t) = I +IlX(X.t). (3.14)
Although the functional forms a(lIx) and a(X) for W(ux) and W(X) are of course diffcrent, weshall use the same symbol a (or hi) for each.
On the other hand. in the problem of longitudinal motion of a rod of incompressible hyper-elastic material (class 2) nonzero displacement gradients occur in the axial and transverse direc-tions. As a measurc of the axial strain we use the stretch X defined in (3.14). As a measure of thetransverse strain we use the transverse stretch µ where µ = X-1/2, Then instead of (3.13),
a(X. µ(X)) = a wax ' (3.15)
where II is the hydrostatic pressure which is to be determined from the transverse stress conditiona w/aµ = 0, i.e.
h = _ ~ a/l/(X,µ)4µ3X2 aµ
It can be shown (see [3] for the demonstration) that for either class I or class 2p-l
a(X) = 6 CkXk .k=r
(3.16)
(3.17)
where,. = 0 for class I and r < 0 for class 2, rand p being integers.From this point on we will concentrate on materials of class I. However. we realize that the
choice is rather arbitrary, and in fact the techniques to be developed in this paper are with minorchanges valid for materials of class 2.
In [31 lemmas establishing strongly monotone and continuity properties for materials of class Iwere proved. We initially give the strongly monotone results:
Theorem 3. J (strongly monotone property). Consider the stress a(X) of (3.17) for cases in
L. C. Wellford. Jr. and J. T. Oden, A theory of discomi/lllOlIS finite elemelll Galerkin appruximatiollS - Part I 7
which Cp_1 > O. Ck = O. k even; Ck ~ 0, k odd. 0,,;;;k ,,;;;p -I. Then there exists a positive constant'Y such that
(a(~) - a(X), X - X) ~ 'YIIX- XII~p(l) ,
whcre (.,.) denotes thc duality pairing between Lp(l) and Lq(/). and q = p/(p-l) .•
Corollary 3. J. Let thc conditions of theorem 3.1 hold. Then
(3.18)
(a(ux) - (ux)' Ux -lix)~'Yllu-UII~vb(I) . (3.19)
where U. Ii E ~V~<n,and (', ')denotes the duality pairing betwecn Lp(I) amI Lq(l)' where q = p/(P-l) .•
Now we give results establishing the continuity property of a:
Theorem 3.2. Let '/I.. XE L/1). s ~ p. Then
II a(~) - a(X) IILq(l) ,,;;;g(~. X) II X - '/I. lII-p(J) •
where q = p/(p -I) and g(X. '/I.) is a nonnegative function on (0.00) X (0.00) given byp-2
g(X,'/I.)=C 6I1X+'/I.II~kPI(P-2)(l)'k=O
in which
C= max (Cj),
o "j<; p-t
and Cj are the material constants appearing in (3.17) .•
Corollary 3.2. Let the conditions of theorem 3.2 hold. Then
where q = p/(p -1),
G (u x' Ux) = g ( I+ Ux' I+ U x) •
andg(',') is defined in (3.21) .•
4. Regularity theory
(3.20)
(3.21)
(3.22)
(3.23)
(3.24 )
While the problem (3.11) and (3.12) is time dependent, we find that an important part of ourformulation here is the following static nonlinear elasticity problem in variational form:
8 L C. Wellford. Jr. and J. 7: Oden, A theory of Jiscolltinuol/s finit(, elemem Galerkin approximations - Part I
(4.1)
where (".) de~otes the Lp - Lq duality pairing (q = pf(p-I)). and I(v) is a continuous linearfunctional on W~(/). We assume that a(Ux) has bounded bclow and continuity properties asgiven by corollaries 3.1 and 3.2 respectively: moreover I(v) is defined by
I(v) = <t. v). v E Ji/~(l) . (4.2)
where ( ... ) is the duality pairing between W~(l) and W;I(I), where q = pf(p-l). 0
We remark that under the stated assumption it can be shown that a solution u E W~(I) exists tothe problem (4.1). Indeed. W;' (1) is a reflexive Banach space and a has been shown to be a stronglymonotone, continuous (and hence hernicontinuolls) operator from a Banach space CU into its dual.The potential energy for a class 2 problem. for example. is
{
p-I }n(u) = J !2 Ck k~ I (I + DU)k+1 - 2IIi dx.
I -
where Du = /lx' The potential energy is lowcr semicontinuous with n(O) = O. For any choice ofdata fE W;I (/) there is a real number E > 0 such that (Da(Du) - f.u) + ll(u) > 0 for IIu II = E. Infact, (a(Du), DTf) - (rTf) is the weak gradient of Il(u). Browder [4.5] has shown that the proper-ties described here are sufficient to guarantee a solution to (4.1). Hence if fE W;I (f). Ii E HOI~(l),
We do not attempt to develop rigorous regularity results for the time dependent problem(3.11) - (3.12). However we do assume here a certain regularity which we intend to be typical ofshock wavc type solutions. We assume that the solution to (3. I I )-(3. I 2) is characterized by
u E L,JW~(l)) .
We make additional assumptions concerning local regularity and the regularity of other time deri-vatives in thc course of this work.
S. Galerkin methods for waves with multiple shocks
We can introduce an altcrnate form for the global energy balance (3.6). This form is the basisfor a va~iational theory of shock propagation and natural Galerkin approximation associated withthe variational theory.
Initially we introduce the global energy balance for a shockless region./; in the form
~ -21 J(pli2 + 2e) dX = {ail +q}lx=y;( t) + Jplli dX ,dt x=Yj_I(t)~ ~
(5.1 )
where the notation in thc first tcrm on the right means the difference between the average valuesof {ali +q} on the singular surfaces Yj and Y1_1. Differentiating the left side of(5.1), introducinge from (3.11 b) and reducing the term involving qx to a boundary integral. we get
I.. C. WeI/ford. Jr. alld J. T. Odell. A theory of discolltinuol/s fillite element Galerkill approximatiollS - Parti 9
f(piiti +olix)dX + V{Hpli2 +e}lxx=yY~(r-)(+ = {oi{} IXX: Yy;(l)( ) - lil/{ + lil/{_J + f pIli dX .-,_It) -,_It
~ ~where
.. X=YiV)liq; = {q}lx=Yj(t) .
Thus. we may now sum to obtain for the entire domain I
(5.2)
N N-I N-I
~ J(piizi + olix) dX + ~ VA p[li2h{ + [eh{) = azi I~:~o - ~ ([qhj - J pIil dX) . (5.3)~ ~
Now, for any field 1/I(X, t) let 1/1j denote the al'erage milic at the surface Yj:
(5.4)
(5.5)
Moreover, since
(5.6)
(3. 12b) can be written as
(5.7)
Now let us assume that (5.7) is satisfied. Then e and q and their jumps can be eliminated from(5.3) to give the final global equation
N N-J N
~ J(piiil+olix)dX+ ~ apVj[it2hi-Oinithj)={oil}I~:~o+~ JpfildX. (5.8)~ ~
This equation is a weak consen'atioJl form of the condition of balance of linear momentum in thematerial body with jump terms. This result is our primary tool in developing Galerkin methods.The central question that arises is the following: if the weak conservation form (5.8) is used forproblems of shock propagation. arc the jump conditions (3.12) satisfied at the shock surfacesYj(t), 1 E;;; i E;;; .IV -I? The form (5.8) is developed from the global encrgy balance (3.6). Thus. thelocal energy jump condition (3. 12b) is certainly satisfied (at least in a weak or average scnse)using the form (5.8) because (3.l2b) is dcveloped from a global encrgy balance by shrinking thevolume to zero at the shock surfacc. In addition. in the dcrivation of (5.8) it was assumed that thealternate jump condition (5.7) was identically satisfied on the surfaces Yj(t). I E;;; i E;;; N -]. But,since (5.7) is satisfied exactly and (3.12b) is satisfied in an average sense. thc local momentum
10 L. C. Wellford. Jr. alld J. T. Oden, A theory of discolI/;,mous finite elemellt Galerkin approximations - Part I
jump condition (3.l2a) is satisfied in an avcrage (see the derivation of (5.7)). In addition the jumpcondition (3.l2c) corresponding to the Clallsius-Duhem inequality is intrinsically satisfied becauseof the constitutive assumptions introduced ill section 3.
Another form of (5.8) can be obtained by integrating by parts in the first ternl on the left side,using (5.6) and the identity
We get
(5.9)
Now we denote by eml the space containing velocities it which satisfy the kinematical constraintsof the problem under consideration. Since it is arbitrary in (5.9).
(5.10)
The first tcrm in (5. 10) is the weak version of the momentum eq ua tion (3.11 a) applied over aunion of domains which do not contain singular surfaces internally. The second term represents aweak version (in the same spirit as the first term) of the momentum jump condition (3. l2a).However. the wcighting function is the average value at the shock surface. Note that ihs an im-plicit part of this formulation that the weight functions v be taken from the space of velocitieswhich are in general discontinuous at the Yi• I ~ i ~ N -I. This point marks the digression fromthe classical weak formulation of the partial differential equations involved.
In the classical weak formulation we choose cm2 as the space of test functions which are ingeneral at least continuously differcntiable everywhere in I. Then the weak form of the problemwould be to find that /I(X, t) such that
f (pi; - ax - pf)v dX = 0 .I
(5.11)
We believe that Galerkin procedures constructed from (5.11) portray too smooth a solution ofthe problem to be useful for shock wave calculations. That is. many of the most useful andeconomical Galerkin schemes constructed from (5.11) do not converge to shock wave solutions.On the other hand. we maintain (and subsequently prove) that a Galerkin procedure consistentwith (5.10) can converge to shock wave solutions.
In order to construct Galerkin approximations consistent with (5.10) and (5.11). we introducefurther notation. In addition to the partition Q described earlier let P denote a partition of 1 de-fined by the set of material nodal points {XI}~O' where
0= Xo < XI < X2 < ...< XM = Lo .Let
l~i..;M
L.e. Welllord. Jr. and J. T. Oden, A theory of disconlinl/ous /inite element Galerkin appmximatio/lS - Part I 11
and
II = [XI_I'XII = {X : X EI, XI_I ~ X ~ XI}
so thatM
1= U II'1=1
We assume that P is quasi-uniform, i.e. there is a constant 'Y > 0 such that
'Y~hl~h.
where h is the mesh parameter
h = sup {hi}'I .. I .. M
We also dcnote by 'Jik(n the space of polynomials of degree ~ k on I.We next introduce two finite element Galerkin subspaces
l~i~M. l{ij=IlnJj}'
Two Galcrkin schemes can now be defined:1. Shock jitling scheme. Find that UE 9l~'(l. P. Q) n C°(l) c em. such that
N N N-I N-t
6 (pU, W))1.2(J.) + 6 {(a, U'X))L (J') - 6 [aW] y. + 6 (p V[ lill +« a]. W)y.1= 1 ' i= 1 2, i= 1 I 1= 1 I
c
'v' WE 9{ ~'(l. P. Q) . (5.12)
2. Shock smearing scheme. Find that U E::m k"(l, P) c em2 such that
«(pU, W))L2(1) + ({a, WX))L2(l) = «pf, W))L2(l)' 'v' W E em ;:' (I. P) . (5.13)
In the above equations. the initial conditions are defined by either an L2 or 11· projec.tion intothe appropriate subspace. It should be understood that the term "shock smearing" in scheme 2 isused only to indicate that the approximation is endowed with.a smoothness which is not charac-teristic of the exact solution to the problem. Also. we use the notation [Ii h.i., == ([ lin,W)y, etc.
, I
A temporal approximation can be developed by introducing a partition R of the intcrval [0, T]defined by R = {to,/!' .. , Ir} where 0 = to < t1 < ...< tr = T and t"+I- til = AI. 11= 0, ... r-l. Thenthe sequence {U"}~,=O represents the values of the function U(t) E 9l~n(l. P. Q) evaluated at thepartition R. We assume that the Galerkin approximation satisfies the kinematical compatibilityequation of the first order (see [6])
12 L.e. Wellford. Jr. and J. T. Oden, A theor)' oj"discontinuous finite element Galerkin approximations - Part 1
Using this relationship. (5.12). (5.14). ami the central difference operator
(5.14)
we obtain a fillly discretized slwck fitting scheme:
N N N-I
6 «p8;Un. W»L (J-) + 6 ((a(U;). Wx», (n - 6 Ua(Ui) W]y.i= I 2, i= I '. I i= 1 '
N-I N
+ 6 (-p(Vn)2[UiD + Ua(Ui)]. W)y. - 6 (pf, h')/dJ') = 0i=1 ' i= I '
o
VWE9(f'(l,P.Q). (5.15)
We now pass on to the development of a mathematical theory for these schemes.
6. Some fundamental approximation theory lemmas
Initially we define a function Z in a subspace d,,(l) C C}f{. the original solution space, by thenonlinear "energy projection"
(6.1 )
In later developments we choose to identify d,,(l) with either 9{~n(l, P. Q) or C}f{:'(l, P). We re-quire that d,,(l) satisfy a finite element interpolation property
I";'i.,;,k+l.
Then a fundamental approximation theory problem is to determine the magnitude of the so-called"interpolation error" E = /I - Z and its temporal derivatives in various norms. Estimates of thiskind for the class I materials of section 3 will be presented in a series of lemmas which are generallytermed approximation theory results.
The first lemma (proved in [3]) establishes the W~(I) norm of the interpolation error:
Lemma 6./. Suppose that /I E W;+I{/), k > 0 is the solution to the problem (3.11-3.12) at anytime and Z is the element in d ,,(1) C W~(/) which satisfies (6.1). In addition suppose that a(·). thefirst Piola-Kirchhoff stress operator. satisfies corollaries 3.1 and 3.2. Then there exists constantsC!. C2> 0 independent of the mesh parameter h. such that
(6.2)
where
L. C. Wellford. Jr. alld J. T. Oden, A theory of discontillllOIlS finite element Galerkin approximatiol/S - Part 1 13
C2
= C1
(G(Ux' ZX))I/(P-I)
and 'Y
£=II-Z.
where G(Ux' Zx) is defincd in (3.24). •The seconcllernma uses a "Nonlinear Nitsche trick" (sec e.g. [7]) and is proved in [31 :
Lemma 6.2. Let the following conditions hold:(i) The conditions of lemma 6.1 are satisfied.(ii) The function Z defined by (6.1) is a member of d ,,(/) C CI (/). _
(iii) 0(') has a Gateaux derivative Dco(·). and DGo(ux+()Ex) > 0 almost everywhere in I.VO~e~1.
Then there exist constants C3. C4> 0 such that
liE II ~ C41101l1l11 1/(kP:I I)Lp(I) IVp (/)'
wherek ..,
0=-+'::"p-I p'
(6.3)
(6.4)
The next lemma establishes an estimate for the first temporal derivative of the interpolationerror. Initially let Dc 0(') be the Gateaux derivative of 0('). Then let
11=
[DCO(U;) ]
~ = infXEI Dco(U;x)
and
[Dco(U;)]. t
Dco(U;) IIL~(1)
Then the following result holds:
(6.5)
(6.6)
Lemma 6.3. Let the conditions of lemma 6.2 hold. Then there exist positive constants ('s. C6,
C7 independent of the mesh parameter 11 such that
liE \I ~ C I1k/(P-t)1I11 1I1/(P-I) + C I k/(P_I)2" 111/(P-I)2r W~(I) 6 t W~+I(I)7 I II w~+t(I)
where
(6.7)
14 L. C. IIlellford. Jr. and J. T. Oden, A Theory of discOlllinliol/s [iniTe elemelll Galerkin approximations - Part 1
Proof: Since a is Gateaux differentiable on nc W~(l). it can be shown (see lemma 3. 12 of [8])that the following Lagrange formula holds:
where
(6.8)
u; = ellx + (1 - 0) Zx . O~O~I.
Now differentiating (6. I) with respect to time:
(6.9)
Using (6.8), we get
(6. 10)
Now let QEd ,,(1) be that element which satisfies
(6.11 )
Since the Gateaux derivative is positive. ~ defined in (6.5) is positive. Then using an analysis similarto the one used to prove lemma 6.1 (see [3J).
(G( Q »)l/(P-J)IIxr' Xt k/(p-l) I/(p-I)
IIl1r-Qtll J ~Cs I- Ii 1I11,lIwk+1(J)'Wp(/) c.'y' p
111en from (6.10) with V = Qt - Z,.
(11(;(11 x' Zx) )J/(P -I) J/(p-l)
IIQ,-ZtIlW~(I) ~ h IIEIIW~(I) .
The theorem follows from (6.12) and (6.13) by using the triangle inequality. •
(6.12)
(6.13)
An estimate for the first temporal derivative of the interpolation error E in the Lp norm can be
I.e. WeI/ford, Jr. alld J. T. Odell, A theory of discOlllilll/OIlS fillite element Galerkill approximatiolls - Part 1 15
derived using the Lagrange formula and an auxiliary problem in much the samc way as in the proofof lemma 6.2. The result follows:
Lemma 6.4. Let the hypotheses of lemma 6.2 hold. Then there exist positive constants C8' C9•
CIO such that
(6.14)
where.,
-----L+.:::...ol=(p_l) p
k JO2 = +.::. ,
(p _I )2 I)
Now let
- . _[DGO<U;)]~ = 1111 ,XEf DGo(U;rx)
and
~ = II lDGU(U. ;) 1.(/ 'ID GO( u.~) 11'00 (l)
•
(6.15 )
Then the second temporal derivative of the interpolation error in the W~(J) norm is given asfollows:
Lemma 6.5. Let thc hypotheses of lemma 6.2 hold. Then there exist positivc constants C12,
C1J, CI 4' and CIS sllch that
liE II ~ C hk/(P-J)21111111/(P-1)2 + C hk/(P-J)31111111/(P-J)311 WI(l) 12 Wk+I(J) IJ II'k+I(J)
P P P
2 2+ C hk/(P-I) 1I11 1I1/(P-1) + C hk/(P-l)lIl1 U1/(P-l)14 r wk+t(J) IS rr II'k+1(J)
P P
where
C_ (,I/(P-1l( 2TjG(U,. Z ) )I/(P-I)
14- 6 /\T'4Xr
~ .'
J-C
t3= C;/(P-l)( -17G(U
xi· Zxr))"(P-t> CIS = C
II (G<u.~r,Zxl1) )I/(P-I)
h •Finally the estimate for the second temporal dcrivative of the interpolation crror in the Lp norm
is given as follows:
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 8 (1976) 17-36© NORTH-HOLLAND PUBLISHING COMPANY
A THEORY OF DISCONTINUOUS FINITE ELEMENT GALERKIN APPROXIMATIONSOF SHOCK WAVES IN NONLINEAR ELASTIC SOLIDS
PART 2: ACCURACY AND CONVERGENCE
L.c. WELLFORD, Jr. and J.T. ODENTexas Insti/llte for Computational Mechanics, The University of Texas at Austill. Austin. Texas
Received 25 Fcbruary 1975
Contents:
7. Accuracy and convergcncc of sCllIidiscrclC discon-tinuous finitc elcmcnt Galerkin approximations forwave propagation
(Part I has bcen published in 8 (1976) 1-16)
17
8. ACl:unlcy, convcrgcncc. and stability for fullydiscretized models
Refcrcnccs2335
7. Accuracy and convergence of semidiscrete discontinuous finite element Galerkin approximationsfor wave propagation
In this section we present studies of accuracy and convergence for semidiscrete Galerkin approxi-mations. As our primary task we wish to discuss and compare the convergence properties of thesemidiscrete shock fitting scheme (5.12) and the semidiscrete shock smearing scheme (5.13). Weshall show by a comparison of the error characteristics for the two schemes that the conjecturesof section 5 regarding the superiority of the shock fitting scheme in resolving dynamic responsewith shocks are correct.
Initially we determine the error in using the semidiscrete shock smearing scheme (5. I 3). We usea method which was developed for the linear case by Dupont [101 and which involves L2 estimates.
Let u be the solution to (5.1 I) and U the solution to (5.13). Let e = II - U be the approximationerror. Let Z be the element ofcrk;l (I, P) defined by the nonlinear energy projection
o
V V E cm.f:'(I, P) . (7.1)
This projection is essentially a generalization of the weighted HI (I) projection introduced byWheeler [7].
Now make the decomposition e = E + c where
E=u-Z, c=Z- U. (7.2)
Then the behavior of the error component c is established through the following:
THEOREM 7.1. Let c = Z - U, where Z is defined by (7.1) and U is the solution to (5.13).
18 LC. Wellford. Jr. and J. T. Oden, A theory of discontinuolls finite dell/em Galerkin approximations - Part 2
Let the Piola-Kirchhoff opL:rator a(·) bc Gateaux diffcrcntiable on a convcx subset n of W~(f)containing II and Z. Then there exist positive constants -y and C such that
IICtll +-yIlCIiP/2
t E;;;C{lIct(Olll +G(Zx(01.Ux(0))"2I1c(0)1I I +IIEttll } .•L~(L2(f)) Loo(lVp(/» 1'2(/) II'p(I) 1'2(L2(1))
(7.3)u
Proof: Evaluating (5. II ) with an c1cment WE em;' (1. P) and integrating by parts, we obtain
(7.4)
o
Similarly. if (5.13) is evaluat.ed for the same element WE em;' (1, P), then
(7.5)
Subtracting (7.5) from (7.4). setting e = E + C. and identifying W with ct'
(p cit' Ct»L2(1) + (a(Zx) - a(Ux)' CXt»L2(1) = -«~pEtro Ct))L2(I) - (a(/lx) - a(Zx), CXt))L2(1) .
(7.6)
Now since 0(') is Gateaux differentiable. the Lagrange formula [8] establishes that for somee E [0, I]
((a(Z x) - a( Ux)· Cx ))t'2(l) = «DGa(U,~) cx, Cx ))L2(1) .
where
U; = eZx + ( I - 8) Ux .
This implies that
(7.7)
Now introducting (7.7) into (7.6) and eliminating the second term on the right using the non-linear energy projection (7.1 ).
Using the Holdcr inequality to simplify the right side of (7.8), using the inequality
b,,;;::TJ 2 I b2 > 0a ""2"0 +Tr1 . TJ •
(7.8)
(E)
L.C. Wellford, Jr. and J. T. Oden. A theory ofdiscolllinuol/s finite element Galerkill approximations - Part 2 19
integrating from 0 to t, and using corollarics 3.1 alld 3.2. wc get
(7.9)
The result (7.5) now follows by using the lemma of Raviart [II], (which gives a result very similarto the Gronwall inequality [12]) and by taking the supremum over all t E [0, T] in the resultingexpression. -
If we nex t lise the triangle ineq lIality and theorem 7.1. we obt ain the final error estimate:
THEOREM 7.2. Let the hypotheses of theorem 7.1 hold. Thcn therc exist positive constantsC. and Cz sllch that
lIetllf'oo(L2(/) + Clllell~~~Lp(I) ~ Cz {lIet(0)IIL2(1) + (G(Zx(O), Ux(0))1/211 C (0) IIwJ,(I)
We assume that thc subspace cik;:' (1, P) has an interpolation property of order j: i.e.
(7.10)
!nf Ilu- VII I ~ Chi-III ull . ,VE CIt( k(l,P) Wp(/) W6(/)
I~j~k+l, O~/~m. (7.11 )
Then we can use the approximation theory results of section 6 and theorem 7.2 to determine theconvergence characteristics of the shock smearing scheme (5.13). The convergence is determinedby the last term in (7.10) (IiEttllf'2(1))' From lemmas 6.1 - 6.6 we can easily deduce that
p~ 2. (7.12)
To obtain convergence to zero of the second temporal derivative of the interpolation error E, wethus require that for each time point t
u(t) E W~(f). u ,(t) E W ~ (/) , lirt(t) E W~(/) . (7.13 )
Thus from (7. I 0) we find that to obtain convergence of the approximation, it is sufficient forthe finite clement interpolation property (7.11) that
u E Loo (W~(J)) , (7.14)
20 I..C. WeI/ford. Jr. and J. T. Oden. A theory of diSCOl/til/liOI/S finite element Galer!..in approximatiollS - Part 2
Clearly neith~r shock nor acceleration wave solutions havc the regularity implicd by (7.14). Thuswe can see that in a very wt:ak norm (the L2 norm in this case) the semidiscrete approximation(5.12) will not converge to shock or acceleration wave solutions of the problem (3.11 )-(3.12).The problem is caused by the reduced global regularity of the shock and acceleration wave solu-tions. Essentially. anytimc we attempt to define an approximation to a solution which is veryirregular globally through a global projection method (of which (5.12) is an example). the conver-gence propcrties will deterioratc. In this case the standard interpolation property of the finiteelement subspace01l t (I. P) given by (7.11) is not sufficient to provide convergence to shock wavesolutions. It is not a question of stability but of approximation.
We have two altcrnate courscs of action to provide a rcmedy. We can increase the global regular-ity of the shock wave solution by introducing artificial viscosity or we can abandon the globalprojection idca in favor of a local projection method. In the latter we project onto the J; shock-Icss subdomains on which the exact solution has increased regularity. The embodiment of thislocal projection idea is the shock fitting schcme (5.11). We now discuss thc convergence andaccuracy properties of this scheme.
In the formulation of an error estimate for the shock fitting scheme (5.12) we let Z be thatclement of 9l ~I (I, P, Q) Jefined by the focal energy projection
VWE9t~/(l,P,Q) ,
j= I, ...,N-l .
i = I, ....N. (7.l5a)
(7.l5b)
Essentially (7.15) represents a scries of local nonlinear HI projections on the J; shock less sub-domains. These projections are constrained at the wavefronts Yj by (7 .15b), which rcquires thatthe average stress be preserved across the wave. Then let the approximation error e be decomposedinto error components e = E + c where E = u - Z and c = Z - U.
The subspace r;X k' (1. P, Q) is assumed to have the following properties:
(i) oinf lIu- VII I 0;;;Chi-III 1111 .VE 'i1(~'(I.p.Q) Wp(Ji) wb(Jj) ,
10;;;/" k+l, 00;;; 10;;; /11, i=I, ... ,N,
(7.16)
j" k+l . (7.17)
We now introduce an auxiliary condition which must bc applied to insure the convergence ofthe shock fitting scheme (5.12). We require that for some positive constant 0: an amplitude condi-tion of the following form be satisfied:
N-I
'Yllc(t)IIPt +6Wp(n ;=1
r
f VI {[a+(Zx) - a+(Ux)] cx - [a-(Zx) - a-(Ux)] c1-} dT;;;:' 0:11c(t)IIP 1 .o Wp(I)
(7. 18)
The second term on the left side disappears in the lincar case due to the Betti-Rayleigh reciprocitytheorem.
The behavior of the error component C for the shock fitting scheme is given by the following:
22 L.c. Wellford. Jr. and J. T. Odell, A theory of discolltinuol/s finite element Galerkin approximations - Part 2
Now using the Gateaux diffcrcn tial in a manner similar to that presentcd in theorem 7.1, weobtain
N N-I
+ E (([Daa(U;)].t cx. CX))/'2(J.) + ~ [(a(Zx) - a(Ux)) ex hY; .;=1 ' ,=1 '
where U; = ()Zx + (1- ())UX'
In addition, from the definition of the L2 norm.
(7.23)
(7.24)
Introducing (7.22). (7.23), and (7.24) into (7.21) and assuming that the intrinsic velocity of thewave is duplicated exactly by the approximate model (V;(u) = V;(Z) = V;(ll)). we get
N N N
= ~ ~«[Daa(U;)],r cx· CX))/'2(J;) - E«(pEw Ct))L2(Ji) - ~ «(a(ux)-a(Zx)' CXt))L2(Ji)
N-I N-I
+ ~ 2(a(ux)-a(Zx).Act)y.- ~ (pVj[Erll,Z"t)Y ..;= I I i= I I
(7.25)
Using the local energy projection (7.15) to eliminate the third and fourth terms on the right sideand estimating the remaining terms using the Holder inequality and inequality E (for positive con-stants T/ and K), we get
(7.26)
L.C. Wellford. Jr. al/d J. T Odel/. A theory of discomil/uous fil/ite e/emem Galerkill approximatiol/s - Part 2 23
But REr]Yj~ 211IEtIllYI(J/.Ji+l) and
112- 112
1/2 ~ 112-lpl12e 12~- sup lip Lt Loo(Jj)
2" ti 2" I"i"N
C2k2In .t: \\2~ - sup lip Lt I'2(Ji) .
2" I" j ... N(7.27)
Eq. (7.27) follows from lise of the inverse property (7.17) and the Sobolev imbedding theorem[ 13). which states that if y E H'" (X), X is defined on an n-dimensional Euclidean space. andm > 11/2. then there exists a positive constant C sllch that
lIyll/,oo(X) ~ CllyIlHm(X) .
Introducing these results into (7.25). we get
N N N
~ 21
~ II [DGo(Ui-)), tilL (J.)II C II~Vl(J') + 712~ IIP 112EttllL2
(J..) +-21 L; Ilpll2 CtllL
2(J.),= 1 .. , p' ,= I 2, 71 ,= 1 2 ,
N-I 2k22"p '" 2 (N -I) C II 112.t: 112
+-DV.2I11Erllly.(JJ)+ 2 sup p Lt1.2(Jj)'h2 j=I' , I· /+1 K • ""j<;;.N(7.28)
Integrating from 0 to t, applying the lemma of Raviart [ 11 J, using the response cond ition (7.18).and taking the supremum over all t E [0, T], we obtain the result (7. 19) .•
The estimate for the approximation error e may be obtained by using theorem 7.3 and thetriangle inequality.
THEOREM 7.4. If e = Ii - U. where u is the solution to (5.] 2), and if the amplitude condition(7. ]8) is satisfied, then there exist positive constants C1 and C2 not depending on II such that
lIerll + C1llellPl2 ~ C2 {\Ier(O)1I (I) + G(Zx(O), UX(0))1I2II c(O)1I I1.oo(L 2(I}) L ...(Lp(I» 1.2 II'p(l)
N
+ IIEtll + IIEIIPl2 + ~IIEtrll +1. sup VjlllEtl1i } .• (7.29)1.00(1.2(1)) L .. (Lp(1) j= I L2(1.2(Jj» h 1<; j<'N-1 L2(Yj(Jj.Jj+l»
24 I..c. Wellford. Jr. alld J. T. Udell, A theory of disCOlllilliiOUSfillite element Galerkill approximations - Purt 2
The question of accuracy and convcrgence of the shock fitting schemc (5. I2) is thus reduced toa problem in approximation theory, We must bound thc interpolation crror E defined by the localprojection method (7.15a). Initially we recognize that (7.15a) rcprcsents a scries of nonlinearelliptic boundary value problems with boundary conditions defined by (7.15b). This implies thatwe can make use of the approximation theory results of this section to define the interpolationerror E on each of the shocklcss domains Jj•
Let Y E 9<. ~"(1.P. Q) be arbitrarily chosen. Then if Jk, I <:;;; k <:;;; N. is a typical shockless region,we have from (7 .15a) with W = Y - Z that
(7.30)
Then using corollaries 3.1 and 3.2 and the Holder inequality.
Taking the inl1mum ovcr all )' on the right hand side of (7.31)
(G(II .. Z.»)I/(P-I) .. I/(p-I)
111:'11 <:;;; - .\ .\ IIlf 1111 -. } 1I1\11(J.) .1V~(Jk) 'Y YE9{'k'(J,P.Q) p k
(7.32)
Using the subspace property (i) [eq. (7. 17)], we obtain estimates for the "interpolation error" Eand its temporal dcrivatives which are the same as developed in lemmas 6.1 - 6.6 with the domainI replaced by Jk, I ..; k <:;;; N. Then by the Minkowski inequality
{N } IIp N
IIExllL (I) = ~ IIEx III (Jk) <:;;; ~ IIExllL (h) .P k=1 P k=) P
Similar relationships hold for the temporal derivatives of E.In addition there exist positive constants C1 and C2 such that
N N
sup VIIIIE,llIr.2(Y.(J .•JI+t»<:;;; C) ~ VIIIErIlL2(L (J.»<:;;; C2~ VjllErllL2(1vl (JI»'t..;; j";;N-1 1 I 1=1 ~ 1 1=) P
(7.33)
(7.34)
From theorem 7.4. (7.33), (7.34), and lemmas 6. I -6.6, we obtain the final error estimate:
THEOREM 7. 6. Let II, lit E L~(W:+' (Ji)) and utI E L2(JV:+1(Jj)) for i = I, ... , N. Then thereexist positive constants C1 and C2 such that
L.e. WeI/ford. Jr. alld J. T. Odell, A theory of discOll/illllOIlS fillite element Galerkill approximatiolls - Part 2 25
(7.35)
8. Accuracy, convergence, and stability for fully discretized models
In this section we demonstrate the convergence of the approximation (5.15) to the solution tothe problem (3.11 )-(3.12), investigate the accuracy of the approximation (5.15), and discussnumerical stability criteria.
The error for the fully discrctized shock fitting scheme (7.15) can be established by a proceduresimilar to the one presented in section 7. Initially we evaluate (5.10) at I = nlit and setv = W E cJl~1l(l, P, Q). Then adding ~~I ((p6; Un' W))L2(J') to each side of the resulting eq uation.and using the first order kinematical compatibility condition [6]. we have
N N N-l
~ «p6;u". W))L2(Jj) + ~ «(a(u1-), WX))/'2(J;) - ~ [a(u1-)Wh;
N-l N N
+ ~ (-PV;[u1-]+[a(u1-)D,W)y;=~«(pf,W))L2(J;)+~«E/I.W))L2(J;)·
where un is the exact solution evaluatcd at timc point 1= nlit and
(8.1)
(8.2)
We assume in this analysis that the regularity property, (j4u/at4 E L2(L2(Jj)). i = I .... N, holdsbetween shocks. Dupont [10] has shown that an estimate for en is
Setting en = u"- U" and subtracting (5.15) from (8.1), we getN N N-IP. «po;ell, W))/-:lVj) + i~ ((a(u1-) - a(U;). WX»)L2(J;) - ~ [(a(u1-) - a(Ux)) J1Ill Yj
N-I N
+ 6 (-pV2[eID+[a(u'I)-a(U")ll W) =6((e W»;=1 n X X x' Y; ;=1 11' L2(Jj)'
(8.3)
(8.4)
26 L.c. Wellford, Jr. and J. T. Oden, A theory of discomilllw/is finite element Galerkin approximations - Part 2
o
I t is convenient to identify an element Z" E 9l :'(1, P, Q) through the discretized local energyproj ection analogous to (7. 15):
Then' we decompose e as follows:
} V WE 91;'(1, p, 0), i = I, ...N, j = I. ...N-l. (8.5)
Ell = u" - ZII . cll = ZII - un , (8.6)
In addition we define certain auxiliary variables by
Ii - 1 ( lI+t11+\12 - 2 It +u") , 0t (X) = Xlt=(II+I)At - Xlt-nA
11+112 - rAt
Initially we introduce a discrete amplitude condition analogous to (7.18). We require that forpositive constants 'Y and A
k = I. ...r-I .~ A II C (k At) II PI'wp(I)
Then the behavior of the error component clI is given in the following:
(8.7)
THEOREM 8. J. Let ctl
be defined by clI = u" - Z". where un is the solution to (5. 10) and Z"is defined by (8.5). Let o4u/at4 E L2(L2(Jj)) for i = I ... " N. In addition. suppose the approxima-tion is numerically stable, the intrinsic wave speed VIIi' i = I .... N. 11 = I .... r, is duplicated exactlyby the approximations (5.15). and the amplitude condition (8.7) is satisfied. Then there existpositive constants µ and v such that
where
1I0tull~ = sup IIOt ull: lIull· =sup lIunll and Vi= sup VII".L",,(X) O';;;II<i;r-1 11+112 X L",,(X) 0';;;11"; r X I<;n"; r I
Proof: It follows from the decomposition (8.6) of en and (8.4) by setting W = Of Cn+t12 + 0t clI_l12
and the assumption that the intrinsic wave speed VII is duplicated exactly by the approximatemodel that
L. C Wellford. Jr. alld J. T. Odell, A theory of discontilluous finite element Ga/erkill approximations - Part 2 27
N N
~ «pli; Cn, lir Cn+11'2 + lir C'.-II'2»L (Jf!) + ~ (o(Z;) - o( Ux)· li 1,,+11'2Cx + li 'n-t/2 Cx)) L2(J!J).=1 2, I-I ,
N-I
~ [(o(Z;) - o( U;))(li t cn+11'2 + lit cII_II'2)] y!J1=1 •
N-t
+ ~ (-p v,; [ c;] + [o(Z;) - o(U;)ll, lit cn+11'2 + 0, cn_I12)yni=l I
N N
= - ~ «pli;En· lir cn+l12 + lit cn-I12» L2(J1') - ~ (o( II x) - o(Z;). litn+112 C X + 0'1/_112 CX»/'2(Jf')
N-l
+ ~ [(o(ux) - o(Z;» (8, Cn+1/2 + 8t Gn-I12)] y!11= 1 I
N-I
- ~ (-pV,;~Ex] + [o(lIx)-o(Z;)D, 0, Gn+I/2+0tC'J-1I2)yf'j=1
N
+ ~«p€I/,0,Cn+II2+0r Cn-I12»L2(Jf') .j= 1
It can be shown that
0, «O(ZX)-O(Ux), Cx) = (o(Z;+I)-o(U;+I)-o(Z;)+o(U;).Or Cx)) nn+1/2 L2(Jj) n+112 L2(Jj )
+ 2«0(Z;) - o( U;). 8r ex» n + (( } [DGo(u~n+l) - DGo(u;.n)]c ~+I, ex)) nn+tl2 L2(Jj ) ~t ' L2(Jj )
(8.9)
(8.10)
and similarly that
+ 2(0(Z;) - o( U;), 0, cx)) n., + «} [DGo( U;'n) - DG o( U;'"-I) 1 CX-I. Cx)) .nn-l/2 L2(JjJ ~t L2(Jj)
+ 1t «O(Z;-I) - O(U;-I), CX-I))L2(Jf'-J['-I). (8.11)
In addition. another useful identity is the following:
«pII2Or CII+112.pI1'20, Cn+t12»L2(1[') - «ptI20t en_I12· pt128t (;n-II2»/'2(1[')
= «p 112or cn+I12' P tl2 8 r Cn+1/2» L2(Jr+t12) - (ptl2 °tC n-II2' P 1120, c,J-tl2» L2(Jr-I12)
- «p 112lit cn+I12' P 1128 r cn+I12» L2(Jr+I12 _ Jf) - «pll2 Or Cn-t12' P 112lir Cn-I12» L2(Jf -Jr-I/2)' (8.12)
28 L.e. Wellford. Jr. and J. T. Oden. A theory of discontilll/ol/s Jillite elemellt Galerkin approximations - Part 2
Introducing (8.10), (8.1 ). and (8.12) into (8.9). we get
N N
+ ! L; 0, (a(Zx) - a(Ux)' cx» +! L; or «a(Zx) - a(Ux)' cx))i= 1 n+t/2 Lz(Ji) 1=1 11-1/2 Lz(Ji>
N
+ L; n(a(ll~y)- a(Zx) (OtCII+lfl + O,C,I-I/2)Dy!li= 1 I
N-l
- L; (-p JI,:UEx] + na(ll~p- a(Zx)ll, O,Cn+1fl + OtCn-l/z)y~1i= 1 ,
N
+ L; «pE". 0t C"+1/2 + o,CII-I12))L2(.I,')i= 1
where
(8.13)
N
" ) ( tl2s:~ pl/ZOC )) n-I12'- L..J - P Ut\"'II_lfl' , 11-112 l,z(Jf-J/ )i= 1 At
N-I JI(31/= ~ 2"i {[a-(Zx)-a-(Ux)]c:Xt/- [a+(Zx)-a+(Ux)]C~"}y~1
,=1 I
N" I .((Z"+I) (Ut/+I) ~"+I»- L..J 2A (a x -a x ' LX I. (J!I+t_J")1=1 ~t z I /
I.. C. Welljord. Jr. alld J, 1: Odell, A rheory of disCOf/tilll/ol/sfillire elemelll Galerkill approximatio/ls - Part 2 29
To obtain this expression. we have used the fact that [orcn+112 +OtC,,_tI2Dy!' = O. i = 1. ... lV-I.In the limit as At -+ O. 1/1", (3n, C;" -+ 0: howcver these quantities are in gderal not positive in
the discrete case. Thus they play an important role in the stability of the approximation. Weproceed to estimate the size of each term. Using an approximate form of the kincmatical compa-tibilityequation VII n cxD ,,= - (Ot cII+t12 - 0t cn-In) ". we find that
1 Y/ YI
N-I
tl/l"= 6 (pV/l(OtC"+ln-Ote/l-1/2),OrCII+II2+0tC/l-II2)y~11=1 I
(8.14)
Initially we decom pose 1/1" by noting that 1/1" = 1/1': + I/I~. where
N-l V N
1/1': = 2 6 (P2/l °r c/l+I12' 0t C,,+II2) ,,- 2 6 2lA ((pt12 Or C,,+II2' p 112Or C,,+II2)) / (J!'+112_J!I) (8.15)1= 1 Y/ 1= I ut ·2 I I
andN-I V N
" - '" ~ s:: J;' ) _ 2 '\' ....!..- « tl2 ° C. 112 0 C ))1/12--4u( 4 Ore.I/_II2·Ut\...II_ll2y~ ~2At P t'n-II2'P r 11-112 /(/;,_J!I-II2)'1=1 I 1=1 u ·2 I I
Now we let or Cn+I12(X/) = 0t Cn+I12(Y/) + XI d(OtCn+1/2(X;))!dXI + higher order terms. Here Xi isthe normal coordinate in a coordinate system imbedded in the shock surface Yi. Thus, withJ == J~+112 - J'.I
I I'
N-I N
1/17= 6 (pV/l 0tCn+II2,OtC/l+II2) 1/-6: 
30 L. C. Wellford, Jr. and J. T. Odell, A theory of discontinuous finite element Galerkin approximations - Part 2
terms, we have
N
1t/l71 ~ ~ Cp V~~ t1.{t12l1c5r CtJ+II2I1:I(JjtJ+II2_Jr) .i= I
Using the inverse propcrty of the subspace (7.17), we get
(8.17)
where
C;l = kV,~/2NpC and VtJ = Slip1<;i"N-l
Through a similar calculation we can show that
Thus (8.17) and (8.18) imply that
We can estimate J3" in a similar fashion. We rewrite (8.13) to obtain
{3'1 = W: + (3'~+ J3~where
(8.18)
(8.19)
(8.20)
N N
J3~ = - E 21A «a(Z;+I) - a(U;+1), C~+I)) ,,+1 11 + E 21A «a(Z;) - a( U;). t:~)) n n-Ij= I ~t L2(Jj -Jj) j= 1 ~t L2(Jj -Ji )
N N
+ ~ 21A «a(Z;)-a(U;). C~))L (JtJ+I J!l)- ~ 2~t «a(Zx-I)-a([f1x1), eX-I))L (-!' J!I-I)'
j= I ~t 2 i-I 1=I ~ 2 Ji - I
I t can be shown using a proccdure similar to the one used to estimate Ij; that for a positiveconstant C2
L.C. Wellford, Jr. alld J. T. Oden, A theory of discontillllOllS fillite element Galerkill approximations - Part 2 3 I
Since the stress is Gateaux differentiable.
---L((o(Zll+t)-O(UlI+l)-O(Zll)+o(U") 0 c: ))D.t x x x X· rll+112 X L2(Jt>
(8.21)
for i=l. ... N. (8.22)
and
--L((o(ZlI)-o(U")-o(ZlI-J)+o(UIl-t).o c: ))D.t x x x X t,,-In X IdJt>(8.23)
Introducing (8.22) and (8.23) into (8.13), we get
+ (([DG o(U;") - DGo(U;1I-1)] C:x· 0tll_tn c:X))/'2(Jt>} .
Now let
(8.24)
'Y? = sup IDGo(U;'Il) I.XEJ?
and also
Y' = sup l'Yrl,l";i";N
~7= sup IDGo(U;n+I) -DGo(Ui-")I,XEJ?
~I/= sup I~;"I.l";i";N
O~n~r-l.
Multiplying ']1/ by D.l, summing from 1 to,. - I. bounding the terms on the right side using theHolder inequality and inequality E, and applying the inverse hypothesis and the Sobolev imbeddingtheorem. we get
3? I.. C. WeI/ford, Jr. and J. T Oden, A theory of discontinuous finite element Galcrkin approximations - Part 2
+ (C6 ~t + C7 ~/2)lIpll2or CII2 I + C8 ~t II C:11I2 I112 H (I) H (I)
r-t r-I
+ ~/3 E Cllllpl120 cII2 + ~/3 E ell IIcn+11I2114 11=1 9 111+112 L2(J) 114 n=t 10 HI(l) ,
where for arbitrary positive constants 17.13. ancl Q
(8.25)
k2~r-1 -r-l ~r-I - ~IC - C - 'Y C - 173 417P '4 ----rp s--r' C6 - 4f3p .
-IC =pf3 Ck2~IIV Ck2;YI/VC - 'Y7--::;-' CII= n+_ II
....p 8 4' 9 4ap 2p
Now we assume that the Gateaux differivative of the stress has a bounded temporal derivative, i.e.for a positive constan t ell
~t [DCO(U;II+I)-Dco(U;II)] ~ ell and' ~t [Dco(U;J')-Dco(U;II-I)] ~ CII
Then it can be shown that
(8.26)
Multiplying (8.13) by ~t, summing on 1/ from I to,. -1. and simplifying the right side using thediscrclt: local energy projection (8.5).
N N
E II P 1/2 or Cr_t1211L2 (J!,-II2) - P IIpl/2 0t CI1211 ~2(Jl:2);= I 2 I I-I I
N
- ~ E Go: E «(a(Z~:-O:)- a( U~-O:), C ~-a:)L2(J"-a:)
1
0:=0.1. ;=1 I lao=ut=-Ir,r-I ar=ar-I=I
r-I /II V r-J r-J
+ E E ~([a+(Z;)-a+(Ux)lc~'- [a-(Zx)-a-(Ux)]cxll}y~+ E ~/lJ;n + E ~tf3n,,=1 ;=1 2 , 11=1 1/=1
r-I r-l r-t N
+ E ~t:J1I + ~t E XII = - E E ~t«po; Ell' or CI/+112 + or C'I/_I12))/ (J!')1/=1 11=1 1/=1/=1 -2 I
r-I N r-J N
+ E E~t(pV,;[Ex],01C:1/+112+0tCI/_112)y!'+ E E ~t((p€,,,OrC.I/+I/2+01C:1I_112)L (J!I)11=1 ;=t I 11= t i= 1 2 I
(8.27)
(8.28)
I..c. Wellford. Jr. and J. T. Oden, A theory of discontilluous finite elemellt Galerkill appruximatiolls - Part 2 33
Estimating the terms on the right side in (8.27) and using (8.19). (8.21). (8.25). and (8.26) aswell as corollaries 3.1 and 3.2. and (8.7) we get
(!:!.t !:!.t2 ( C!:!.t. (C fi.()I -c - - C -)lIpI/20 C 112 + A - ~ - C !:!.t)IIC'I( + A - ~ IIC,.-IIiP
3/2 4 / 2 I '-1/2 L (I) /2 5 1\11(1) / 2 II'I([)I I 2 I PIP
;;;;;(l+C6!:!.t+C7!:!./2)lIpI/20t CII21 +C1111C01l2 I + (CI2+CS!:!.t)IICI1l2 11/2 H (I) - II'p(I) Wp(I)
~I N N
+ !:!.I ~ [-y ~ lip 1/2o;E,,1I 2 + ~ ~lp1l2 V,~. [Ex] ,,12"=1 i= I L2(Ji) i= I 2112 I Yi
N 2
+ ~ -211 Ip"2U.ltC,,+1I2 + O,C,,_1I2)y1I12 + -YllpI/2€1111[2 ([)J1=1 K "2
r-I [( 2 112) ( I !:!. 112)+!:!.t ~ ....L+C" fi.t + C ~ IIpl120 CII2 + -+ C ~ IIpl120 CII2n= I 2-y 9 1z4 I 112 tn+tl2 1'2(f) 2-y 1 112 t,,-tl2 1'2(f)
( 2) C !:!./1/2,+ C +C" fi.t IIc,,+1112 +(C + 2 ) II C/11 2 +C IICI/-1112 J
II 10 /4 1\11(1) It 2 II't(l) II Will)I P Iz· P P
where e12 = G(Zx(O), Ux(O))/2 + C2.
As conditions ofstabilUy we require that there exist positive constants Q. {3 such that
I - C !:!.t _ C !:!.t2 > Q • (8.29a)3 1z2 4 112
C2!:!.t(8.29b)A--- C I1t> Q2 5 •II
1 + C611t + C7!:!.t2 < {3. (8.29c)
CI2 + Csl1t < ~, (8.29d)
-L+ Cn I1t2 + C !:!./112 < {3 . (8.2ge)2-y 9 1z4 I 112
--L+ C I1tl12 < e . (8.29f), 2-y 1 112
"C + C I1tl/2
+ C" fi.t2 < i3 (8.29g)
- 1 I 2 112 10 114 .
For stable schemes the result (8.8) follows by using the Sobolev imbedding theorem as intheorem 7.4, the discrete version of the Gronwall inequality [131. and (8.3) .•
We obtain the estimate for the approximation error e" from theorem 8.2 and the triangle in-equality:
(8.30)
34 L.C. Wellford. Jr. alld J. T Odell, A theory of disconril1l1ousfillite element Galerkill approximatiol1s - Part 2
THEOREM 8.2. If e" = lin - un, where u" is the solution to (5.10) and un is the solution to(5. I 5), and if the hypotheses of thcorem 8.1 are satisfied, then there exist positive constants wand ¢ such that
I/8tetll- +wl/ell~!2 ..:;;¢ (1I8t (:11 I +11(:°11 IL ...(I~2(l)) L ... (Lp(l» In H (1) wp(l)
N
+11(:11/ 1 + liE, 1/ +I/EIIP!2 +6 IIErtllWp(J) L ...(I'2(l)) L ...(Lp(l)) j=) L2(L2(Jj»
N 4
+1.. sup V? IIIExllJ + 6 t:J.t2 I/~II ) .•h I';;; j<;;N-] L ... (Yj(Jj,Ji+l» j= I at4 L2(L2(Jj»
From theorcm 8.2 and lemmas 6.1 - 6.6, we obtain the final error estimate:
THEOREM 8.3. Let u, lit E L...(W;+I(J)), litt E L2(W;+) (Jj)), and lirttt E L2(L2(Jj» fori = ] .... N. In addition, suppose that the hypotheses of theorem 8.1 are satisfied. Then there existpositive constants wand r such that
118tetIlZ ...(L2(1) + wllellt~Lp(1)":;; r (1I8tll2eIlHl(J) + II (:°11 W~(I)
N ~+~ N k +2+ 611(P-l)2 p IIIi II + 6 h(p-I) P IIIi 111!(P-I)
t k+l) tt k+1j=) L ... (Wp (lj) j=1 L2(Wp (Jj»(8.3 I)
•Now consider the practical implementation of the stability constraints (8.29). Initially we set
Then let there be a specific real number v such thatX":;; v.
Then choose a t:J.t and select a: so that
a: < I - C3 t:J.t v - C4 t:J.t3fZ v2•
(8.32)
(8.33)
(8.34)
We can always adjust Cs so that this choicc of a: is less than X - C2 t:J.tl12 V - Cs t:J.t. For this choiceof t:J.t, Cl2 + C8t:J.t can always be made less than some nllmber (3, as can 1+ t:J.t C6 + t:J.t2 C7. Thus
L.c. Wellford. Jr. and J. T. Oden. A theory of discontinuous finite element Galcrkin approximations - Part 2 35
(8.29a)-(8.29d) can always be satisfied: otherwise. if C~' < O. then (8,29f) is satisfied whenever(8.2ge) is satisfied. It is thus necessary to choose v so that (8.29g) is also satisfied.
In summary, we use the following procedureI. Pick At.2. Calculate the right sides of (8.29c) and (8.29d). Choose a specific small ~ that just satisfies
these two inequalities.3. Then choose v to satisfy (8.2ge)-(8.29g). i.e.
2C~'
0< v < min {3-1/2'YC,
yP-2C11 Ci C2
Cn + 4(C" )2 - 2C"10 10 10
(8.35)
The stability constraints (8.29) can now be satisfied by choosing an acceptable Q. A sufficientcondition for stability is then to choose II so that
(8.36)
In the following theorem we give the stability criteria for (5.15) resulting from the aboveprocedure:
THEOREM 8.4. If the hypothesis of theorem 8.1 are satisfied, then a sufficient condition toinsure the numerical stability of the solution to (5.15) (in the L,,.{L2(J)) sense in the discretevelocities 0rVII+112 and in the L,,.,CLp(J)) sense in the discrete displacements V") is to choose Atand II so that (8.34). (8.35), and (8.36) are simultaneously satisfied. -
Acknowledgemen t
We are pleased to acknowledge support of this work by the Army Research Office in Durhamunder Contract No. DAHC04-75-G-0025.
References
1101 T. IJupont. L2 estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10 (1973)880-889.
[II J P.A. Raviart. SlIT I'approximation de certaines equations d'cvolutionlincaires et nonlincaires, J. Math.Pures App\. 46(967)II-11l3.
[121 R. Bellman. Stability theory of differential equations (McGraw-lIil1. 1952).II 3 J S.L. Sobolev. Applicalions of functional analysis in mathematical physics (American Mathematieal Society. l'rovidence.1963).[14) M. Lees, A priori estimates for the solution of difference approximations to parabolic differential equations. Duke Ma tho J.
27 (1960) 247-311.
36 /.. C. It't'llford. Jr. a"d J. T Od£'I/.A theory of discontin/lol/s finite demellt Galerkil/ approximations - Part 2
A NOTE ilDj)t.D IN PROOF: In thc proof of Lcmma 6.1. an auxiliary problem is used. Thisauxiliary problem takes the following form:
o((Dcu(/{x)li'x'ux» = u 1JI. v» VuE W~(/).
where 1JI Le//)' q> I. In order to obtain the results of Lcmma 6.2. we assumed a regularity result
We now realize that this assumption limits the class of physical problems to which our resultsapply. Howcver, the techniql,lcs demonstrated arc very general and. in fact. can be used to obtainsimilar results for a much broader class of problems.
