cohen - mixed finite elements with mass-lumping for the transient wave equation

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Journal of Computational Acoustics, Vol. 8, No. 1 (2000) 171–188c© IMACS

MIXED FINITE ELEMENTS WITH MASS-LUMPING FOR THE

TRANSIENT WAVE EQUATION

GARY COHEN and SANDRINE FAUQUEUX

INRIA, Domaine de Voluceau, Rocquencourt, B.P. 105,78153 Le Chesnay Cedex, France

E-mail : [email protected]; [email protected]

Received 15 June 1999Revised 30 September 1999

Solving the acoustics equation by finite elements with mass-lumping requires the use of spectralelements. Although avoiding the inversion of a mass-matrix at each time-step, these elementsremain expensive from the point of view of the stiffness-matrix. In this paper, we give a mixedfinite element method which provides a factorization of the stiffness-matrix which leads to a gainof storage and computation time which grows with the order of the method and the dimensionin space. After proving the equivalence between classical spectral elements and this method, wegive a dispersion analysis on nonregular periodic meshes. Then, we analyze the accuracy and thestability of Q3 and Q5 approximations on numerical tests in 2D.

1. Introduction

For a long time, the use of finite elements, in particular in their higher-order form, for

solving the transient wave equation, raised the important problem of the inversion of the

mass-matrix, which appears after discretization. This difficulty has been overcome by the

use of Legendre–Gauss–Lobatto quadrature points as points of interpolation. This choice,

currently called spectral element,1 ensures mass-lumping without affecting (and even by

increasing) the accuracy of the method.2−6 However, the FEM approach remains expensive

in terms of storage since the stiffness-matrix depends on space and needs to be stored for each

degree of freedom of the mesh. This storage is an important barrier to the applications.7,8

In this paper, we present a new mixed FEM approach with mass-lumping, in the spirit of

that used in Ref. 9, which realizes over the classical spectral elements an important gain

of storage and, in the same time, a significant gain in the time of computation. Both gains

of storage and time increase with the dimension of the problem and the order of the method.

The extension of this approach to the system of elastic waves can easily be realized and is

being studied.

171

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172 G. Cohen & S. Fauqueux

2. Position of the Problem

We want to solve the heterogeneous transient wave equation in dimension N = 2, 3:

Find u: Ω× [0, T ]→ R so that:

ρ∂2u

∂t2(x, t)−∇ · (µ∇u(x, t)) = f(x, t) in Ω× [0, T ]

u(x, 0) = 0 in Ω

∂u

∂t(x, 0) = 0 in Ω

u(x, t) = 0 on ∂Ω

(2.1)

where Ω ⊂ RN , x = (x1, . . . , xN ) ∈ Ω, by using a classical C0 FEM.

Let us set: Ω = ∪Nej=1Kj , Kj : quadrilateral or hexahedron of any shape,a

Ukh = w ∈ C0(Ω) so that w|Kj Fj ∈ Qk(K) and w = 0 on ∂Ω (2.2)

where K = [0, 1]N , Fj = (F j1 , . . . , FjN ) is the transform so that Fj(K) = Kj and Qk(K) is

the space of polynomials on K of degree less or equal to k in each variable.

Then, the approximation of (2.1), after variational formulation, can be written:

Find uh ∈ L∞(0, T ;Ukh ) so that:

d2

dt2

∫Ωρuhϕh, dx +

∫Ωµ∇uh · ∇ϕh dx =

∫Ωfϕh dx ∀ϕh ∈ Ukh

uh(x, 0) = 0

∂uh∂t

(x, 0) = 0

(2.3)

The decomposition of (2.3) on the canonical basis BU of Ukh provides the discrete problem:

d2

dt2DhU +KhU = Fh (2.4)

where U is the vector of the components of uh on BU , Dh the mass-matrix and Kh the

stiffness-matrix.

Let NU = dim(Ukh ). If ϕj is a basis function of Ukh , Sj its support and if we set Kh =

(kpq)(NU ,NU )(p,q)=(1,1), we have:

kpq =

∫Ω∇ϕp · ∇ϕq dx

=∑

K`∈Sp∩Sq

∫K`

∇ϕp · ∇ϕq dx

=∑

K`∈Sp∩Sq

∫K

|J`|DF−1` µDF ∗−1

` ∇ϕp F` · ∇ϕq F`dx

(2.5)

aIn all the following, the boundaries of Kj can even be supposed curved.

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Mixed Finite Elements with Mass-Lumping for the Transient Wave Equation 173

where x = (x1, . . . , xN ) ∈ K, ∇ = (∂/∂x1, . . . , ∂/∂xN )T , DF` is the Jacobian matrix

corresponding to F` and J` = det(DF`).

Let ν = (k + 1)N and xpνp=1 the Lagrange points of interpolation on K. By choosing

xp = ξp, where ξpνp=1 are the Legendre–Gauss–Lobatto quadrature points on K and by

using this rule to compute all the integrals of (2.3), one shows6 that Dh becomes diagonal

without loss of accuracy for the method.

However, formula (2.5) shows that one must store all the interactions between the basis

functions on each degree of freedom of each quadrilateral or hexahedron,b which is a huge

storage.

For this reason, we are going to introduce a new formulation of the problem which will

lead to an important saving of storage.

3. A mixed Finite Element Approach

3.1. Formulations of the wave equation as systems

In order to formulate the mixed FEM, we first rewrite the wave equation (2.1) as a first

order system in the pressure u and an auxiliary variable v:

ρ∂u

∂t= ∇ · v + F in Ω× [0, T ]

µ−1∂v

∂t= ∇u in Ω× [0, T ]

u(x, 0) = 0 in Ω

v(x, 0) = 0 in Ω

u(x, t) = 0 on ∂Ω

(3.1)

in which F is a primitive of the right-hand side used in (2.1).

One can easily see that (2.1) and (3.1) are equivalent.

3.2. Variational formulation of the system

In order to give the variational formulation of the system (3.1), we must first define the

functional spaces in which the unknowns will be sought. The classical mixed formulation of

this problem should be done by setting ∀ t ∈ ]0, T [, u(., t) ∈ L2(Ω) and v(., t) ∈ H(div,Ω)c

(i.e., discontinuous finite elements for U and elements with continuous fluxes or normal

components for v.10,11 However, such a choice would complicate the algorithm in our case.

bActually, since the matrix DF−1` µDF ∗−1

` is symmetric, one must store just (k + 1)N ((k + 1)N + 1)/2

interactions per element instead of (k + 1)2N interactions.cWe remind that this space is defined as:

H(div,Ω) = w ∈ [L2(Ω)]2, so that div w ∈ L2(Ω)

where div w is taken in the sense of distributions.

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So, we shall search u(., t) in

H10 (Ω) = w ∈ L2(Ω), so that ∇w ∈ L2(Ω), u|∂Ω = 0

which is its natural functional space and we shall take v(., t) just in [L2(Ω)]N .

In these spaces, the variational formulation of (3.1) can be written as follows:

Find u ∈ L∞(0, T ;H10 (Ω)) and v ∈ L∞(0, T ; [L2(Ω)]N ) so that:

d

dt

∫Ωρuϕ dx = −

∫Ω

v · ∇ϕ dx +

∫ΩFϕ dx ∀ϕ ∈ H1

0(Ω)

d

dt

∫Ωµ−1v ·ψ dx =

∫Ω∇u ·ψ dx ∀ψ ∈ [L2(Ω)]N

u(x, 0) = 0 in Ω

v(x, 0) = 0 in Ω

(3.2)

3.3. Definition of the approximation

3.3.1. Approximation on K

Let us first define the polynomial approximation on K for each variable u and v in the same

way as in Ref. 9.

We shall define on K two kinds of degrees of freedom:

• As in the first model given in (2.3), we define scalar degrees of freedom for u which

correspond to the points of interpolation of Lagrange in Qk(K), taken at the Legendre–

Gauss–Lobatto quadrature points.

• At the same points, we define, for v, a N -dimensional vector corresponding to the value

of a function of [Qk(K)]N at this point.

The basis functions for u are the classical Lagrange interpolation basis functions, that is

functions (ϕp)νp=1, so that ϕp(ξq) = δpq, where δpq is the Kronecker symbol.

For v, the basis functions will be N -dimensional vectorial functions ψ`

p, p = 1, . . . , ν,

` = 1, . . . , N , so that the `th component of ψ`

p will be equal to ϕp and the other ones to

zero.

3.3.2. Approximation on Ω

On the basis of this interpolation, we can define the spaces of approximation in which u and

v will be sought.

For u, the space of approximation will remain Ukh defined in (2.2).

A natural space of approximation V kh for v would have been:

V kh = w ∈ [L2(Ω)]N so that w|Kj Fj ∈ [Qk(K)]N (3.3)

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However, for reasons which will appear later, we shall define this space as follows:

V kh = w ∈ [L2(Ω)]N so that JjDF

−1j w|Kj Fj ∈ [Qk(K)]N (3.4)

From a more practical point of view, the fact that w ∈ [L2(Ω)]N provides functions

discontinuous at the interfaces between two elements of the mesh. On the other hand,

JjDF−1j w|Kj Fj ∈ [Qk(K)]N means that w is obtained by using the H(div)-conform

transform (1/Jj)DFj which transforms a vector normal to ∂K into a vector normal to ∂Kj .

3.3.3. The approximated problem in space

We can now define the approximated problem in space, in the above defined spaces of

approximation:

Find uh ∈ L∞(0, T ;Ukh

)and vh ∈ L∞

(0, T ;V k

h

)so that:

d

dt

∫Ωρuhϕh dx = −

∫Ω

vh · ∇ϕh dx +

∫ΩFϕh dx ∀ϕh ∈ Ukh

d

dt

∫Ωµ−1vh ·ψh dx =

∫Ω∇uh ·ψh dx ∀ψh ∈ V k

h

uh(x, 0) = 0 in Ω

vh(x, 0) = 0 in Ω

(3.5)

3.4. Features of this approximation

By taking in (3.5) ϕh and ψh as basis functions of Ukh and V kh derived from the basis

functions on K defined in 3.3.1 and after computing all the integrals by the Gauss–Lobatto

quadrature rule, we get the semi-discrete problem in space:Dh

dU

dt= −RhV + Fh

BhdV

dt= R∗hU

(3.6)

where U is the vector of the components of uh on the basis of Ukh and V, the vector of the

components of vh on the basis of V kh . Dh is the mass-matrix involved in (2.4).

The definition of the degrees of freedom and the use of the Gauss–Lobatto quadrature

rule provides a N ×N block-diagonal mass-matrix Bh. To each point Fj(ξp) of an element

Kj corresponds a N ×N block Bj,ph whose elements bj,p`m are:

bj,p`m = ωpµ−1 Fj(ξp)

N∑q=1

∂F j`∂xq

(ξp)∂F

jm

∂xq(ξp) ` = 1, . . . , N, m = 1, . . . , N (3.7)

where (ωp)νp=1 are the weights of the Gauss–Lobatto quadrature formula.

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Since Dh is diagonal and Bh block-diagonal, both D−1h and B−1

h can be stored and (3.6)

can be rewritten as: dU

dt= D−1

h (RhV + Fh)

dV

dt= B−1

h (R∗hU)

(3.8)

which leads to a very fast algorithm of resolution in time.

However, the main feature of this approximation lies in the definition of V kh since if

ϕh ∈ Qk(K) and ψh ∈ [Qk(K)]N so that ϕh Fj = ϕh and ψh Fj = (1/Jj)DFjψh, we

have the following relation:∫Kj

∇ϕh ·ψh dx =

∫K

∇ϕh · ψhdx (3.9)

This relation implies that all the terms of the matrix Rh can be computed locally on K.

That means that the whole information on Rh is given by the knowledge of the integrals of

the basis functions on K:∫K

∇ϕp · ψqdx p = 1, . . . , ν, q = 1, . . . , 2ν (3.10)

This property implies that just the 2(k+ 1)2N integrals defined in (3.10) must be stored for

the matrices Rh and R∗h, which realizes a huge gain of storage!

In fact, the whole information about geometry and physics is contained in the mass-

matrices Dh and Bh only.

Fig. 1. The ratios between the storages required by classical spectral FEM and mixed FEM in 2D (left) and3D (right). The abscissae represent the order of the polynomial approximation. One can notice that the newmethod is more expensive for the first-order approximation and that the gain increases with the order andthe dimension.

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The gain of storage taking into account both matrices and variables is represented in

Fig. 1.

Besides the gain of storage, this approach presents another feature. One can easily show

that the use of degrees of freedom for u and v at the same points and the properties of

orthogonality of these degrees of freedom on K provide remarkable sparsity properties for

the stiffness matrices Rh and R∗h. Actually, all the interactions between degrees of freedom

on a line nonparallel to the axes are equal to zero. Moreover, the sparsity of the matrices

increase with the order of the method and the dimension in space. As we shall see later, this

property of sparsity will lead to a significant gain of time over classical spectral elements.

All these features would provide a very interesting algorithm if the accuracy properties

of the method are comparable to those of the spectral elements. This question is the purpose

of the following paragraph.

4. A Theorem of Equivalence

The comparison of mixed FEM with spectral FEM will be fixed in

Theorem 4.1. If Bh, Kh, Rh and R∗h are the matrices defined in (2.4) and (3.6), we have

the factorization:

Kh = RhB−1h R∗h (4.1)

which ensures that

U = U (4.2)

U and U given by (2.4) and (3.6) respectively.

Proof. Let us first give some definitions

• Let τ = Kjj=1,...,Ne , Kj ⊂ Ω

• ∀ j = 1, . . . , Nd, where Nd is the number of degrees of freedom in Ω− ∂Ω, we set

Sj = i so that Ki ⊂ Supp(ϕj)= index of all the elements in the support of ϕj

(4.3)

and ∀ i = 1, . . . , Ne, the functions loci and globi so that:

∀ j = 1, . . . , Nd, loci(j) = p and globi(p) = j with Fi(ξp) = xj . (4.4)

• V kh will be defined in a more general frame which will provide a better understanding of

the theorem:

V kh = w ∈ (L2(Ω))2 so that P−1

i w|Ki Fi ∈ [Qk(K)]N (4.5)

where, ∀ i, Pi is an invertible N ×N -matrix.

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• Let (ϕ`)`=1,..., Nd be the basis functions of Ukh and (ϕj)j=1,..., ν the basis functions of Qk(K)

and (ψj,l)j=1,..., ν; l=1,...,N those of [Qk(K)]N , as defined in 3.3.1.

From these functions, we define ϕij and ψij,l so that ϕij = ϕglobi(j)|Ki , ϕij Fi = ϕj and

ψij,l Fi = Piψj,l with:

Ki = Supp(ϕij) = Supp(ψij,l) .

• With these notations, we get:

uh =

Nd∑i=1

uiϕi =Ne∑i=1

ν∑j=1

uglobi(j)ϕij (4.6)

vh =Ne∑i=1

ν∑j=1

N∑l=1

vij,lψij,l (4.7)

Remark 4.1.

(1) These notations take into account the facts that vh is discontinuous and that uh vanishes

on the boundary of Ω.

(2) In order to simplify the proof, we shall set ψp,l so that:

∀ ξq, q = 1, . . . , ν, ψj,l(ξq) = δjqel where (e1, . . . , eN ) is the canonical basis of RN and

δjq is the Kronecker symbol.

• Decomposition of∫

Ω vh · ψip,m dx:∫Ω

vh ·ψip,m dx =Ne∑`

∫K`

vh ·ψip,m dx (4.8)

By using (4.7), we get:∫Ω

vh ·ψip,m dx =

∫Ki

ν∑j=1

N∑l=1

vij,lψij,l ·ψip,m dx (4.9)

Let us set: x = Fi(x).∫Ω

vh ·ψip,m dx =ν∑j=1

N∑l=1

vij,l

∫K

|Ji|Pi ψj,l · Piψp,m dx (4.10)

After transposition, we get:∫Ω

vh ·ψip,m dx =ν∑j=1

N∑l=1

vij,l

∫K

|Ji|P ∗i Pi ψj,l · ψp,m dx (4.11)

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The use of the Gauss–Lobatto quadrature rule combined with the properties of

orthogonality of the functions ψp,m leads to:

∫Ω

vh ·ψip,m dx 'ν∑j=1

N∑l=1

vij,l

ν∑q=1

ωq |Ji(ξq)| P ∗i (ξq)

×Pi(ξq) ψj,l(ξq) · ψp,m(ξq)

'ν∑j=1

N∑l=1

vij,l

ν∑q=1


×Pi(ξq) δj,q el · δp,q em

' ωpN∑l=1

vip,l|Ji(ξp)|P ∗i (ξp) Pi(ξp)el · em

(4.12)

Remark 4.2. The above formula shows that the matrix Bh is N ×N -block diagonal.


Ω∇uh ·ψip,m dx:

We use the same method as for∫

Ω vh ·ψip,m dx. This method provides:

∫Ω∇uh ·ψip,m dx =

ν∑j=1

uglobi(j)

∫K

|Ji| P ∗i DF ∗−1i ∇ϕj · ψp,m dξ

'ν∑j=1

uglobi(j)

ν∑q=1


×DF ∗−1i (ξq)∇ϕj(ξq) · ψp,m(ξq)

'ν∑j=1

uglobi(j)ωp |Ji(ξp)|P ∗i (ξp)

×DF ∗−1i (ξp) ∇ϕj(ξp) · em

(4.13)

Finally, we get the following relation:

N∑l=1

vip,l|Ji(ξp)|P ∗i (ξp)Pi(ξp)el · em

=ν∑j=1

uglobi(j)|Ji(ξp)|P ∗i (ξp)DF

∗−1i (ξp)∇ϕj(ξp) · em

(4.14)

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Ω vh · ∇ϕn dx:

We use the equality:

ϕn =∑i∈Sn

ϕn|Ki =∑i∈Sn

ϕiloci(n)

The same method leads to:∫Ω

vh · ∇ϕn dx =∑i∈Sn

ν∑p=1

N∑l=1

vip,l

∫Ki

ψip,l · ∇ϕiloci(n) dx

=∑i∈Sn

ν∑p=1

N∑l=1

vip,l

∫K

|Ji|DF−1i Piψp,l · ∇ϕloci(n)dx

'∑i∈Sn

ν∑p=1

N∑l=1

vip,lωp|Ji(ξp)|DF−1i (ξp)

×Pi(ξp)el · ∇ϕloci(n)(ξp)

(4.15)

Now, we are going to introduce P ∗i (ξp)Pi(ξp) in order to use (4.14).

We can write:∫Ω

vh · ∇ϕn dx '∑i∈Sn

ν∑p=1

ωp DF−1i (ξp) P

∗−1i (ξp)

×N∑l=1

vip,l |Ji(ξp)| P ∗i (ξp) Pi(ξp) el · ∇ϕloci(n)(ξp)

(4.16)

Let us set

∇ϕloci(n)(ξp) =N∑m=1

∂ϕloci(n)

∂xm(ξp)em

∫Ω


ν∑p=1

ωp DF−1i (ξp) P

∗−1i (ξp)

×N∑m=1

∂ϕloci(n)

∂xm(ξp)

N∑l=1

vip,l |Ji(ξp)| P ∗i (ξp)Pi(ξp)el · em

(4.17)

Now, let us set Api = |Ji(ξp)| P ∗i (ξp)DF∗−1i (ξp). By using (4.14), we get:∫

Ωvh · ∇ϕn dx '

∑i∈Sn

ν∑p=1

ωpDF−1i (ξp) P

∗−1i (ξp)

×N∑m=1

∂ϕloci(n)

∂xm(ξp)

ν∑j=1

uglobi(j)Api ∇ϕj(ξp) · em

(4.18)

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By switching the sums in p and j and after simplication, we have:

∫Ω


ν∑j=1

uglobi(j)

ν∑p=1

ωp |Ji(ξp)|DF−1i (ξp)

×DF ∗−1i (ξp) ∇ϕloci(n)(ξp) · ∇ϕj(ξp)

(4.19)

which is the approximation by the Gauss–Lobatto quadrature rule of∫K

|Ji|DF−1i DF ∗−1

i ∇ϕloci(n) · ∇ϕj dx (4.20)

So, if we denote∫ GLQ f(x)dx the approximation by a Gauss–Lobatto quadrature rule

of∫Q f(x)dx, we get

∫ GL

Ωvh · ∇ϕn dx =

∑i∈Sn

ν∑j=1

uglobi(j)

∫ GL

K

|Ji|DF−1i DF ∗−1

i ∇ϕj · ∇ϕloci(n) dx

=∑i∈Sn

ν∑j=1

uglobi(j)

∫ GL

K

|Ji|DF ∗−1i ∇ϕj ·DF ∗−1

i ∇ϕloci(n) dx

=∑i∈Sn

ν∑j=1

uglobi(j)

∫ GL

Ki

∇ϕij · ∇ϕiloci(n) dx

which finally provides:∫ GL

Ωvh · ∇ϕn dx =

∑i∈Sn

∫ GL

Ki

∇uh|Ki · ∇ϕin|Ki dx

=

∫ GL

Ω∇uh · ∇ϕn dx

(4.21)

So, by taking for ϕn the elements of BU , we get (4.1).

Remark 4.3.

(1) For reasons of simplicity, this proof was done when ρ = µ = 1 but the proof remains

the same when these parameters depend on x.

(2) The definition (4.5) of V kh shows that this theorem holds for any approximation of

[L2(Ω)]N .

(3) Actually, this proof holds for any quadrature rule, as soon as the quadrature points

coincide with the degrees of freedom.

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(4) The factorization given in 4.1 can be interpereted as a discrete expression of:

ρ∂2u

∂t2= ∇ · v + f in Ω× [0, T ]

µ−1v = ∇u in Ω× [0, T ]

u(x, 0) = 0 in Ω

v(x, 0) = 0 in Ω

u(x, t) = 0 on ∂Ω

(4.22)

5. Dispersion Analysis of the Method

A complete dispersion analysis of the method for Q1, Q2 and Q3 approximation was made

in Ref. 6 in 2D. However, this analysis was made for regular meshes only. In this section,

we propose a dispersion analysis for more general elements in 2D, as in Ref. 9.

Of course, a dispersion analysis is based on plane wave analysis which must be performed

on a periodic mesh. For this purpose, we define a periodic mesh composed of square cells of

size 2h divided into four quadrilaterals. If S = Sii=1,..., 4 are the summits of the square,

M = Mii=1,..., 4, the centers of its edges and A an interior point, each quadrilateral will

have two summits in S, one summit in M and A as fourth summit (Fig. 2).

The dispersion analysis was done for Q3 finite elements. We first define U and V as

a periodic plane wave solutions in (3.6), then we inject the second equation of (3.6) in

the first one and we get a generalized eigenvalues problem of dimension 36. The eigen-

values correspond to the velocities of the physical wave and of 35 parasitic waves which

vanish when h → 0. The coefficients of the matrices of the problem, computed exactly

Fig. 2. The square cell. In the analysis and the experiments, A = (εh, ηh) will be moved along the straightline (D) of equation η = (3ε− 1)/2 (0.6 ≤ ε ≤ 1.4).

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by Maple, are then transferred to a double precision FORTRAN program which gives a

pointwise solution of the problem. For more details on this kind of analysis, one can see in

Ref. 6.

In Fig. 3, we give the dispersion curves, that is, the ratio between the numerical velocity

ch of the physical wave and the exact velocity c. One can see that the loss of accuracy

remains reasonable, even for important distortions.

Fig. 3. Dispersion curves (ch/c) versus ε for some angles of propagation (0o, 15o, 30o, 45o) and threeelements per wavelength. ε = 1 corresponds to a regular orthogonal mesh ε = 0.6 or ε = 1.4 correspond todegenerations into triangles.

Fig. 4. The physical domain.

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Remark 5.1. This analysis can be interpreted as a numerical Bloch waves analysis.12

6. Numerical Study of the Method

We studied Q3 and Q5 approximations on a ]0, 12[× ]0, 12[ bounded domain with

homogeneous Dirichlet conditions on the boundary (Fig. 4). The right-hand side is:

Fig. 5. Periodic mesh for Q5 with 16× 16 elements and ε = 0.7.

Fig. 6. CFL for a leapfrog scheme versus ε for Q3 and Q5 approximations. The lower CFL curve is thatof Q5.

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Table 1. Number of elements in Q3 and Q5

for given CPU times.

CPU Ne in Q3 Ne in Q5

23s 34× 34 16× 1632s 38× 38 18× 1844s 42× 42 20× 2059s 46× 46 22× 22

f(x, t) = g(t)e−7[(x1−6)2+(x2−6)2] where

• g(t) = 2γ(2γ(t − a)2 − 1)e−γ(t−a)2

• γ =(πb

)2, a = 1.35, b = 1.31.

In the first step, we studied the behavior of periodic meshes (Fig. 5) constructed by using

cells defined as in Fig. 2. We moved the point A along the straight line (D) of equation

η = (3ε − 1)/2 (0.6 ≤ ε ≤ 1) and we drew the curves giving the stability condition (CFL)

(i.e., the ratio between the time-step ∆t and h/k where k is the number of points per

element) versus the parameter ε (Fig. 6). In particular, this study shows that the method

remains stable for important distortions when one divides the CFL by 2.

In the second step, we compared Q3 and Q5 when ε = 1, ε = 0.85 and ε = 0.7 for

equivalent CPU times, as shown in Table 1. The solution was computed for t ∈ ]0, 50[. The

discretization in time was made by using a leapfrog scheme with ∆t corresponding to a CFL

divided by 2. We divided the CFL by 2 in order to reduce the effect of the second-order

character of the leapfrog method. For each experiment, we computed the l2-error on the

seismogram at the center of the domain (where the source is located), given by(Nc∑i=1

(uci − uri )2

/Nr∑i=1

(uri )2

)1/2

(6.1)

where ur is the reference solution computed with a 60× 60 elements mesh and a time-step

equal to 10−3. uc is the computed solution and Nr and Nc are the numbers of points for

the two corresponding seismograms (the time-step of a computed solution is a multiple of

10−3). In Fig. 7, we draw the l2-errors for Q3 and Q5 versus the CPU time. One can see

that Q5 is much more accurate than Q3 and that the distortion has very few influence on

the error. However, large distortions can generate some ripples for coarse meshes in Q5 as

shown in Fig. 8.

The comparison of our method with classical spectral elements (the leapfrog method is

strictly equivalent to centered second-order finite difference) shows that the mixed approach

is 1.1 faster in Q3 and 1.9 faster in Q5 than a classical spectral element method. These

March 24, 2000 11:53 WSPC/130-JCA 0006


Fig. 7. l2-errors versus the CPU times for different values of ε. The lower curves are those of Q5.

March 24, 2000 11:53 WSPC/130-JCA 0006


Fig. 8. Seismograms at the center of the domain on the time interval ]25, 50[ for Q5 when ε = 1 (above left),ε = 0.85 (above right) and ε = 0.70 (below) compared with the reference solution. One can notice someripples for ε = 0.70.

ratios are about 25% better than expected. This additional gain is due to the computation

of array addresses, since the arrays are larger for classical elements.

7. Conclusion

We gave a factorization of the stiffness-matrix of a spectral element method which leads

to an important reduction of storage and a significant reduction of time. Numerical 2D

experiments showed that the use of higher order elements can provide a substantial reduction

March 24, 2000 11:53 WSPC/130-JCA 0006


of time for a reasonable deformations of the elements. This method is being extended to the

elastics system and the preliminary studies gave very promishing results.

References

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2. P. G. Ciarlet “The finite element method for elliptic problems,” North-Holland 1987, exercises4.1.4 and following.

3. G. Cohen, P. Joly, N. Tordjman, “Higher-order finite elements with mass lumping for the 1-Dwave equation,” Finite Elements in Analysis and Design 17 (1994), 329–336.

4. J.-P. Hennart, E. Sainz, and M. Villegas, “On the efficient use of the finite element method instatic neutron diffusion calculations,” Computational Methods in Nuclear Engineering 1 (1979),3–87.

5. J.-P. Hennart, “Topics in finite element discretization of parabolic evolution problems,” LectureNotes in Mathematics 909 (1982), 185–199.

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8. G. Seriani, E. Priolo, “Spectral element method for acoustic wave simulation in heterogeneousmedia,” Finite Elements in Analysis and Design 16(3 4) (1994), 337–348.

9. G. Cohen, P. Monk, “Mur-Nedelec finite element schemes for Maxwell’s equations,” Comp. Meth.in Appl. Mech. Eng. 169(3) (1999), 197.

10. E. Becache, P. Joly, C. Tsogka, “Elements finis mixtes et condensation de masse enelastodynamique lineaire. (I) Construction,” C.R.A.S., 325 srie I (1997), 545–550.

11. J-M. Thomas, “Sur l’analyse numerique demethodes d’elements finis hybrides et mixtes,” Thesed’Etat, Mai 1977, Universite de Paris VI.

12. J. B. Keller, F. Odeh, “Partial differential equations with periodic coefficients and Bloch wavesin crystals,” J. Mathematical Physics, 5 (1964), 1499–1504.

cohen - mixed finite elements with mass-lumping for the transient wave equation

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