7,~ ; |" I

ELSEVIER Finite Elements in Analysis and Design 16 (1994) 329--336

FINITE ELEMENTS IN ANALYSIS A N D DESIGN

Higher-order finite elements with mass-lumping for the 1 D wave equation

Gary Cohen*, Patrick Joly, Nathalie Tordjman INRIA Domaine de Voluceau-Rocquencourt B. P 1(15 78153 Le Chesnay C~dex. France

Abstract

This paper is devoted to the construction and analysis of a method, higher order in space and time, for solving the one-dimensional wave equation. This method is based on P3 Lagrange finite elements with mass-lumping which avoids the inversion of a mass matrix at each time-step. The mass-lumping implies to make the abscissae of the interior points coincide with these of the Gauss-Lobatto quadrature rule. A Fourier analysis of the method for a regular mesh points out a superconvergence result. The gain of accuracy is illustrated by numerical experiments.

I . I n t r o d u c t i o n

Solving the wave equation in time by numerical methods is a difficult but basic problem for modeling many physical phenomena (acoustics, elastics, Maxwell equations...). Such methods are necessary for two kinds of models for which analytical methods do not work: - non-homogeneous media, - domains of arbitrary shape.

For non-homogeneous media in rectangular domains, as in geophysics, centered finite difference methods (FDMs) give satisfactory results. These methods were studied, first at the second order [1,2], then methods fourth order in space [3] and in time [4-8]. These studies showed a significant gain in accuracy and computing time with a fourth-order FDM. For domains of arbitrary shape, even in the homogeneous case, FDMs are, of course, not so efficient. But, on the other hand, finite element methods (FEMs), although more suited to such domains, raise some troublesome problems as the inversion of a mass matrix and the existence of parasitic waves for higher-order methods. The aim of this study is to construct and analyze by Fourier transform Lagrange finite elements in one-dimensional (1 D) which will provide mass-lumping and an accuracy comparable to that of fourth-order FDM. This accuracy is obtained for P3 finite elements. For these elements, one gets mass-lumping by using a Gauss-Lobatto quadrature rule which implies a modification of the location of the interior nodes of the elements. Fourier analysis of this method points out a superconvergence phenomenon in the case of regular meshes which

" Corresponding author.

0168-874X/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSD1 0 1 6 8 - 8 7 4 X ( 9 3 ) E 0 I 0 2 - 7

3 3 0 G. ('ohen et al. / Finite Elements in ,.lnah'.~i.s and Desi~tn 16 (1994~ 329 336

is confirmed by numerical results• Moreover, the expcriments prove the gain of computing time and accuracy over classical Lagrange P~ finite elements.

2. Presentation of the method

We shall consider the lbllowing model problem:

Find u : IR ×]0. T[--, IR so that:

?:u ?2u ~ ( x , t ) - (~x:(X.t) = 0 in IRx]0, T[,

u(x, 0) = uo(x), ~t(x, 0) = ul(x) in IR

the variational formulation of which is

(1)

d 2 ~ /~ ?u dv u r d x + ,,-~ ~ dx = 0 Vr E H ' ( N ) . (2)

We shall deal, in this section, with the semi-discretization in space which is the main point of this study. Let Vj~(IR) = {v E H~(IR) [ Vi E 7/ vii,,..,,. I E Pk} be the Lagrange finite element space ofkth-order associated to a mesh {x,} of IR. The semi-discretized tbrmulation of the problem can be written as follows:

Find u~(-,t) E Vj~(IR), t E]0, T[ so that:

at'd2 /~ / 3uh dch_;_(x,t)~_r(x) Vt'~, E Vj~(IR)5~ uh(x,t)Vj,(x) dx + dx = O, (3) • . ( ' X

Uh(X, 0) = UOj,(X), ~t-(x,O) -- UI.h(X) in IR.

Let (2,)iel be a basis of V~(IR). Then (3) is equivalent to the following (infinite) ordinary differential equations system:

A ' , , . 2

Mh d~- ( t ) + Khuh(t) = 0, with

(Mh)I., --- / . 2t(x);t,(x) dr , (4)

~?).1 g)., (K,)/., = . ~K~-x(X)?~x(X)dx, (i,l) E 7/".

l fonc chooses the canonical basis for V~, the mass matrix Mh is heptadiagonal tbr k = 3. This implies that, after discretization in time, an inversion of such a (symmetric positive) matrix will be required at

G. Cohen et al./ Finite Elements in Analysis and Desi¢ln 16 f1994 ) 329 336 331

1 2 3 4 1 2 3 4

I i i I = I i i i Classical Gauss-Lobatto

Fig. I.

each time-step. Of course, this matrix will be even larger in the 2D and 3D cases. At this stage of the method, the interest of mass-lumping, that we wish to reach thanks to the use of adequate quadrature

formulas, is quite obvious. In order to keep the same accuracy of the scheme, one must use, for the kth order, a quadrature rule

exact for P:k-~ [9]. So, for P3, polynomials of P~ must be exactly integrated. A necessary condition to get mass-lumping is that the degrees of freedom of the finite elements

coincide with the quadrature points. However, if one keeps the standard nodes of P3, one can only get a quadrature rule exact for P3. So, in order to reach the required accuracy, we shall make the interior points of the elements coincide with the quadrature points of the Gauss-Lobatto quadrature rule (which uses the end points of the elements), see Fig. 1. That means that, on the reference element [0, 1], the abscissae of the interior points will be It -- (5 - x/5)/lO and v = (5 + v'r5)/lO instead of I/3 and 2,/3. The weights of the Gauss-Lobatto quadrature rule are:

( ' )1 --- 0 ) 4 : 112 and o)2 = o)3 = s_,.

This idea was first introduced by Hennart for steady [10] and parabolic [l I] problems.

3. Fourier analysis o f the semi-discret ized equation

Let us consider a uniform mesh (i.e. x/ = jh, .j E 77, where h is the space-step). On such a mesh, finite elements Ps will provide three kinds of nodes periodically repeated: the ends of the intervals and the two interior nodes, as one can see in Fig. 2.

The Fourier analysis of this approximation will enable us to estimate the error committed at each kind of node of the mesh. For this purpose, we shall introduce three auxiliary functions hA, i~h_~, and hh., which interpolate uh at the points x , x,_ t, and x, . , . Then, after having written the system (4) in a continuous form (i.e. uh(x, t), uh(x + lth, t) and Uh(X + vh, t) will take the place of u,, u,.~, and ui_,), we replace u~ by £'h, hj,_~, and hh~, in (4). Then, by applying the Fourier transform to the new equation, we get an ODE system in hh, hj,.~, and hA+, which are the Fourier transforms of the three auxiliary functions. By comparing the solutions of this system with the Fourier transform h of the solution of the continuous problem ( 1 ), one gets, by using Plancherel's formula, the error estimates given in Theorem ! for which we shall not give detailed proof due to lack of space.

Th eorem I. Let us suppose that (uo, ul) E H7(IR) x H¢'(IR),(~l,,hh.i,,fi~.,) can he decomposed as ./ollows:

12 312 312 312 3, I J l i l i i l i I

Fig. 2.

332 G. ('ohen et al. / Finite Eh,ments in AnalysL~" and Desigln 16 r 1994) 320 336

=

- - Uh-t

where, by definition: (~np ~np ~np tfl, , u~_,, u h., ) is the non-parasitic part o f the sohttion.

(h~, ~ . , , h~.,) is the parasitic part o f the solution.

Moreover, !f II • II denotes the L '--norm, one can get the.following error estimates:

for the non-parasitic ware.

-.p C h ~, [ I de'u0 , d~'uo ' U ~ 1 ul ~< = 7 , I I - t ( (LX"'

f :'"P - ul <~ Ct¢1 llh t I~

np ,, I uh. , . - ur <~ Ch ~ k

./or the parasitic wave:

-p, { d~'u° Iluhl~-< c/1 ~ I I d ~

d~'u0 !~L,, ~ C h 5 ll-dV; II

d6uo I I~+,ll ~< ch ~ II-dT II

d°/'/~ ' } + 2t :,i ~Vr~, ,

d~u° i i d V lt° dt'll l

d~'uo dVuo dalai G7, II ~- tllG~-r: I - 2t!l Gz-r~

,

/-

d 6 u i

d', / + t dx 6 ,

d~'tt I } 4- t i i ~ - •

Remark I. This theorem shows that the superconvergence phenomenon is more important at the end points o f the elements than at its interior points.

Remark 2. The proof of Theorem 1 as well as analogous results for P~ and Pz elements may be found in [12].

Remark 3. The proof of Theorem 1 is based on the computation of the eigenvalues and eigenvectors ot" Nh which is the symbol (operator got by Fourier transform) of Nh = M~;-~Kh (Mr, and K, defined in (4).

G Cohen et al./Finite Elements in Analysis and Desiqn 16 (1994) 329 336 333

These quantities, sought as Taylor expansions in h were obtained with the help of MAPLE software. This computation pointed out the existence of one physical wave for whose velocity one gets an error in O(h ~) and two parasitic waves, whose velocities tend to infinity when h tends to zero.

4. D i s c r e t i z a t i o n in t i m e

The most natural way to get a fourth-order time-discretization would be to discretize the time derivative by using a centered fourth-order finite difference scheme. Unfortunately, such a scheme is unconditionally unstable. So two solutions remain: either use a standard second-order finite difference scheme,

rt~ I p~ rl-- I ~"2t lh( tn) ~'~--- llh - - 21"1t' -~- Uh ( 5 )

~,l 2 A t +̀ '

which is stable but reduces the convergence of the method to second order or apply a modified equation approach described, for instance, in [6] but in a slightly different way, as described below.

By writing down the Taylor expansion of (5), we get

~2Uh(tn ) U~ +1 -- 2U'~ + U~ -I A t 2 C4Uh(ln) - - - + O ( h 6) . (6 )

?12 ~ l 2 12 ~t 4

At this step, ?4uh/?t~ is replaced in [6] by A2u by using (I), then discretized in space by a second-order centered FDM (this process can be written (A2)hUh). Of course, this approach does not work in our case for in V~(IR), which is a subset of H~(IR), we cannot approximate A 2 which operates in H2(IR). For that reason, instead of replacing ~.'4Uh/?14 by (A2)hUh, we shall replace it by NZuh (which corresponds to (Ah)2Uh) by using (4).

This new formulation can then be written as

t , ; " = 2u~ - u',:-" 'A"4n- Nh u'~ - ~ N h u h . (7)

Of course, such an approach is more expensive than that described in [6] in which (A2)s had a second- order FDM stencil. In fact, formulation (7) is equivalent, from the point of view of computing time, to the computation of two time-steps of the scheme, second order in time. However, the stability coefficient of the second-order approximation in time is

2 / [ ~ 6 ( 7 + v /~) ] (~ 0.232)

and for fourth order in time, wc get

2v /3 / [~6 (7 + v /~) ] (-~ 0.4018),

so that one can use a time-step about two times as large for the fourth-order scheme as for the second- order one.

A natural idea given by the superconvergence result would be to try to get sixth order in time. By using the same process as fourth order in time, we get the following formulation:

334 G. Cohen et al./ Finite Elements in Analvsi.s and Desiqn 16 f1994) .¢29 336

0 .4

0 . 2

0 .1

O'

- O . l

-o,, I

- O . J

- O . t

- 0 . $

- 0 . 6

Fig. 3. Scheme. second-order in time lbr classical P~ left) and with mass-lumping (right) meshes: 3.5 elements per

wavelength.

. . . . . . , ,_ .I 2N,, L," u h ,--j, - uh /72 Nh u;, ~' + ~-"-Nhuh3u " (8)

Such a schcme is three times morc expensive than the second-order-in-time schemc. Morcovcr, its stability coefficicnt is

+ l0 - - - - ( ~ _ 0.319) / e

\ /6(7 .-. v / ~ )

and does not balance, as for the fourth-order scheme, the increase in computing time coming from A~. In fact, numerical experiments show that thc gain of accuracy provided by a schemc sixth order in time is not significant enough and its use is too expensive.

5. N u m e r i c a l r e s u l t s

We tested these different approaches for the following problem:

~.2 It ~2 U - - - ( x , t ) - - , ~ ( x , t ) = C , q ( x ) f ( t ) ~t 2 ~x-

in ]0, 12[ x ]0 ,50[ .

j ' ( l - ( x 6)2.:4) ~ i n [ 4 . 8 ] . t/(X, O) C2uo 0 in ]0,4[ U ]8.12[,

(9)

~-,t(x.0) -- 0 in ]0. 12[.

u(O,t)-=- u(12,/) = 0 .

where .q is a Gaussian function centered at x = 6 and./( / ) is the second derivative o fa Gaussian function (a Ricker function).

G. Cohen et al./ Finite Elements in Analrsis and Desi$In 16 /1994) 329 336 335

ii o.

01

-0.2

-o.)

-m4

-o,5

-0.6 ~ 1 2

J Fig. 4. Mass-lumping and lburth-order in time for unitbrm (left) and non-unilbrm (right) meshes: 3.5 elements per wavelength.

0.2

0.1

0

-0.1 i i i i /

2 4 6 I 1

0.4

0.$

0.4

0.1

0.6

0.~

O.t

0.1

0.2

0.1

0

-0.1 I * I I a

; t I I 1

Fig. 5. Second experiment: Mass-lumping and second-ordcr in time (Ict't) Iburth-ordcr in time (right) meshes: 5 elements per wavelength.

In the following, we present two experiments. - I n a first experiment, we set C~ = 1 and C~ .... 0, which corresponds to a solution with right-hand side. We use uniform and non-uniform meshes with 3.5 elements per wavelength.

Non-uniform meshes werc obtained by random modification o f the uniform mesh so that the dis- placement o f the nodes should not exceed 20% of the length of the elements. In Figs. 3 -5 the curves in dotted lines represent the exact solution (obtained by using a vcry fine mesh) and those in continuous lines, the computed solution. We can make the fol lowing comments on these figures. (1) The time o f computation is thrcc times greater for the scheme without mass-lumping (z¢ = 0.15)

versus the one with mass-lumping (z¢ = 0.19) and we get almost the same accuracy. (2) On the other hand, we get an almost perfect accuracy for a scheme with fourth order in time (z¢ =

0.38), with the same number o f elemcnts. To get the same accuracy, one must use 25 elements for the same scheme in spacc and second order in time and 16 elements for a schemc, P~ in space and tburth order in time.

- In a second experiment, we set C~ = 0 and C2 = 1, which corresponds to a solution with an initial value. Wc use a mesh with 5 elements per wavelength (:~ = 0.19 for a scheme second order in time and .z¢ = 0.38 for a scheme fourth order in time). This mesh is required to have an "exact" solution

336 G. Cohen et al. / Finite Fh'ments in Analysis and Desiqn 16 ~ 1994 j 329 336

for the scheme fourth order in time. We notice, in this case also, the gain of accuracy provided by a fourth-order discretization in time.

R e f e r e n c e s

[I] A. Bambcrgcr, G. Chavcnt and P. Lailly, "Etude dc schemas numeriqucs pour Ics dquations de I'elasdodynamique lindairc", Rapport INRIA RR41, 1980.

[2] 1,. Nicoletis, "'Simulation numerique de la propagation d'ondcs sismiqucs". Thesis, Unix. Pierre ct Marie (.'uric, 1981. [3] R.M. Alford, KR. Kelly and D.M. Boorc, "'Accuracy of finite difference modeling of the acoustic wave equation",

Geoph.vsics 39 (6), pp. 834 842, 1974. [4] M.A. Dablain, The application ~?/ hi~lh m'der dif]~'remin~l /~r the scalar ware equation, Geophysics, n~'51, l, pp.

54-66, 1986. [5] G. Cohen "'A class of schemes, fourth order in space and time, for the 2D wave equation", Proc. 6th IMA ('S lnternat.

,S~vmp, on ()mtputer Metho~A/~," Partial Di(J~,rential Equation.s. Bethlehem, PA, USA. pp. 23 27, June 1987. [6] (i. Cohen, P. Joly "Fourth order schemes for the heterogeneous acoustics equation", ('omput. Methodx Appl. Mech.

l:n¢t. 80. pp. 397~4.07, 1990. [7] (;.R. Shubin and J.B. Bell, "'A modified equation approach to construction fourth order methods for acoustic wave

propagation", S IAM J. SoL Statit. ()re(put. 2, pp. 135 151, 1987. [8] J. Tuomcla, Analyse de terrains problcmes lies a la r6solution num,Sriquc des 6qua(ions aux d,,Sriv6cs partiellcs

hyperboliqucs lin6aircs, Thesis, Univ. Paris 7, 1992. [9] P.(i. Ciarlct, Fhe Finite Element Method./i~r Elliptic Probh'm~, Exercise 4.1.7, North-Holland, Amsterdam. p. 204.

[10] J-P. tlennart, E. Sainz and M. Villcgas, "'()n the efficient use of the finite clcrncnt method in static neutron diffusion calculations", ('ompt~t. Metho¢ts Nuclear En,q. I, p. 3.87, 1979.

[ l lJ J-P. Hennart, Topics in .finite eh'ment di,~cretizati~m ~?/paraholic orohm(ion prohh'ms, l,ccturc Notes in Mathematics, 909, pp. 185 199, 1982.

[12] (i. Cohen, P. Joly and N. Tordjman, "Elements finis d'ordrc 61eve avcc condensation de masse pour 1",Squation des ondcs". INRIA Report, to appear.