A POSTERIORI FINITE ELEMENT ERRORESTIMATION FOR

SECOND-ORDER HYPERBOLIC PROBLEMS

Slimane Adjerid

Department of Mathematicsand

Interdisciplinary Center for Applied Mathematics

Virginia Polytechnic Institute and State UniversityBlacksburg, VA 24061

March 7, 2002

Abstract

We develop a posteriori finite element discretization error estimates for thewave equation. In one dimension, we show that the significant part of the spatialfinite element error is proportional to a Lobatto polynomial and an error estimateis obtained by solving a set of either local elliptic or hyperbolic problems. Intwo dimensions, we show that the dichotomy principle of Babuska and Yu holds.For even-degree approximations error estimates are computed by solving a set oflocal elliptic or hyperbolic problems and for odd-degree approximations an errorestimate is computed using jumps of solution gradients across element boundaries.This study also extends known superconvergence results for elliptic and parabolicproblems [24] to second-order hyperbolic problems.

1 Introduction

A posteriori error estimates are an integral part of adaptive procedures for solving partialdifferential equations. They are used to steer the adaptive process by indicating regionswhere more or less resolution is needed and to control solution accuracy and stop theadaptive process. In summary, error estimation when combined with adaptivity, pro-duces efficient and reliable methods. Several a posteriori error estimates are availablefor linear second-order elliptic [11, 9, 10, 23, 25, 26] and parabolic [4, 6, 2, 7] problems.More recently a posteriori error estimates have been developed for first-order hyperbolicproblems [21, 5, 8]. To the best knowledge of the author, this is the first rigorous studyof asymptotically correct a posteriori error estimates for second-order hyperbolic prob-lems. Ideal error estimates should be (i) asymptotically correct, i.e., they converge to

1

the true error under mesh refinement, (ii) robust when applied to linear and nonlinearproblems and different meshes, (iii) reliable in estimating the error in several norms and(iv) computationally inexpensive relative to the solution cost.

In this paper, we present a posteriori estimates for the spatial bi − p finite elementapproximations and prove that they are asymptotically correct under mesh refinement.Furthermore, these odd-even error estimators apply to a larger set of more cost effectivefinite element spaces. Although the theory is not developed for nonlinear problems andgeneral quadrilateral meshes, computational results indicate they hold for a larger classof problems and meshes. More recent and interesting approaches to error estimationinclude guaranteed bounds [12, 13], goal-oriented estimates [14, 18, 19] and bounds forfunctionals [20] (and references therein).

An outline of the paper is as follows: In §2 we present an a posteriori error analysisfor the one-dimensional wave equation and show several computational results. In §3 wedevelop odd- and even-degree error estimators with their convergence proofs and presentcomputational results for several linear and nonlinear second-order hyperbolic problems.We conclude, in §4, with a few remarks.

2 The One-dimensional Wave Equation

2.1 Problem formulation and preliminary results

We consider the one-dimensional initial-Dirichlet boundary value problem for the gener-alized wave equation

∂ttu + Lu = f(x, t) , 0 < x < 1, t > 0. (2.1a)

Lu = −∂x(a(x)∂xu) + b(x)u, (2.1b)

u(x, 0) = u0(x), ∂tu(x, 0) = u1(x), 0 ≤ x ≤ 1. (2.1c)

u(0, t) = 0, u(1, t) = 0, t ≥ 0, (2.1d)

where ∂ttu is the partial derivative ∂2u/∂t2. The operator L is positive definite, i.e.,a(x) > 0 and b(x) ≥ 0. We assume that u0(x), u1(x), a(x) and b(x) are smooth functionswhere u0 and u1 are consistent with the boundary data.We denote by L2 the space of square integrable functions on [0, 1] and by Hs the usualSobolev spaces of functions defined on [0, 1] with distributional derivatives of order s ≥ 0in L2 and equipped with inner product

(u, v)s =

s∑i=0

(∂iu

∂xi,∂iv

∂xi) (2.2a)

2

and the norm

||u||s =√

(u, u)s, (2.2b)

where

(v, u) =

∫ 1

0

vudx. (2.2c)

We denote by H10 the space of functions in H1 that satisfy homogeneous Dirichlet bound-

ary conditions.The Galerkin formulation for (2.1) consists of determining u(·, t), belonging to H1

0

for each t ≥ 0, such that

(v, ∂ttu) + A(v, u) = (v, f), t > 0, (2.3a)

A(v, u) = A(v, u0), t = 0, (2.3b)

A(v, ∂tu) = A(v, u1), t = 0, ∀v ∈ H10 . (2.3c)

where

A(v, u) =

∫ 1

0

a(x)∂xv∂xu + b(x)vudx. (2.3d)

In our analysis we will need the following normed spaces for T > 0

L∞(Hs) = u | sup0≤t≤T ||(u(., t)||s < ∞ (2.4a)

and

L2(Hs) = u |∫ T

0

||(u(., t)||2sdt < ∞, (2.4b)

equipped with the norms

||u||L∞(Hs) = sup0≤t≤T ||(u(., t)||s, ||u||L2(Hs) =

√∫ T

0

||(u(., t)||2sdt. (2.4c)

We use (2.3d) to define the energy norm

||u||A =√A(u, u). (2.4d)

Local norms, e.g., ||u||s,j, are defined by restricting the integration in (2.2) to [xj−1, xj ].We obtain a discrete weak formulation by partitioning [0, 1] into N subintervals

[xi−1, xi], i = 1, 2, . . . , N , and constructing a finite dimensional space

SN,p = w ∈ H1 |w(x) ∈ Qp([xi−1, xi]), i = 1, 2, . . . , N , (2.5)

3

where Qp is the space of polynomials of degree p. We use the hierarchical basis given onthe canonical element [−1, 1] by

ψ0(ξ) = (1− ξ)/2, ψ1(ξ) = (1 + ξ)/2 (2.6a)

ψi(ξ) =Pi(ξ) − Pi−2(ξ)√

2(2i− 1), i = 2, . . . , p, (2.6b)

where Pi and ψi, respectively, are Legendre and Lobatto polynomial of degree i, i > 1.Furthermore, ψi,j(x) is obtained by mapping ψi(ξ) to [xj−1, xj ].Using (2.6) we may construct a hierarchical basis for SN,p which consists of the piecewise

linear functions φ(1)j (x) such that φ

(1)j (xi) = δij, 0 ≤ i, j ≤ N , where δij is the Kronecker

delta

δij =

1, if i = j0, otherwise.

(2.7a)

Higher-order basis functions are defined as

φ(k)j (x) =

ψk,j(x), if x ∈ [xj−1, xj]0, elsewhere

, 2 ≤ k ≤ p, 1 ≤ j ≤ N. (2.7b)

An arbitrary function of SN,p0 = SN,p ∩H1

0 can be written as

U(x, t) =N−1∑j=1

c(1)j (t)φ

(1)j (x) +

p∑k=2

N∑j=1

c(k)j (t)φ

(k)j (x). (2.7c)

We determine U(·, t), belonging to SN,p0 for each t ≥ 0, such that

(V, ∂ttU) + A(V, U) = (V, f), t > 0, (2.8a)

A(V, U) = A(V, u0), t = 0, (2.8b)

A(V, ∂tU) = A(V, u1), t = 0, ∀ V ∈ SN,p0 . (2.8c)

In our analysis we will use the elliptic projection U(·, t) ∈ SN,p0 of u(·, t) ∈ H1

0 for eacht ≥ 0 [22] defined by

A(U − u, V ) = 0, t ≥ 0, ∀ V ∈ SN,p0 , (2.9a)

and the L2 projection U of u defined by

(U − u, V ) = 0, t ≥ 0, ∀ V ∈ SN,p0 . (2.9b)

We further assume a method of lines approach with exact temporal integration and statethe following standard a priori finite element error estimates and a superconvergenceresult in H1 for the generalized wave equation.

4

Theorem 2.1. Let u and U be solutions of (2.1) and (2.8), respectively, and let U be theelliptic projection of u defined by (2.9a). If u, ∂tu ∈ L∞(Hp+1) and ∂ttu ∈ L2(Hp+1),then there exist positive constants, all denoted by C(u), such that if

e = u− U, (2.10a)

then

||e||L∞(L2) + ||∂te||L∞(L2) < C(u)hp+1, t > 0, (2.10b)

where

h = max1≤i≤Nhi = max1≤i≤N (xi − xi−1). (2.10c)

Furthermore,

||U − U ||L∞(H1) < C(u)hp+1, t > 0, (2.11)

and

||e||L∞(H1) < C(u)hp, t > 0. (2.12)

Proof. Cf. [16].

In the previous theorem we stated that ||e||L∞(L2) and ||∂te||L∞(L2) are O(hp+1). In

the next Lemma we will state and prove a similar result for ||∂tte||L∞(L2).

Lemma 2.1. Let u, U be solutions of (2.1) and (2.8), respectively. If u, ∂tu, ∂ttu ∈L∞(Hp+1) and ∂tttu ∈ L2(Hp+1), then

(∂tte, V ) = 0, t = 0, ∀ V ∈ SN,p0 , (2.13)

and there exists C(u) > 0 such that

||∂tte||L∞(L2) < C(u)hp+1, t > 0. (2.14)

Proof. Subtracting (2.3a) with v = V from (2.8a) we obtain

(∂tte, V ) + A(e, V ) = 0, t ≥ 0, ∀ V ∈ SN,p0 . (2.15)

Using (2.8b) completes the proof of (2.13). Differentiating (2.8a) with respect to t andusing (2.13) we show that W = ∂tU satisfies the following weak problem

(V, ∂ttW ) + A(V,W ) = (V, ∂tf), t > 0, (2.16a)

A(V,W ) = A(V, u1), t = 0, (2.16b)

(V, ∂tW ) = (V, ∂ttu), t = 0, ∀ V ∈ SN,p0 . (2.16c)

Applying Theorem 2.1 completes the proof of (2.14).

5

The analysis of the a posteriori error estimation follows the same lines as that forparabolic problems [7]. Some of the proofs from [7] are included here for completeness.We begin by stating a few preliminary results in a series of lemmas.

Let πju be the polynomial of degree p that interpolates u at the roots of ψp+1,j. Letπju be a polynomial of degree p + 1 such that πj πju = πju and interpolate u at oneadditional point in (xj−1, xj). First we show that the interpolation error u− πju can bedivided into a significant and a less significant part.

Lemma 2.2. If u ∈ Hp+2, t ≥ 0, then

u− πju = φj + γj, x ∈ [xj−1, xj], t ≥ 0, (2.17a)

where

φj(x, t) = βj(t)ψp+1,j(x), (2.17b)

γj(x, t) = u(x, t)− πju(x, t), (2.17c)

||φj||s,j < Chp+1−s||u||p+1,j, ||γj||s < Chp+2−s||u||p+2,j, t ≥ 0, s = 0, 1. (2.17d)

Proof. Cf. [7].

Next, we split the finite element error into a significant and a less significant part.

Lemma 2.3. Let u ∈ L∞(H10 ∩Hp+2) and U ∈ SN,p

0 be solutions of (2.1) and (2.8),respectively. If ut ∈ L∞(H1

0 ∩Hp+2), ∂ttu ∈ L2(Hp+2) , U is the elliptic projection of udefined by (2.9a) and Πu|[xj−1,xj] = πju, then

||U − Πu||1 ≤ C(u)hp+1, (2.18)

e(x, t) = φj(x, t) + ωj(x, t), x ∈ [xj−1, xj], (2.19a)

where

ωj(x, t) = γj + πju − U, (2.19b)

with φj(x, t) and γj(x, t) defined as in Lemma 2.2. Furthermore,

N∑j=1

||ωj(., t)||21,j < C(u)h2(p+1), t ≥ 0. (2.19c)

Proof. Consult [7] for a proof of (2.18). Additing and subtracting πju to e(x, t) as

e = u− πju+ πju− U, x ∈ [xj−1, xj ] (2.20)

and using (2.17a) lead to (2.19a) and (2.19b).Adding and subtracting U and applying the triangle inequality we obtain

||ωj(x, t)||21,j < C(||γj||21,j + ||πju − U ||21,j + ||U − U ||21,j). (2.21)

Summing over j and using (2.11), (2.17d) and (2.18) completes the proof.

6

The results of Lemma 2.3 show that finite element solutions using piecewise-polynomialsof degree p have a higher O(hp+2) rate of convergence (superconvergence) at the roots ofLobatto polynomials of degree p+ 1 than they do elsewhere, which is generally O(hp+1)in, e.g., the L2 norm.

2.2 A posteriori error estimation

If we substitute u = e + U in (2.3) we obtain the weak problem for the finite elementerror

(v, ∂tte) + A(v, e) = g(t, v), t > 0, ∀ v ∈ H10 , (2.22)

where

g(t, v) = (v, f) − (v, ∂ttU) − A(v, U). (2.23)

Neglect the higher-order term ett, approximate e [xj−1, xj ] by

e ≈ E(x, t) = bj(t)ψp+1,j(x) (2.24)

and test against ψp+1,j to obtain the elliptic estimator

Aj(ψp+1,j, E) = gj(t, ψp+1,j). (2.25)

Solving for bj yields

bj =gj(t, ψp+1,j)

||ψp+1,j||2A,j

. (2.26)

Substituting (2.19a) in (2.22) and replacing v by ψp+1,j leads to

βj =gj(t, ψp+1,j)

||ψp+1,j||2A,j

−Hj(t), (2.27)

where

Hj(t) =(ψp+1,j, ∂tte)j + Aj(ψp+1,j, ωj)

||ψp+1,j||2A,j

. (2.28)

Using (2.24) in (2.22) and testing against ψp+1,j, we obtain the local initial value problems

(ψp+1,j, ∂ttE) + A(ψp+1,j, E) = g(t, ψp+1,j), t > 0, j = 1, 2, · · · , N, (2.29a)

with initial conditions

A(ψp+1,j, E) = A(ψp+1,j, u0 − U), t = 0, j = 1, 2, · · · , N, (2.29b)

A(ψp+1,j , ∂tE) = A(ψp+1,j , u1 − ∂tU), t = 0, j = 1, 2, · · · , N. (2.29c)

Prior to proving the convergence of the elliptic error estimate to the true error undermesh refinement we need to prove the following lemma.

7

Lemma 2.4. Let u, ut, utt ∈ L∞(H10 ∩Hp+2) and uttt ∈ L2(H1

0 ∩Hp+2) . Let βj(t)and bj(t), j = 1, 2, . . . N be as defined in (2.17b) and (2.26), respectively. Then,

N∑j=1

[bj(t)− βj(t)]2||ψp+1,j||21,j < C(u)h2p+2, (2.30)

N∑j=1

[b2j (t) + β2j (t)]||ψp+1,j||21,j < C(u)h2p, t ≥ 0. (2.31)

Proof. Subtracting (2.27) from (2.26) we obtain

αj = bj(t) − βj(t) = Hj(t). (2.32)

Applying Schwarz inequality and combining

C1h2(p−s)+3j ≤ ||ψp+1,j||2s,j ≤ C2h

2(p−s)+3j , s = 0, 1, (2.33)

with the equivalence of ||.||1 and ||.||A norms leads to

α2j ≤ C

||ψp+1,j||20,j||∂tte||20,j + ||ψp+1,j||2A,j||ωj||2A,j

||ψp+1,j||4A,j

≤ C||∂tte||20,j + ||ωj||21,j

||ψp+1,j||21,j. (2.34)

Multiplying by ||ψp+1,j||21,j and summing over j while using (2.14) and (2.19c) yields(2.30). Combining (2.26), (2.22), and (2.33), Schwarz inequality and the equivalence ofnorms leads to

b2j ||ψp+1,j||21 = ||ψp+1,j||21|(ψp+1,j, ∂tte)j + Aj(ψp+1,j, e)|2

||ψp+1,j||4A,j

≤ C(||∂tte||20,j + ||e||21,j), t > 0. (2.35)

Using (2.12), (2.14) and (2.17d) completes the proof.

The main result of this section is stated in the following theorem.

Theorem 2.2. Let u ∈ L∞(H10 ∩Hp+2) and U ∈ SN,p

0 be solutions of (2.1) and (2.8),respectively. Under the conditions of Lemma 2.4

||e(·, t)||21 =N∑j=1

||E(·, t)||21,j +O(h2p+1), t ≥ 0, (2.36)

where E is the elliptic estimator (2.25).

8

Proof. The proof is the same as that of Theorem 3.1 in [7].

Using the estimates (2.12) and (2.36) for smooth solutions the effectivity index

η = ||E||1/||e||1 → 1 (2.37)

under mesh refinement.

2.3 Temporal integration

In matrix notation the problem (2.8) may be expressed as a linear system of second-orderordinary differential equations

Md2Y

dt2(t) +KY(t) = F, (2.38a)

subject to the initial conditions

Y(0) = Y0,dY

dt(0) = Y0, (2.38b)

where, for instance, when p = 1, M = (mij) is the mass matrix with elements mij =

(φ(1)i , φ

(1)j ), K = (kij) is the stiffness matrix with kij = A(φ

(1)i , φ

(1)j ), F = (fi) with

fi(t) = (f(., t), φ(1)j ) and Y = (yj) with yj = c

(1)j (t). Y0 and Y0 are obtained from the

initial conditions u0 and u1, respectively, using either the elliptic or L2 projection.To integrate (2.38) in time, we introduce a partition in t with a step size ∆t > 0,tn = n ∆t, n = 0, 1, . . . , and let Yn be an approximation of Y(tn), n = 1, 2, . . . .

When the system (2.38), with an exact or consistent mass matrix M, is solved exactlyin time it yields optimal convergence rates (2.10b), these full rates of convergence aremaintained if we approximate M using a quadrature rule that integrates exactly allpolynomials of degree 2p − 2 for a finite element approximation of degree p. Moreover,such quadrature rules may be constructed such that the mass matrix M in (2.38a) isreplaced by a diagonal mass matrix M to obtain

Md2Y

dt2(t) +KY(t) = F (2.39a)

where

mjj =N−1∑l=1

mjl. (2.39b)

The lumped mass formulation (2.39) leads to explicit and very efficient time-integrationschemes with better stability properties than those resulting from the consistent massformulation (2.38). For a detailed discussion of mass lumping consult [17]. We integratethe system (2.39) using a central difference method in time and a piecewise linear finiteelement approximation in space to obtain

mjj(yn+1j − 2ynj + yn−1

j ) = −∆t2(KYn)j + ∆t2fj(tn),

9

j = 1, 2, . . . , N − 1, n ≥ 1, (2.40a)

or

yn+1j = 2ynj − yn−1

j − ∆t2[(KYn)j − fj(tn)]

mjj

,

j = 1, 2, . . . , N − 1, n ≥ 1. (2.40b)

For instance, when a(x) = c2 and and b(x) = 0, this scheme is stable provided thefollowing CFL condition is satisfied

λ =c∆t

h≤ 1. (2.41)

We solve (2.29) for the hyperbolic estimator using the implicit second-order scheme

(bn+1j − 2bnj + bn−1

j )

∆t2||ψp+1,j||20,j + bn+1

j ||ψp+1,j||2A,j = gj(tn+1, ψp+1,j),

n ≥ 1, j = 1, 2, . . . N. (2.42)

A simple linear stability analysis [15] reveals that the scheme (2.42) is unconditionallystable. In order to start the schemes (2.40) and (2.42) we compute second-order accurateY1 ≈ Y(t1) and b

1j ≈ bj(t1) using Taylor series.

A second method consists in transforming (2.38) into a system of first-order ordinarydifferential equations and applying the fourth-order Runge-Kutta method [15] to performthe temporal integration.

2.4 A computational example

Example 2.1.

We consider the linear wave equation

∂ttu = ∂xxu, 0 < x < 2π, t > 0 (2.43a)

with initial and Dirichlet boundary conditions selected such that the exact solution is

u(x, t) = cos(x− t). (2.43b)

We use this simple example to study the convergence of elliptic and hyperbolic estimatorswith lumped mass for p = 1 under mesh refinement using the method (2.40). We alsostudy the convergence of the error estimates under h− and p− refinements for the consis-tent mass formulation using the fourth-order Runge-Kutta method to solve (2.38). In allcomputational experiments we assume the temporal discretization error to be negligibleand compute estimates of the spatial discretization error.

10

We solve (2.43) on 0 < t ≤ 7π, using (2.40) for the solution and the elliptic errorestimation procedure (2.25) on uniform meshes having 8, 16, 32, 64 and 128 elements.We approximate the initial conditions using the elliptic projection (2.9a) and integrate intime with a fixed CFL number λ = ∆t/∆x = 0.8. We plot the effectivity indices versustime in Figure 1 (left). We perform a second experiment using L2 projection (2.9b)for the initial conditions with all other parameters having the same values as before,solve (2.40) for the solution and (2.25) for the ellitpic error estimates. We present globaleffectivity indices as a function of time in Figure 1 (right). We observe that while errorestimates for the elliptic and L2 projections both converge to the true error under meshrefinement, results obtained using the elliptic projection are slightly better than thoseobtained using the L2 projection.

In order to compare the performance of elliptic and hyperbolic estimators we usean elliptic projection of the initial conditions to solve (2.40) for the solution and thehyperbolic error estimation procedure (2.42) with λ = 0.8 on uniform meshes havingN = 8, 16, 32, 64, 128 elements with piecewise linear finite element approximation. Weplot the effectivity indices versus time for elliptic and hyperbolic estimators in Figure 2.As expected, the computational results of Figure 2 indicate that the hyperbolic estimatorperforms slightly better than its elliptic counterpart but not without additional cost sincewe need to integrate a set of local hyperbolic problems. However, we note that hyperbolicestimators may be useful in an adaptive setting where local error estimates are used tomove the mesh [3].Next, we solve (2.38) using the fourth-order Runge-Kutta method on 0 < t ≤ 3π ona 16-element uniform mesh with piecewise polynomial finite element approximations ofdegrees p = 1 to 6. Since no analytic CFL formulas exist in the literature, we used thefollowing (p, λ) pairs (1, 0.3), (2, 0.08), (3, 0.05), (4, 0.03), (5, 0.02), (6, 0.01) based on trialand error. We present the errors e(x, t) versus x at t = 3π for N = 16 and p rangingfrom 1 to 6 in Figure 3. Errors vanish at points close to the Lobatto points of degreep+1. Superconvergence at Lobatto points becomes stronger as p increases with the rootsof e(x, t) on [xj−1, xj] getting closer to those of ψp+1,j(x).We solve (2.38) using the fourth-order Runge-Kutta method on 0 < t ≤ 3π on uniformmeshes having N = 8, 16, 32, 64 and 128 elements and piecewise polynomial finiteelement approximations of degrees p = 1 to 4 with all other parameters having thesame values as in the previous experiment. We plot the effectivity indices for the ellipticestimators versus time in Figure 4 for p = 1 to 4. Effectivity indices stay above 0.8 at alltimes for all meshes and degrees except for the coarsest mesh with p = 1. Furthermore,these results indicate that the effectivity indices converge to one under h−refinementwhich is in full agreement with the results of Theorem 2.2. Finally, we solve (2.38) usingthe fourth-order Runge-Kutta method on 0 < t ≤ 3π on an eight-element mesh forp ranging from 1 to 6 with all other parameters having the same values as before andpresent the effectivity indices as a function of time in Figure 5. The effectivity indicesstay above 0.97 at all times for p > 1 and they appear to converge to one for all t ∈ [0, 7π]under p-refinement.

The present theory does not cover the effect of mass lumping or time discretizationerrors. We will report on error estimation for the fully discrete case with mass lumpingin a forthcoming paper [1].

11

0 5 10 15 20 250.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t

η

N=8

N=16

N=32

N=128

N=64

0 5 10 15 20 250.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t

η

N=8

N=16

N=32

N=64

N=128

Figure 1: Effectivity indices using the elliptic (left) and L2 (right) projections for Example2.1 with p = 1 and N = 8, 16, 32, 64, 128.

0 5 10 15 20 250.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t

η

N=8

N=16

N=32

N=64

N=128

Figure 2: Hyperbolic (solid) and elliptic (dash-dot) effectivity indices versus time forExample 2.1 with p = 1 and N = 8, 16, 32, 64, 128.

3 The Two-dimensional Wave Equation

3.1 Problem formulation

We consider the linear generalized wave equation

∂ttu + Lu = f(x), x = [x1, x2]T ∈ Ω, t > 0, (3.1)

Lu = −2∑

j=1

2∑k=1

∂xj(aj,k(x)∂xk

u) + b(x)u, (3.2)

u(x, 0) = u0(x), x ∈ Ω ∪ ∂Ω, (3.3)

∂tu(x, 0) = u1(x), x ∈ Ω ∪ ∂Ω, (3.4)

12

0 2 4 6−0.1

−0.05

0

0.05

0.1

x

e(x,

t)

0 2 4 6−1

−0.5

0

0.5

1x 10

−3

x

e(x,

t)

0 2 4 6−2

0

2x 10

−5

x

e(x,

t)

0 2 4 6−4

−2

0

2

4x 10

−7

x

e(x,

t)0 2 4 6

−4

−2

0

2

4x 10

−9

x

e(x,

t)

0 2 4 6−1

−0.5

0

0.5

1x 10

−10

x

e(x,

t)

p=1 p=2

p=3 p=4

p=5 p=6

Figure 3: Discretization errors of Example 2.1 at t = 3π using the fourth-order Runge-Kutta method with p = 1 to 6 and N = 16. Plus signs indicate the roots of ψp+1,j.

0 2 4 6 8 100.6

0.8

1

1.2

t

η

0 2 4 6 8 100.97

0.98

0.99

1

t

η

0 2 4 6 8 100.985

0.99

0.995

1

1.005

t

η

0 2 4 6 8 10

0.995

1

t

η

N=32

N=8

N=16

N=32 N=16

N=8

N=32

N=16

N=8

N=32

N=16

N=8

N=64 N=128 N=64, 128

N=64,128 N=64,128

Figure 4: Effectivity indices as a function of time for Example 2.1 using N =8, 16, 32, 64, 128 for p = 1 to 4 (upper left to lower right).

13

0 1 2 3 4 5 6 7 8 9 10 110.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

t

η

p=1

p=2

p=3

p=4 p=5,6

0 1 2 3 4 5 6 7 8 9 10 110.97

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

t

η

p=2

p=3

p=4

p=6

p=5

Figure 5: Effectivity indices versus time for Example 2.1 using an eight-element meshwith p = 1 to 6 (left) and with p = 2 to 6 (right).

u(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (3.5)

where aj,k(x), j, k = 1, 2, and b(x) are smooth functions and L is a positive-definiteoperator.The space of square-integrable functions L2 is equipped with the inner product and norm

(v, u) =

∫ ∫Ω

uv dx1dx2, ||u||0 =√(u, u). (3.6a)

We use (3.6a) to define Sobolev inner products and norms

(u, v)s =∑

τ1+τ2≤s

(∂τ1+τ2 u

∂xτ11 ∂xτ22

,∂τ1+τ2 v

∂xτ11 ∂xτ22

), s ≥ 0, (3.6b)

and

||u||s =√

(u, u)s, s ≥ 0. (3.6c)

Sobolev spaces Hs are defined as

Hs = u | ||u||s <∞, s ≥ 0. (3.7)

Let H10 denote the set of functions in H1 that satisfy homogeneous Dirichlet boundary

conditions.The weak Galerkin formulation consists of determining u(·, t), belonging to H1

0 for eacht ≥ 0, such that

(v, ∂ttu) + A(v, u) = (v, f), t > 0, (3.8a)

14

A(v, u) = A(v, u0), t = 0, (3.8b)

A(v, ∂tu) = A(v, u1), t = 0, ∀ v ∈ H10 , (3.8c)

A(v, u) =

∫ ∫Ω

[2∑

j=1

2∑k=1

aj,k(x)∂xjv∂xk

u + b(x)vu] dx1dx2 (3.8d)

We discretize in space by partitioning Ω into N square elements, κi, i = 1, 2, . . . , Nand defining a finite dimensional space

SN,p = w ∈ H1 |w(x) ∈ Qp(κi), x ∈ κi, i = 1, 2, . . . , N , (3.9)

where Qp is the space of bi− p polynomials. We determine U(·, t), belonging to SN,p0 =

SN,p ∩H10 for each t ≥ 0, such that

(V, ∂ttU) + A(V, U) = (V, f), t > 0, (3.10a)

A(V, U) = A(V, u0), t = 0, (3.10b)

A(V, ∂tU) = A(V, u1), t = 0, ∀ V ∈ SN,p0 . (3.10c)

We further note that the results of Theorem 2.1 and Lemma 2.1 hold in two dimensionswith h = max1≤i≤Ndiam(κi).We use (3.8d) to define the energy norm

||u||2A = A(u, u). (3.11)

The local norms, e,g., ||u||s,j, s ≥ 0, are defined by restricting the integrals in (3.6) toκj .

The even-odd-degree principle discovered in [25, 4, 2] applies to second-order hyper-bolic problems and the analysis follows the same lines as that for parabolic problems [4].We define a two-dimensional interpolation operator denoted by π such that on a squareelement κ, πu ∈ Qp and πu(xl, xk) = u(xl, xk), l, k = 1, . . . , p + 1, where xl arethe roots of the Lobatto polynomial ψp+1. Let π be an interpolation operator such thatππu = πu and interpolate u at two additional points in κ. It takes a few steps and aseries of lemmas to develop error estimates and establish their convergence to the trueerror. First, we show that the interpolation error u−πu on an element κ may be dividedinto a significant and a less significant part.

Lemma 3.1. Let u ∈ Hp+2, t ≥ 0, then

u(x, t) − πu(x, t) = φ(x, t) + γ(x, t), x ∈ κ, (3.12a)

15

where

φ(x, t) = β1(t)ψp+1(x1) + β2(t)ψp+1(x2), (3.12b)

γ(x, t) = u − πu (3.12c)

||∂xjφ||0,κ ≤ Chp||u||p+1,κ, j = 1, 2, (3.12d)

and

||γ||s,κ ≤ Chp+2−s||u||p+2,κ, s = 0, 1, t ≥ 0. (3.12e)

Proof. Cf. [4].

Next, we split the finite element error into a significant and a less significant part

Lemma 3.2. Let u ∈ L∞(H10 ∩Hp+2) and U ∈ SN,p

0 be solutions of (3.8) and (3.10),respectively. Under the conditions of Lemma 2.3

e = u− U = φ + θ, (x1, x2) ∈ κ, (3.13)

where φ is defined in (3.12b)

θ(x, t) = γ + πu− U, (3.14)

N∑j=1

||θ||21,j < C(u)h2(p+1), t > 0. (3.15)

Proof. The proof is the same as that of Lemma 2.3.

3.2 A posteriori error estimation of even-degree approximations

The analysis parallels that of §2.2 with a lemma and a theorem to prove the convergenceresult. Substitute u = U + e into (3.8) to obtain

(v, ∂tte) + A(v, e) = g(t, v), t > 0, (3.16)

where

g(t, v) = (v, f) − (v, ∂ttU) − A(v, U). (3.17)

Neglect the higher-order term ∂tte, approximate e on κj by

e ≈ E = b1jψp+1,j(x1) + b2jψp+1,j(x2), (x1, x2) ∈ κj, (3.18)

16

and test against

vi(x) = (xi − xi)δj(x1)δj(x2), i = 1, 2, (3.19a)

where

δj(z) =ψp+1,j(z)

z − z. (3.19b)

We also use the function

σj(z) = ψ′p+1,j(z). (3.19c)

A hyperbolic error estimator is obtained by solving

(vi, ∂ttE(., t))j + Aj(vi, E) = gj(t, vi), i = 1, 2, t > 0, (3.20a)

with initial conditions

Aj(vi, U + E) = Aj(vi, u0), t = 0, i = 1, 2, (3.20b)

Aj(vi, Ut + Et) = Aj(vi, u1), t = 0, i = 1, 2. (3.20c)

An elliptic estimator is obtained by solving the following system for b1j and b2j

Aj(vi, E) = gj(t, vi), i = 1, 2, t > 0, (3.21)

which yields

bij =gj(t, vi)

aii∫ ∫

κjσ2j (xi)δj(x(i mod 2)+1)dx1dx2

, t > 0, i = 1, 2. (3.22)

Substituting (3.13) with (3.12b) in (3.16) leads to

βij aii

∫ ∫κj

σ2j (xi)δj(x(i mod 2)+1)dx1dx2 = gj(t, vi)− Fij(t), (3.23a)

where

Fij(t) =

∫ ∫κj

[2∑

k=1

2∑l=1

akl∂xkvi∂xl

θ +2∑

k=1

2∑l=1

(akl − akl)∂xkvi∂xl

φ

+bvie+ vi∂tte]dx1dx2 (3.23b)

and aii = aii(x) evaluated at the center x of κj . Thus,

βij(t) =gj(t, vi) − Fij

aii∫ ∫

κjσ2j (xi)δj(x(i mod 2)+1) dx1dx2

, i = 1, 2. (3.24)

17

Lemma 3.3. Let βij and bij be defined by (3.22) and (3.24, respectively. Under theconditions of Lemma 2.4

N∑j=1

(βij − bij)2 ≤ C(u), i = 1, 2, (3.25a)

N∑j=1

(β2ij − b2ij) ≤

C(u)

h, i = 1, 2, t ≥ 0. (3.25b)

Proof. Subtracting (3.22) from (3.23) we obtain

αij(t) = βij − bij = − Fij(t)∫ ∫κjσ2j (xi)δj(x(i mod 2)+1) dx1dx2

, t ≥ 0, i = 1, 2.

(3.26)

Applying Schwarz and triangle inequalities to (3.26) we obtain

αij(t)2 ≤ C||vi||21,j

[||θ||21,j + h2||φ||21,j + ||e||0,j + ||∂tte||20,j]2[∫ ∫

κjσ2j (xi)δj(x(i mod 2)+1) dx1dx2]2

. (3.27)

Summing over all elements and using Lemma 4.1 of [4] leads to

N∑j=1

αij(t)2 ≤ Ch−2(p+1)[||Θ||21 + h2||Φ||21 + ||e||20 + ||∂tte||20],

t ≥ 0, i = 1, 2. (3.28)

Using (2.10b), (2.14), (3.12d) and (3.15) establishes (3.25a). Combining (3.22), (3.16),(2.10b), (2.14) and applying Schwarz and triangle inequalities we obtain

bij(t)2 ≤ Ch−2(p+1)[||∂tte||20,j + ||e||21,j] ≤ C

h2, i = 1, 2, t ≥ 0. (3.29)

Differentiating (3.12b), using (3.12d) and Lemma 4.1 of [4] yields

β2ij(t) =

||φxi||20,j

||σj(xi)||20,j≤ C

h2||u||2p+1,j, i = 1, 2, t ≥ 0. (3.30)

We complete the proof by combining (3.29), (3.30) and (3.25a) and applying Schwarzinequality

N∑j=1

[β2ij(t) − b2ij(t)] ≤ C[

N∑j=1

(βij(t) − bij(t))2]

12 [

N∑j=1

(β2ij(t) + b2ij(t))]

12 . (3.31)

18

The elliptic error estimate (3.21) converges to the true error under mesh refinement asstated in the following theorem.

Theorem 3.1. Let u ∈ L∞(H10∩Hp+2) and U ∈ SN,p

0 be solutions of (3.8) and (3.10),respectively, where p > 0 is a even positive integer. If the domain Ω is partitioned into asquare mesh, then under the conditions of Lemma 2.4

||e(·, t)||21 =

N∑j=1

||E(·, t)||21,j + O(h2p+1) (3.32a)

where E is the elliptic estimator defined by 3.21.

Proof. The proof is the same as that of Theorem 4.1 of [4].

3.3 A posteriori error estimation of odd-degree approximations

Error estimates for odd-degree piecewise polynomial approximations are constructed byassuming that u is smooth and using the jumps in derivatives of U at the interior vertices.We compute jumps in the derivatives of e(x, t) at the vertices pk, k = 1, 2, 3, 4 of κj toobtain

e = u − U ≈ E, (3.33)

[∂xie(pk, t)]i = −[∂xi

U(pk, t)]i ≈ (3.34)

b1j(t)[∂xiψp+1,j(p1k)]i + b2j(t)[∂xi

ψp+1,j(p2k)]i,

i = 1, 2, k = 1, 2, 3, 4, x ∈ κj, (3.35)

where [w(p)]i denotes the jump in w at the point p in the xi direction.At each vertex pk we obtain a system for b1 and b2 in terms of the known quantities

[∂xiU(pk, t)]i. When solved, this system yields an estimate E. In order to obtain an

error estimate on an element κj we use the average of the estimates computed from allinterior element vertices. Finally, by summing over all elements a global error estimateis obtained as

||E(·, t)||21 =N∑j=1

||E(·, t)||21,j. (3.36)

On square elements we have the following result.

Theorem 3.2. Under the conditions of Theorem 3.1 and if p is an odd positive integer,then

||e(·, t)||21 =

N∑j=1

||E(·, t)||21,j +O(h2p+1), (3.37a)

whereN∑j=1

||E(·, t)||21,j =h2

16(2p + 1)

N∑j=1

2∑i=1

4∑k=1

[∂xiU(pk, t)]

2i . (3.37b)

19

Proof. The proof follows the same lines as that of Theorem 4.1 in [4].

The estimates (3.32) and (3.37) extend to more general finite element spaces as definedin the following theorem.

Theorem 3.3. Under the conditions of Theorems 3.1 and 3.2, let Qp(κi) be the restric-tion of SN,p to κi and let Mp(κi) be a space of complete polynomials of degree p on κi. IfQp satisfies

Mp ⊂ Qp, Mp+1 ⊂ Qp ∩ xp+11 , xp+1

2 , (3.38)

then the error estimates (3.32) and (3.37) apply when p is even and odd, respectively.

Proof. The proof follows the same lines as those of Theorem 3.1 and Theorem 3.2.

3.4 Computational examples

We consider three two-dimensional examples that illustrate the performance of the errorestimation procedures for both odd- and even-degree approximations. For even-degreeapproximations we use the elliptic error estimator on square and quadrilateral meshesusing bi−p and hierarchical finite element approximations. Finally, we present results fora nonlinear wave equation where the assumptions of Theorems 3.2-3.3 are violated; thus,indicating that the estimation procedures apply more widely than the theory suggests.If E is an asymptotically correct estimate of e then η should converge to unity as themesh is refined. The estimate is, furthermore, robust if η does not appreciably differ fromunity for a wide range of mesh spacings and polynomial degrees.

Example 3.1.

Consider the linear wave equation

∂ttu = ∂x1x1u + ∂x2x2u , x ∈ Ω = [0, 2]× [0, 2], t > 0, (3.39a)

with initial and Dirichlet boundary conditions selected such that the exact solution is

u(x, t) = sin(x1√2+x2√2− t). (3.39b)

Lumped mass:We solve (3.39) using a bilinear finite element approximation on uniform square mesheshaving 16, 64, 256, and 1024 elements with a lumped mass matrix and integrate in timefrom t = 0 to t = 3 using a fourth-order Runge-Kutta method with λ = 0.5. Theeffectivity indices shown in Figure 6 converge to unity under mesh refinement with λkept fixed.Consistent mass:In the following examples we select the time step ∆t such that the temporal discretizationerror is much smaller than the spatial error.

We solve (3.39) on uniform square meshes having 16, 64, 256, and 1024 elementsusing bi − p and hierarchical piecewise polynomial approximations of orders 1 to 4 with

20

0 0.5 1 1.5 2 2.5 30.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

t

effe

ctiv

ity in

dex

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

t

Effe

ctiv

ity in

dex

Figure 6: Elliptic effectivity indices for Example 3.1 (left) and for Example 3.3 (right)for 16 (dash-dot), 64 (dash), 256 (dot) and 1024 (solid) squares elements.

a consistent mass matrix. Errors and global effectivity indices for piecewise bi − p andhierarchical approximations are presented in Tables 1 and 2, respectively. The results areidentical for p = 1 and the effectivity indices converge to unity under mesh refinement.

Table 1: Errors and effectivity indices for Example 3.1 at t = 1 on N -element uniformsquare meshes with piecewise bi− p polynomial approximations.

p 1 2N ||e||1 η ||e||1 η16 0.1461(+0) 0.91266 0.6792(-2) 0.9891164 0.7086(-1) 0.97803 0.1694(-2) 0.99733256 0.3515(-1) 0.99450 0.4223(-3) 0.999341024 0.1753(-1) 0.99862 0.1058(-3) 0.99983p 3 4N ||e||1 η ||e||1 η16 0.1912(-3) 0.97512 0.4507(-5) 0.9988864 0.2390(-4) 0.99367 0.2817(-6) 0.99970256 0.2988(-5) 0.99841 0.1760(-7) 0.999921024 0.3735(-6) 0.99959 0.1100(-8) 0.99998

Example 3.2.

In order to show that our error estimates apply to more general meshes, let us considerthe problem (3.39) on the quadrilateral Ω = P1P2P3P4, where P1 = (0, 0), P2 = (4, 0.5),P3 = (5, 3.5), and P4 = (−1, 6).

21

Table 2: Errors and effectivity indices for Example 3.1 at t = 1 on N -element uniformsquare meshes with piecewise hierarchical polynomial approximations.

p 1 2N ||e||1 η ||e||1 η16 0.1460(+0) 0.91266 0.6792(-2) 0.9891164 0.7086(-1) 0.97803 0.1694(-2) 0.99733256 0.3515(-1) 0.99450 0.4233(-3) 0.999341024 0.1753(-1) 0.99862 0.1058(-3) 0.99984p 3 4N ||e||1 η ||e||1 η16 0.1912(-3) 0.97513 0.4507(-5) 0.9988864 0.2390(-4) 0.99367 0.2817(-6) 0.99970256 0.2988(-5) 0.99841 0.1760(-7) 0.999921024 0.3735(-6) 0.99959 0.1100(-8) 0.99998

We solve (3.39) in Ω on uniform quadrilateral meshes have 16, 64, 256, and 1024 elements.We use the fourth-order Runge-Kutta method to integrate in time from t = 0 to t = 1with a time step ∆t = 10−3 for p = 1, 2, 3 and ∆t = 10−4 for p = 4. We present theexact errors and global effectivity indices at t = 1 in Table 3. These computationalresults again indicate that the results of Theorem 3.2 and Theorem 3.1 extend to generalquadrilateral meshes.

Table 3: Errors and effectivity indices for Example 3.2 at t = 1 on N -element uniformquadrilateral meshes with piecewise hierarchical polynomial approximations.

p 1 2N ||e||1 η ||e||1 η16 0.7888(+0) 0.87951 0.6911(-1) 0.9561864 0.3658(+0) 0.95618 0.1713(-1) 0.98884256 0.1786(+0) 0.98879 0.4274(-2) 0.997201024 0.8872(-1) 0.99722 0.1068(-2) 0.99930p 3 4N ||e||1 η ||e||1 η16 0.4774(-2) 0.99186 0.2252(-3) 1.0237764 0.5942(-3) 0.99997 0.1414(-4) 1.00581256 0.7420(-4) 1.00058 0.8848(-6) 1.001461024 0.9273(-5) 1.00024 0.5532(-7) 1.00037

Example 3.3.

22

Consider the nonlinear wave equation

∂ttu = ∂x1x1u + ∂x2x2u + u(2u2 − 1), x ∈ [0, 2]× [0, 2], t > 0, (3.40a)

where we select the initial and Dirichlet boundary conditions such that the exact solutionis

u(x, t) = sech(cosh(1)√

2x1 +

cosh(1)√2

x2 − sinh(1)t). (3.40b)

We solve (3.40) on the same meshes as in Example 3.1 using bilinear approximationwith lumped mass. We integrate in time from t = 0 to t = 3 using the fourth-orderRunge-Kutta method with λ = 0.5 and present the effectivity indices in Figure 6.We solve (3.40) on the same meshes as in Example 3.1 using bi−p and hierarchical finiteelement approximations with a consistent mass matrix. We use a fourth-order Runge-Kutta method to integrate the system from t = 0 to t = 1 with a time step ∆t = 10−3 forp = 1, 2, 3 and ∆t = 10−4 for p = 4. We present the exact errors and global effectivityindices at t = 1 in Table 3. The computational results again indicate that the results ofTheorem 3.2 and Theorem 3.1 extend to some nonlinear problems.

Table 4: Errors and effectivity indices for Example 3.3 at t = 1 on N -element uniformsquare meshes with piecewise hierarchical polynomial approximations.

p 1 2N ||e||1 η ||e||1 η16 0.2425(+0) 0.78503 0.2898(-1) 0.9263964 0.1153(+0) 0.92711 0.7243(-2) 0.98019256 0.5632(-1) 0.97972 0.1811(-2) 0.995011024 0.2797(-1) 0.99479 0.4527(-3) 0.99876p 3 4N ||e||1 η ||e||1 η16 0.3147(-2) 0.82380 0.3254(-3) 0.9650964 0.3978(-3) 0.93473 0.2066(-4) 0.99488256 0.4986(-4) 0.98352 0.1299(-5) 0.998731024 0.6236(-5) 0.99589 0.8131(-7) 0.99967

4 Conclusions

We developed a posteriori error estimates for the wave equation in one dimension and twodimensions on rectangular meshes. We show how to construct elliptic and hyperbolic errorestimators. Both elliptic and hyperbolic a posteriori error estimates are shown to convergeto the true error under mesh refinement. Although the theory is not developed for non-rectangular meshes, we expect good results with bi− p and hierarchical approximations

23

when the solution is smooth and the mesh is not severely distorted. Indeed, effectivityindices for the bi − p and hierarchical bases are in excess of 0.90 for virtually all mesh-order combinations. The effectivity indices of both of these approximations appear toconverge to unity under mesh refinement. The data of Tables 1, 2 and 4 indicate abetter performance for even-degree polynomials than for odd. Furthermore, the even-degree estimator does not involve communication with neighboring elements and thusis ideal for parallelization. This work extends the superconvergence results for ellipticand parabolic problems [24], to second-order hyperbolic problems, i.e., the semi-discretefinite element solution is superconvergent at the Lobatto points. We also expect theseerror estimates to apply to the more efficient higher-order finite element approximationswith lumped mass matrix. Computational results indicate that the hyperbolic errorestimates are slightly more accurate than their elliptic counterparts with some additionalcomputational effort. Extending these results to hexahedral meshes is straight forward.Currently we are investigating a posteriori finite element error estimates on triangularand tetrahedral meshes.

Acknowledgement

Portions of this research were supported by the National Science Foundation (GrantNumber ASC 9720227) and Sandia National Laboratories (Contract Number AW5657).

References

[1] S. Adjerid. A posteriori error estimates for second-order hyperbolic problems withmass lumping. in preparation, 2002.

[2] S. Adjerid, B. Belguendouz, and J. Flaherty. A posteriori finite element error esti-mation for diffusive systems. SIAM Journal on Scientific Computing, 21:728–746,1999.

[3] S. Adjerid and J. Flaherty. A moving finite element method with local refinementfor parabolic partial differential equations. Computer Methods in Applied Mechanicsand Engineering, 55:3–26, 1986.

[4] S. Adjerid, J. Flaherty, and I. Babuska. A posteriori error estimation for the finiteelement method-of-lines solution of parabolic problems. Mathematical Modelling andMethods in Applied Science, 9:261–286, 1999.

[5] S. Adjerid, J. Flaherty, and L. Krivodonova. A posteriori discontinuous Galerkinerror estimation for hyperbolic problems. Computer Methods in Applied Mechanicsand Engineering, 191, 2002. 1097–1112.

[6] S. Adjerid, J. Flaherty, P. Moore, and Y. Wang. High-order adaptive methods forparabolic systems. Physica D, 60:94–111, 1992.

24

[7] S. Adjerid, J. Flaherty, and Y. Wang. A posteriori error estimation with finiteelement methods of lines for one-dimensional parabolic systems. Numer. Math.,65:1–21, 1993.

[8] S. Adjerid and T. C. Massey. A posteriori discontinuous Galerkin error estimationfor two-dimensional hyperbolic problems. in preparation, 2002.

[9] M. Ainsworth and J. Oden. A procedure for a posteriori error estimation for h-pfinite element method. Computer Methods in Applied Mechanics and Engineering,101:72–96, 1992.

[10] M. Ainsworth and J. Oden. A unified approach to a posteriori error estimation usingelement residual methods. Numer. Math., 65:23–50, 1993.

[11] M. Ainsworth and J. Oden. A Posteriori Error Estimation in Finite Element Anal-ysis. John Wiley, New York, 2000.

[12] I. Babuska, T. Strouboulis, and S. Gangaraj. Guaranteed computable bounds for theexact error in the finite element solution - Part I: One-dimensional model problem.Computer Methods in Applied Mechanics and Engineering, 176:51–79, 1999.

[13] I. Babuska, T. Strouboulis, and S. Gangaraj. Guaranteed computable bounds forthe exact error in the finite element solution - Part II: Bounds for the energy normof the error in two-dimensions. Int. J. Numer. Meth. Eng., 47:427–475, 2000.

[14] R. Becker and R. Rannacher. A feedback approach to error control in finite elementmethods: Basic analysis and examples. EAST-WEST J. Num. Math., 4:237–264,1996.

[15] R. L. Burden and J. D. Faires. Numerical Analysis. Brooks/Cole, Pacific Grove,2001.

[16] T. Dupont. l2−estimates for Galerkin methods for second order hyperbolic equa-tions. SIAM Journal on Numerical Analysis, 10:392–410, 1973.

[17] T. J. R. Hughes. The Finite Element Method. Dover Publications, New York, 2000.

[18] J. Oden and S. Prudhomme. Goal-oriented error estimation and adaptivity for thefinite element method. Computers & Mathematics with Applications, 41:735–756,2001.

[19] J. Oden and K. Vermaganti. Estimation of local modeling error and goal-orientedadaptive modeling of heterogeneous materials; part: Error estimates and adaptivealgorithms. J. Comp. Physics, 164:22–47, 2000.

[20] M. Paraschivoiu, J. Preraie, Y. Maday, and A. T. Patera. Fast bounds for outputsof partial differential equations. In J. Borggaard, J. Burns, E. Cliff, and S. Schreck,editors, Computational Methods for Optimal Design and Control, Boston, 1998.Birkhauser.

25

[21] E. Suli and P. Houston. Finite element methods for hyperbolic problems: A poste-riori error analysis and adaptivity. In I. Duff and G. Watson, editors, The State ofthe Art in Numerical Analysis, pages 441–471, Oxford, 1997. Clarendon Press.

[22] V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin, 1997.

[23] R. Verfurth. A Review of a Posteriori Error Estimation and Adaptive Mesh Refine-ment Techniques. Teubner-Wiley, Teubner, 1996.

[24] L. B. Whalbin. Superconvergence in Galerkin Finite Element Methods. SpringerVerlag, New York, 1995.

[25] D. Yu. Asymptotically exact a-posteriori error estimator for elements of bi-evendegree. Math. Numerica Sinica, 13:89–101, 1991.

[26] D. Yu. Asymptotically exact a-posteriori error estimator for elements of bi-odddegree. Math. Numerica Sinica, 13:307–314, 1991.

26