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arXiv:1003.3641v1 [math.NA] 18 Mar 2010

A POSTERIORI L∞(L2)-ERROR BOUNDS IN FINITE ELEMENT

APPROXIMATION OF THE WAVE EQUATION

EMMANUIL H. GEORGOULIS, OMAR LAKKIS, AND CHARALAMBOS MAKRIDAKIS

Abstract. We address the error control of Galerkin discretization (in space)

of linear second order hyperbolic problems. More specifically, we derive a pos-

teriori error bounds in the L∞(L2)-norm for finite element methods for the

linear wave equation, under minimal regularity assumptions. The theory is

developed for both the space-discrete case, as well as for an implicit fully dis-

crete scheme. The derivation of these bounds relies crucially on carefully con-

structed space- and time-reconstructions of the discrete numerical solutions,

in conjunction with a technique introduced by Baker (1976, SIAM J. Numer.

Anal., 13) in the context of a priori error analysis of Galerkin discretization of

the wave problem in weaker-than-energy spatial norms.

1. Introduction

In computing approximate solutions of evolution initial-boundary value prob-

lems mesh-adaptivity plays an important role, in that it drives variable resolu-

tion requirements, thereby contributing reduction in computational cost. Adaptive

strategies are often based on a posteriori error estimates, i.e., computable quantities

which estimate the error of the finite element method measured in a suitable norm

(or other functionals of interest).

A posteriori error bounds are well developed for stationary boundary value prob-

lems (e.g., [Ver96, AO00, BS01, CB02, D96, Ste07, CKNS08] and the references

therein). Adaptivity and error estimation for parabolic problems has also been an

active area of research for the last two decades (e.g., [EJ95, Ver03, Pic98, HS01,

MN03, BBM05, BV04, LM06, GLV08] and the references therein).

Surprisingly, there has been considerably less work on the error control of fi-

nite element methods for second order hyperbolic problems, despite the substantial

amount of research in the design of finite element methods for the wave problem

(e.g., [Bak76, BB79, BDS79, BD80, DS81, Joh93, Mak92, BJR90, CJT93, BJT00,

KM05] and the references therein). A posteriori bounds for standard implicit time-

stepping finite element approximations to the linear wave equation have been pro-

posed and analyzed (but only in very specific situations) by Adjerid [Adj02]. Also,

Bernardi and S¨ uli [BS05] derive rigorous a posteriori L∞(H1)-error bounds, using

Date: March 19, 2010.

2010 Mathematics Subject Classification. 65M60,65M15.

E.H.G. acknowledges the support of the Nuffield Foundation, UK, and of the Foundation for

Research and Technology-Hellas, Heraklion, Greece.

O.L. acknowledges the partial support of the Royal Society UK and of the Foundation for

Research and Technology-Hellas, Heraklion, Greece, where the initial steps of this work were

made.

C.M. acknowledges the support of the London Mathematical Society, Universities of Leicester

and Sussex, UK, and supported in part by the European Union grant No. MEST-CT-2005-021122.

1

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2 E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

energy arguments, for the same fully-discrete method we analyze in this work. We

note that goal-oriented error estimation for wave problems (via duality techniques)

is also available [BR99, BR01], while some earlier work on a posteriori estimates

for first order hyperbolic systems have been studied in the time semidiscrete set-

ting [MN06], as well as in the fully discrete one [Joh93, S96, S99].

In this work, we derive a posteriori bounds in the L∞(L2)-norm of the error,

which appear to be unavailable in the literature so far. The theory is developed

for both the space-discrete case, as well as for the practically relevant case of an

implicit fully discrete scheme.

The derivation of these bounds relies crucially on reconstruction techniques, used

earlier for parabolic problems [MN03, LM06, AMN06]. Another key tool in our

analysis is the special testing procedure due to Baker [Bak76], who used it in the a

priori error analysis of Galerkin discretization of the wave problem in weaker-than-

energy spatial norms.

While for the proof of a posteriori bounds for the semidiscrete case, the elliptic

reconstruction previously considered in [MN03, LM06] suffices, the fully discrete

analysis necessitates the careful introduction of a novel space-time reconstruction,

satisfying a crucial local vanishing moment property in time.

based on the one-field formulation of the wave equation and, thus, non-trivial three-

point time reconstructions are required. A further challenge presented by the wave

equation is the special treatment of deriving bounds for the “elliptic error” of the

reconstruction framework, to obtain practically implementable residual estimators.

The derived a posteriori estimators are formally of optimal order, i.e., of the same

order as the error on uniform space- and time-meshes.

The rest of this work is organized as follows. In §2 we present the model problem

and the necessary basic definitions along with the finite element methods for the

wave equations considered in this work. In §3 we consider the case of a posteriori

bounds for the space-discrete problem. In §4, we derive abstract a posteriori error

bounds for the fully-discrete implicit finite element method, while in §5 the case

of a posteriori bounds of residual type are presented. In §6, we draw some final

concluding remarks.

Our approach is

2. Preliminaries

2.1. Model problem and notation. We denote by Lp(ω), 1 ≤ p ≤ +∞, ω ⊂ Rd,

the Lebesgue spaces, with corresponding norms ? · ?Lp(ω). The norm of L2(ω),

denoted by ?·?ω, corresponds to the L2(ω)-inner product ?·,·?ω. We denote by

Hs(ω), the Hilbertian Sobolev space of order s ≥ 0 of real-valued functions defined

on ω ⊂ Rd; in particular H1

on the boundary ∂ω (boundary values are taken in the sense of traces). Negative

order Sobolev spaces H−s(ω), for s > 0, are defined through duality. In the case

s = 1, the definition of ?·,·?ω is extended to the standard duality pairing between

H−1(ω) and H1

X being a real Banach space with norm ?·?X, consisting of all measurable functions

v : (0,T) → X, for which

??T

0

?v?L∞(0,T;X):= esssup0≤t≤T?v(t)?X< +∞,

0(ω) signifies the space of functions in H1(ω) that vanish

0(ω). For 1 ≤ p ≤ +∞, we also define the spaces Lp(0,T,X), with

(2.1)

?v?Lp(0,T;X):=?v(t)?p

Xdt

?1/p

< +∞, for1 ≤ p < +∞,

p = +∞. for

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L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION3

Let Ω ⊂ Rdbe a bounded open polygonal domain with Lipschitz boundary ∂Ω.

For brevity, the standard inner product on L2(Ω) will be denoted by ?·,·? and the

corresponding norm by ?·?.

For time t ∈ (0,T], we consider the linear second order hyperbolic initial-

boundary value problem of finding u ∈ L2(0,T;H1

and utt∈ L2(0,T;H−1(Ω)) such that

(2.2)utt− ∇ · (a∇u) = f

where f ∈ L2(0,T;L2(Ω)) and a is a scalar-value function in ∈ C(¯Ω), with 0 <

αmin≤ a ≤ αmax, such that

u(x,0) = u0(x) on Ω × {0},

ut(x,0) = u1(x) on Ω × {0}

u(0,t) = 0 on ∂Ω × (0,T],

where u0∈ H1

We identify a function v ∈ Ω × [0,T] → R with the function v : [0,T] → H1

and we use the shorthand v(t) to indicate v(·,t).

2.2. Finite element method. Let T be a shape-regular subdivision of Ω into dis-

joint open simplicial or quadrilateral elements. Each element κ ∈ T is constructed

via mappings Fκ: ˆ κ → κ, where ˆ κ is the reference simplex or reference square, so

that¯Ω = ∪κ∈T¯ κ [Cia78].

For a nonnegative integer p, we denote by Pp(ˆ κ) either the set of all polynomials

on ˆ κ of degree p or less, when ˆ κ is the simplex, or the set of polynomials of at most

degree p in each variable, when ˆ κ is the reference square (or cube). We consider p

fixed and use the finite element space

0(Ω)), with ut∈ L2(0,T;L2(Ω))

in (0,T) × Ω,

(2.3)

0(Ω) and u1∈ L2(Ω).

0(Ω)

(2.4)Vh:= {v ∈ H1

0(Ω) : v|κ◦ Fκ∈ Pp(ˆ κ), κ ∈ T }.

Further, we denote by Γ := ∪κ∈T(∂κ\∂Ω), i.e., the union of all (d−1)-dimensional

element edges (or faces) e in Ω associated with the subdivision T excluding the

boundary. We introduce the mesh-size function h : Ω → R, defined by h(x) =

diamκ, if x ∈ κ and h(x) = diam(e), if x ∈ e when e is an edge.

The semidiscrete finite element method for the initial-boundary value problem

(2.2)–(2.3) consists in finding U ∈ L2(0,T;Vh) such that

(2.5)?Utt,V ? + a(U,V ) = ?f,V ?

where the bilinear form a is defined for each z,v ∈ H1

?

Ω

and the corresponding energy norm is defined for v ∈ H1

(2.7)

∀V ∈ L2(0,T;Vh),

0(Ω) by

(2.6)a(z,v) =a∇z · ∇v dx,

0(Ω) by

?v?a= ?√a∇v?.

To introduce the fully-discrete implicit scheme approximating (2.2)–(2.3), we

consider a subdivision of the time interval (0,T] into subintervals (tn−1,tn], n =

1,...,N, with t0= 0 and tN= T, and we define kn:= tn−tn−1, the local time-step.

Associated with the time-subdivision, let Tn, n = 0,...,N, be a sequence of meshes

which are assumed to be compatible, in the sense that for any two consecutive

meshes Tn−1and Tn, Tncan be obtained from Tn−1by locally coarsening some

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4 E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

of its elements and then locally refining some (possibly other) elements. The finite

element space corresponding to Tnwill be denoted by Vn

We consider the fully discrete scheme for the wave problem (2.2), (2.3)

for each n = 1,...,N, find Un∈ Vn

?∂2Un,V ? + a(Un,V ) = ?fn,V ?

where fn:= f(tn,·), the backward second and first finite differences

∂2Un:=∂Un− ∂Un−1

h.

hsuch that

∀V ∈ Vn

h,

(2.8)

(2.9)

kn

,

with

(2.10)∂Un:=

Un− Un−1

kn

V0:= π0u1

, for n = 1,2,...,N,

for n = 0,

where U0:= π0u0, and π0: L2(Ω) → V0

element space (e.g., the orthogonal L2-projection operator).

ha suitable projection onto the finite

3. A posteriori error bounds for the semi-discrete problem

We derive here a posteriori error bound for the error ?u − U?L∞(0,T;L2(Ω))be-

tween the exact solution of (2.2), (2.3) and that of the semidiscrete scheme 2.5.

3.1. Definition (elliptic reconstruction and error splitting). Let U be the (semidis-

crete) finite element solution to the problem (2.5). Let also Π : L2(Ω) → Vhbe the

orthogonal L2-projection operator onto the finite element space Vh. We define the

elliptic reconstruction w = w(t) ∈ H1

elliptic problem

0(Ω), t ∈ [0,T], of U to be the solution of the

(3.1)a(w,v) = ?g,v?∀v ∈ H1

0(Ω)

where

(3.2)g := AU − Πf + f,

and A : Vh→ Vhis the discrete elliptic operator defined by

(3.3)for q ∈ Vh,

We decompose the error as follows:

?Aq,χ? = a(q,χ)∀χ ∈ Vh.

(3.4)e := U − u = ρ − ǫ, where ǫ := w − U, and ρ := w − u.

3.2. Lemma (error relation). With reference to the notation in (3.4) we have

(3.5)?ett,v? + a(ρ,v) = 0∀v ∈ H1

0(Ω).

Proof. We have, respectively,

(3.6)

?ett,v? + a(ρ,v) = ?Utt,v? + a(w,v) − ?f,v?

= ?Utt,Πv? + a(w,v) − ?f,v?

= −a(U,Πv) + a(w,v) + ?Πf − f,v? = 0,

observing the identity a(U,Πv) − ?Πf − f,v? = a(w,v), due to the construction of

w.

?

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L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION5

3.3. Theorem (abstract semidiscrete error bound). With the notation introduced

in (3.4), the following error bound holds:

(3.7)

?e?L∞(0,T;L2(Ω))≤?ǫ?L∞(0,T;L2(Ω))+

√2

?

?u0− U(0)? + ?ǫ(0)?

?

+ 2

?T

0

?ǫt? + Ca,T?u1− Ut(0)?,

where Ca,T := min{2T,?2CΩ/αmin}, where CΩ is the constant of the Poincar´ e–

Friedrichs inequality ?v?2≤ CΩ?∇v?2, for v ∈ H1

Proof. We use a testing procedure due to Baker [Bak76]. Let ˜ v : [0,T] × Ω → R

with

?τ

t

from some fixed τ ∈ [0,T]. Clearly ˜ v ∈ H1

that:

0(Ω).

(3.8)˜ v(t,·) =

ρ(s,·)ds,t ∈ [0,T],

0(Ω) as ρ ∈ H1

0(Ω). Also, we observe

(3.9)˜ v(τ,·) = 0,∇˜ v(τ,·) = 0, and˜ vt(t,·) = −ρ(t,·), a.e. in [0,T].

Set v = ˜ v in (3.5), integrate between 0 and τ with respect to the variable t and

integrate by parts the first term on the left-hand side, to obtain

?τ

0

Using (3.9), we have

?τ

0

2

0

2

which implies

1

2?ρ(τ)?2−1

Hence, we deduce

(3.13)

1

2?ρ(τ)?2−1

Now, we select τ = ˆ τ such that ?ρ(ˆ τ)? = max0≤t≤T?ρ(t)?, and we present two

alternative, but complementary, ways to complete the proof.

In the first way, we start by observing that ?˜ v(0)? ≤ τ?ρ(ˆ τ)?, gives

(3.14)

2?ρ(0)?2≤

Using the bound ?ρ(0)? ≤ ?e(0)?+?ǫ(0)?, e(0) = U(0)−u0and et(0) = Ut(0)−u1,

we conclude that

?e?L∞(0,T;L2(Ω))≤?ǫ?L∞(0,T;L2(Ω))+√2

??T

0

The second alternative, described next, consists in a different treatment of the

last term on the right-hand side of (3.13). The Poincar´ e–Friedrichs inequality and

the positivity of the diffusion coefficient a imply ?˜ v(0)?2≤ CΩα−1

(3.10)−

?et, ˜ vt? + ?et(τ), ˜ v(τ)? − ?et(0), ˜ v(0)? +

?τ

0

a(ρ, ˜ v) = 0.

(3.11)

1

d

dt?ρ?2−

?τ

1

d

dta(˜ v, ˜ v) =

?τ

0

?ǫt,ρ? + ?et(0), ˜ v(0)?,

(3.12)

2?ρ(0)?2+1

2a(˜ v(0), ˜ v(0)) =

?τ

0

?ǫt,ρ? + ?et(0), ˜ v(0)?.

2?ρ(0)?2+1

2a(˜ v(0), ˜ v(0)) ≤ max

0≤t≤T?ρ(t)?

?τ

0

?ǫt? + ?et(0)??˜ v(0)?.

1

4?ρ(τ)?2−1

??τ

0

?∂tǫ? + τ?et(0)?

?2

.

(3.15)

?

?u0− U(0)? + ?ǫ(0)?

?

+ 2

?ǫt? + T?u1− ∂U(0)?

?

.

min?˜ v(0)?2

a, for

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6 E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

some constant CΩ depending on the domain Ω only. Combining this bound with

(3.13), we arrive to

(3.16)

1

2?ρ(τ)?2−1

2?ρ(0)?2≤ max

0≤t≤T?ρ(t)?

?τ

0

?ǫt? +1

2CΩα−1

min?et(0)?2,

which implies

(3.17)

?e?L∞(0,T;L2(Ω))≤?ǫ?L∞(0,T;L2(Ω))+

√2

?

?u0− U(0)? + ?ǫ(0)?

?

+ 2

?T

0

?ǫt? +

?

2CΩ/αmin?u1− Ut(0)?.

Taking the minimum of the bounds (3.15) and (3.17) yields the result.

?

3.4. Remark (short and long integration times). The use of two alternative argu-

ments in the last step of the proof of Lemma 3.2 improves the “reliability constant”

Ca,T that works for both the short-time and the long-time integration regimes.

3.5. Remark (Completing the a posteriori estimation). To obtain a practical a

posteriori bound, we need to estimate the norms involving the elliptic error ǫ.

By construction, the elliptic reconstruction w is the exact solution to the elliptic

boundary-value problem (3.1) whose finite element solution is U. Indeed, inserting

v = V ∈ Vhin (3.1), we have

(3.18)a(w,V ) = ?AU − Πf + f,V ? = a(U,V ),

which implies the Galerkin orthogonality property a(w − U,V ) = 0. Therefore,

by construction, ǫ is the error of the finite element method on Vh for the elliptic

problem

(3.19) − ∇ · (a∇w) = g,

with homogeneous Dirichlet boundary conditions, with g defined by (3.2).

3.6. Definition. For every element face e ⊂ Γ, we define the jump across e of a

field w, defined in an open neighborhood of e, by

(3.20)[ [w] ](x) = lim

δ→0

?w(x + δne) − w(x − δne)

?

· ne,

for x ∈ e, where nedenotes one of the two normal vectors to e (the definition of

jump is independent of the choice).

3.7. Theorem (elliptic a posteriori residual bounds [Ver96, AO00]). Let z ∈ H1

be the solution to the elliptic problem:

0(Ω)

(3.21)− ∇ · (a∇z) = r

r ∈ L2(Ω) and Ω convex, and let Z ∈ Vhbe the finite element approximation of z

satisfying

(3.22)a(Z,V ) = ?r,V ?∀V ∈ Vh.

Then, there exists a positive constant Cel, independent of T , h, z and Z, so that

(3.23)?z − Z?2≤ CelE(Z,r,T ),

where

(3.24)E(Z,r,T ) :=

??

κ∈T

?

?h2(r + ∇ · (a∇Z)?2

κ+

?

e⊂Γ

?h3/2[ [a∇Z] ]?2

e

??1/2

.

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L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION7

3.8. Corollary (semidiscrete residual-type a posteriori error bound). Assume that

the hypotheses of Theorems 3.3 and 3.7 hold. Assume further that f is differentiable

with respect to time. Then the following error bound holds:

(3.25)

?e?L∞(0,T;L2(Ω))≤Cel?E(U,g,T )?L∞(0,T)+ 2Cel

+√2CelE(U(0),g(0),T )

+

?T

0

E(Ut,gt,T )

√2?u0− U(0)? + Ca,T?u1− Ut(0)?.

Proof. Using (3.18), ?ǫ? and ?ǫt? can be bounded from above using (3.23).

3.9. Remark. A bound of the form (3.23) is only required to to hold for Corollary

3.8 to be valid. Therefore, other available a posteriori bounds for elliptic prob-

lems [Ver96, AO00] can be also used.

?

4. A posteriori error bounds for the fully discrete problem

The analysis of §3 is now extended to the case of a fully-discrete implicit scheme

with the aid of a novel three point space-time reconstruction, satisfying a crucial

vanishing moment property in the time variable.

4.1. Definition (space-time reconstruction). Let Un, n = 0,...,N, be the fully dis-

crete solution computed by the method (2.8), Πn: L2(Ω) → Vn

L2-projection, and An: Vn

hto be the discrete operator defined by

(4.1)for q ∈ Vn

We define the elliptic reconstruction wn∈ H1

elliptic problem

a(wn,v) = ?gn,v?

with

gn:= AnUn− Πnfn+¯fn,

where¯f0(·) := f(0,·) and¯fn(·) := k−1

need to define the elliptic reconstruction ∂w0∈ H1

the elliptic problem

a(∂w0,v) = ?∂g0,v?

with

∂g0:= A0V0− Π0f0+ f0.

The time-reconstruction U : [0,T] × Ω → R of {Un}N

U(t) :=t − tn−1

kn

kn

for t ∈ (tn−1,tn], n = 1,...,N, noting that ∂U0is well defined. We note thatˆU

is a C1-function in the time variable, withˆU(tn) = UnandˆUt(tn) = ∂Unfor ,

n = 0,1,...,N.

We shall also use the time-continuous elliptic reconstruction w, defined by

w(t) :=t − tn−1

kn

kn

hbe the orthogonal

h→ Vn

h,?Anq,χ? = a(q,χ)∀χ ∈ Vn

h.

0(Ω), of Unto be the solution of the

(4.2)∀v ∈ H1

0(Ω),

(4.3)

n

?tn

tn−1f(t,·)dt for n = 1,...,N. Finally, we

0(Ω), of V0to be the solution of

(4.4)∀v ∈ H1

0(Ω),

(4.5)

n=0, is defined by

(4.6)

Un+tn− t

Un−1−(t − tn−1)(tn− t)2

kn

∂2Un,

(4.7)

wn+tn− t

wn−1−(t − tn−1)(tn− t)2

kn

∂2wn,

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8E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

noting that ∂w0is well defined. By construction, this is also a C1-function in the

time variable.

We decompose the error as follows:

(4.8)e := U − u = ρ − ǫ, where ǫ := w − U, and ρ := w − u.

4.2. Remark (notation overload). In this section we use symbols, e.g., U,w,e,ǫ,ρ,

that where used in §3, but with a slightly different meaning. Indeed, these are

now fully-discrete constructs, corresponding in aim and meaning, but different, to

their semidiscrete counterpart. It is hoped that this overload of notation should

not create any confusion.

4.3. Proposition (fully-discrete error relation). For t ∈ (tn−1,tn], n = 1,...,N,

we have

(4.9) ?ett,v?+a(ρ,v) = ?(I−Πn)Utt,v?+µn?∂2Un,Πnv?+a(w−wn,v)+?¯fn−f,v?,

for all v ∈ H1

operator onto Vn

0(Ω), with Πn: L2(Ω) → Vn

h, I is the identity mapping in L2(Ω), and

µn(t) := −6k−1

2 :=1

hdenoting the orthogonal L2-projection

(4.10)

n(t − tn−1

2),

where tn−1

2(tn+ tn−1).

Proof. Noting that Utt(t) = (1 + µn(t))∂2Un, for t ∈ (tn−1,tn], n = 1,...,N, and

the identity a(Un,Πnv) − ?Πnfn−¯fn,v? = a(wn,v), we deduce

?ett,v? + a(ρ,v) = ?Utt,v? + a(w,v) − ?f,v?,

= ?(I − Πn)Utt,v? + ?Utt,Πnv? + a(w,v) − ?f,v?,

= ?(I − Πn)Utt,v? + µn(t)?∂2Un,Πnv?

− a(Un,Πnv) + a(w,v) + ?Πnfn− f,v?

= ?(I − Πn)Utt,v? + µn(t)?∂2Un,Πnv? + a(w − wn,v) + ?¯fn− f,v?.

(4.11)

?

4.4. Remark (vanishing moment property). The particular form of the remainder

µn(t) satisfies the vanishing moment property

?tn

which appears to be of crucial importance for the optimality of the a posteriori

bounds presented below.

(4.12)

tn−1µn(t)dt = 0,

4.5. Definition (a posteriori error indicators). We define in a list form the error

indicators which will form error estimator the fully discrete bounds.

mesh change indicator: η1(τ) := η1,1(τ) + η1,2(τ), with

(4.13)η1,1(τ) :=

m−1

?

j=1

?tj

tj−1?(I − Πj)Ut? +

?τ

tm−1?(I − Πm)Ut?,

and

(4.14)η1,2(τ) :=

m−1

?

j=1

(τ − tj)?(Πj+1− Πj)∂Uj? + τ?(I − Π0)V0(0)?,

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L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION9

evolution error indicator:

(4.15)η2(τ) :=

?τ

0

?G?,

where G : (0,T] → R with G|(tj−1,tj]:= Gj, j = 1,...,N and

(4.16)Gj(t) :=(tj− t)2

2

∂gj−

?(tj− t)4

4kj

−(tj− t)3

3

?

∂2gj− γj,

with gjas in Definition 4.1 and γj:= γj−1+

with γ0= 0;

data error indicator:

k2

2∂gj+

j

k3

12∂2gj, j = 1,...,N,

j

(4.17)η3(τ) :=

1

2π

m−1

?

j=1

??tj

tj−1k3

j?¯fj− f?2?1/2

+

??τ

tm−1k3

m?¯fm− f?2?1/2

;

time reconstruction error indicator:

(4.18)η4(τ) :=

1

2π

m−1

?

j=1

??tj

tj−1k3

j?µj∂2Uj?2?1/2

+

??τ

tm−1k3

m?µm∂2Um?2?1/2

.

4.6. Theorem (abstract fully-discrete error bound). Recalling the notation of Def-

inition 4.1 and the indicators of Definition 4.5 we have the bound

(4.19)

?e?L∞(0,tN;L2(Ω))≤?ǫ?L∞(0,tN;L2(Ω))+

√2

?

?u0− U(0)? + ?ǫ(0)?

?

+ 2

??tN

0

?ǫt? +

4

?

i=1

ηi(tN)

?

+ Ca,N?u1− V0?,

where Ca,N := min{2tN,?2CΩ/αmin}, CΩ is Poincar´ e–Friedrichs inequality con-

stant.

Proof. The proof of Theorem 4.6, is spread in this and the following paragraphs up

to

Next we set v = ˜ v in (4.9) with ˜ v defined by (3.8) where ρ is defined as in (4.8)

(i.e., the fully discrete ρ), assuming that tm−1< τ ≤ tmfor some integer m with

1 ≤ m ≤ N. We integrate the resulting equation with respect to t between 0 and

τ, to arrive to

(4.20)

?τ

0

?ett, ˜ v? +

?τ

0

a(ρ, ˜ v) = I1(τ) + I2(τ) + I3(τ) + I4(τ),

Page 10

10E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

where

(4.21)

I1(τ) :=

m−1

?

j=1

?tj

tj−1?(I − Πj)Utt, ˜ v? +

?τ

tm−1?(I − Πm)Utt, ˜ v?,

I2(τ) :=

m−1

?

j=1

?tj

tj−1a(w − wj, ˜ v) +

?τ

tm−1a(w − wm, ˜ v)

I3(τ) :=

m−1

?

j=1

?tj

tj−1?¯fj− f, ˜ v? +

?τ

tm−1?¯fm− f, ˜ v?,

I4(τ) :=

m−1

?

j=1

?tj

tj−1µj?∂2Uj,Πj˜ v? +

?τ

tm−1µm?∂2Um,Πm˜ v?.

In Lemmas 4.7, 4.8, 4.9, and 4.11 we will derive bounds of the form

(4.22)Ii(τ) ≤ ηi(τ) max

0≤t≤T?ρ(t)?,

for i = 1,2,3,4. With the help of these, we will conclude the proof in §4.12.

4.7. Lemma (mesh change error estimate). Under the assumptions of Theorem 4.6

and with the notation (4.21) we have

?

(4.23)I1(τ) ≤ η1(τ) max

0≤t≤T?ρ(t)?.

Proof. Observing that the projections Πj, j = 1,...,N, commute with time-

differentiation, we integrate by parts with respect to t, arriving to

(4.24)

I1(τ) =

m−1

?

j=1

?tj

tj−1?(I − Πj)Ut,ρ? +

?τ

tm−1?(I − Πm)Ut,ρ?

+

m−1

?

j=1

?(Πj+1− Πj)Ut(tj), ˜ v(tj)? − ?(I − Π0)Ut(0),v(0)?.

The first two terms on the right-hand side of (4.24) are bounded by

(4.25) max

0≤t≤T?ρ(t)?

?m−1

j=1

?

?tj

tj−1?(I − Πj)Ut? +

?τ

tm−1?(I − Πm)Ut?

?

.

Recalling the definition of ˜ v and that U(tj) = ∂Uj, j = 0,1,...,N, we can bound

the last two terms on the right-hand side of (4.24) by

(4.26)max

0≤t≤T?ρ(t)?

?m−1

j=1

?

(τ − tj)?(Πj+1− Πj)∂Uj? + τ?(I − Π0)V0(0)?

?

.

?

4.8. Lemma (evolution error bound). Under the assumptions of Theorem 4.6 and

with the notation (4.21) we have

(4.27)I2(τ) ≤ η2(τ) max

0≤t≤T?ρ(t)?.

Page 11

L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION11

Proof. First, we observe the identity

(4.28)w − wj= −(tj− t)∂wj+

on each (tj−1,tj], j = 2,...,m. Hence, from Definition 4.1, we deduce

?

k−1

j(tj− t)3− (tj− t)2?

∂2wj,

(4.29)a(w − wj, ˜ v) = ?−(tj− t)∂gj+

The integral of the first component in the inner product on the right-hand side

of (4.29) with respect to t between (tj−1,tj] is then given by G. The choice of

constants in G implies that G is continuous on tj, j = 1,2,...,N and G(0) = 0.

Hence, integrating by parts on each interval (tj−1,tj], j = 1,...,m, we obtain

?τ

which already implies the result.

?

k−1

j(tj− t)3− (tj− t)2?

∂2gj, ˜ v?

(4.30)I2(τ) =

0

?G,ρ?,

?

4.9. Lemma (data approximation error bound). Under the assumptions of Theo-

rem 4.6 and with the notation (4.21) we have

(4.31)I3(τ) ≤ η3(τ) max

0≤t≤T?ρ(t)?.

Proof. We begin by observing that

(4.32)

?tj

tj−1(¯fj− f) = 0,

for all j = 1,...,m − 1. Hence, we have

m−1

?

j=1

(4.33)

?tj

tj−1?¯fj− f, ˜ v? =

m−1

?

j=1

?tj

tj−1?¯fj− f, ˜ v −¯˜ vj?,

where¯˜ vj(·) := k−1

j

?tj

tj−1˜ v(t,·)dt. Using the inequality

?tj

(4.34)

tj−1?˜ v −¯˜ vj?2≤

k2

4π2

j

?tj

tj−1?˜ vt?2,

and recalling that ˜ vt= ρ, we have, respectively,

?tj

(4.35)

m−1

?

j=1

tj−1?¯fj− f, ˜ v? ≤

m−1

?

j=1

??tj

tj−1?¯fj− f?2?1/2??tj

tj−1?˜ v −¯˜ vj?2?1/2

≤

1

2π

m−1

?

j=1

??tj

tj−1?¯fj− f?2?1/2??tj

tj−1k2

j?ρ?2?1/2

≤

1

2π

m−1

?

j=1

??tj

tj−1k3

j?¯fj− f?2?1/2

max

0≤t≤T?ρ(t)?.

For the remaining term in I3, we first observe that

(4.36)

?τ

tm−1?˜ v?2dt ≤

?τ

tm−1km

?τ

t

?ρ?2dsdt ≤ k3

mmax

0≤s≤T?ρ(t)?2,

which implies

(4.37)

?τ

tm−1?¯fm− f, ˜ v? ≤

??τ

tm−1k3

m?¯fm− f?2?1/2

max

0≤t≤T?ρ(t)?.

Page 12

12E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

Recalling η3from Definition 4.5 we conclude the proof.

?

4.10. Remark (the order of the data approximation indicator). The choice of the

particular combination of functions involving the right-hand side data f in the

definition of gnin the elliptic reconstruction, results to the property (4.32). When

f is differentiable, we have η3(τ) = O(k2) as k := max1≤j≤mkj → 0, and the

convergence is of second order with respect to the maximum time-step. In this

case, η3is, therefore, a higher order term.

4.11. Lemma (time-reconstruction error bound). Under the assumptions of Theo-

rem 4.6 and with the notation (4.21) we have

(4.38)I4(τ) ≤ η4(τ) max

0≤t≤T?ρ(t)?.

Proof. The method of bounding I4(τ) is similar to that of Lemma 4.9, so we shall

only highlight the differences.

Recalling the vanishing moment property (4.12) and noting that ∂2Ujis piece-

wise constant in time, we have

(4.39)

m−1

?

j=1

?tj

tj−1µj?∂2Uj,Πj˜ v? =

m−1

?

j=1

?tj

tj−1µj?∂2Uj,Πj(˜ v −¯˜ vj)?,

where¯˜ vj(·) = k−1

we obtain

(4.40)

m−1

?

j=1

j

?tj

tj−1˜ v(t,·)dt. Hence, since Πjcommutes with time integration,

?tj

tj−1µj?∂2Uj,Πj(˜ v −¯˜ vj)? ≤

1

2π

m−1

?

j=1

??tj

tj−1?µj∂2Uj?2?1/2??tj

tj−1k2

j?Πjρ?2?1/2

≤

1

2π

m−1

?

j=1

??tj

tj−1k3

j?µj∂2Uj?2?1/2

max

0≤t≤T?ρ(t)?.

For the remaining term in I4, upon using an argument similar to (4.36), we have

?τ

(4.41)

tm−1?µm∂2Um,Πm˜ v? ≤

??τ

tm−1k3

m?µm∂2Um?2?1/2

max

0≤t≤T?ρ(t)?.

Recalling the definition of η4in §4.5 we conclude.

?

4.12. Concluding the proof of Theorem 4.6. Starting from (4.20), integrating

by parts the first term on the left-hand side, and using the properties of ˜ v, we arrive

to

(4.42)

?τ

0

1

2

d

dt?ρ?2−

?τ

0

1

2

d

dta(˜ v, ˜ v) =

?τ

0

?ǫt,ρ? + ?et(0), ˜ v(0)? +

4

?

i=1

Ii(τ),

which implies

(4.43)

1

2?ρ(τ)?2−1

2?ρ(0)?2+1

2a(˜ v(0), ˜ v(0)) =

?τ

0

?ǫt,ρ?+?et(0), ˜ v(0)?+

4

?

i=1

Ii(τ).

Page 13

L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION13

Hence, we deduce

(4.44)

1

2?ρ(τ)?2−1

2?ρ(0)?2+1

2a(˜ v(0), ˜ v(0))

≤ max

0≤t≤T?ρ(t)?

??τ

0

?ǫt? +

4

?

i=1

ηi(τ)

?

+ ?et(0)??˜ v(0)?.

We select τ = ˆ τ such that ?ρ(ˆ τ)? = max0≤t≤tN ?ρ(t)?.

?˜ v(0)? ≤ τ?ρ(ˆ τ)?, gives

1

4?ρ(τ)?2−1

First, observing that

(4.45)

2?ρ(0)?2≤

??τ

0

?ǫt? +

4

?

i=1

ηi(τ) + τ?et(0)?

?2.

Using the bound ?ρ(0)? ≤ ?e(0)?+?ǫ(0)? and observing that e(0) =ˆU(0)−u(0) =

U0− u0and that et(0) =ˆUt(0) − ut(0) = V0− u1, we arrive to

?e?L∞(0,tN;L2(Ω))≤?ǫ?L∞(0,tN;L2(Ω))+

??tN

0

The second way is completely analogous to the proof of the semidiscrete case.

(4.46)

√2

?

?u0− U0? + ?ǫ(0)?

?

?

+ 2

?ǫt? +

4

?

i=1

ηi(tN) + tN?u1− V0?.

5. Fully-discrete a posteriori estimates of residual type

To arrive to a practical a posteriori bound for the fully-discrete scheme from

Theorem 4.6, the quantities involving the elliptic error ǫ should be estimated in

an a posteriori fashion: this is the content of Lemmas 5.1 and 5.3 below, when

residual-type a posteriori estimates are used.

5.1. Lemma (estimation of the elliptic error). With the notation introduced in

Definition 4.1, we have

√2?ǫ(0)? ≤ δ1(tN) +

where

?8k1

?35

27kj−1

with Ej:= E(Uj,AjUj− Πjfj+ fj,Tj), j = 0,1,...,N.

Proof. For t ∈ (tj−1,tj], j = 1,...,N, we have

(5.3)

ǫ =t − tj−1

kj

kj

from which, we can deduce

??35

27kj−1

noting that

(5.1)?ǫ?L∞(0,tN;L2(Ω))+

√2CelE0,

(5.2)

δ1(tN) := max

27CelE(V0,∂g0,T0),

kj

27+31

max

1≤j≤N

?

max

0≤j≤N

?CelEj+ CΩα−1

min?¯fj− fj???

,

(wj− Uj) +tj− t

(wj−1− Uj−1) −(t − tj−1)(tj− t)2

kj

(∂2wj− ∂2Uj),

(5.4)?ǫ? ≤ max

27+31

max

1≤j≤N

kj

?

max

0≤j≤N?wj− Uj?,8k1

27?∂w0− V0?

?

,

(5.5)max

t∈(tj−1,tj]

(t − tj−1)(tj− t)2

kj

=4k2

j

27.

Page 14

14 E.H. GEORGOULIS, O. LAKKIS, AND C. MAKRIDAKIS

It remains to estimate the terms ?wj− Uj? and ?∂w0− V0?. To this end, recalling

the notation of Definition 4.1, we define wj

problem

∗∈ H1

0(Ω) to be the solution of the elliptic

(5.6)a(wj

∗,v) = ?AjUj− Πjfj+ fj,v?∀v ∈ H1

0(Ω),

for j = 0,1,...,N. Note that, due to the fact that¯f0= f0, we have w0

By construction, we have a(wj

V ∈ Vj

elliptic boundary-value problem (5.6). In view of Theorem 3.7, this implies that

∗= w0.

∗,V ) = ?AjUj− Πjfj+ fj,V ? = a(Uj,V ) for all

h, j = 0,1,...,N. Hence, Ujis the finite element solution (in Vj

h) of the

(5.7)?wj

∗− Uj? ≤ CelEj,

for j = 0,...,N. Similarly, by construction, we have a(∂w0,V ) = ?A0V0−Π0f0+

f0,V ? = a(V0,V ) for all V ∈ V0

(5.8)?∂w0− ∂U0? ≤ CelE(V0,∂g0,T0).

Moreover, since wj− wj

¯fj− fj, standard elliptic stability results yield

(5.9)?wj− wj

for j = 1,...,N. Finally, using the triangle inequality

h. Hence,

∗is the solution of an elliptic problem with right hand-side

∗? ≤ CΩα−1

min?¯fj− fj?,

(5.10)?wj− Uj? ≤ ?wj− wj

∗? + ?wj

∗− Uj?,

along with the bounds (5.9), (5.8) and (5.7), already implies the result.

?

5.2. Remark. The bound (5.1) contains both the elliptic estimators E(·,·,·) and the

data-oscillation terms ?¯fj− fj? which are, in general, of first order with respect to

the time-step. The data-oscillation terms are expected to dominate the data error

indicator η3(cf. Remark 4.10). On the other hand, if the numerical scheme (2.8) is

altered so that fj=¯fj(as done, e.g., in [Bak76]), then the data-oscillation terms

in (5.1) vanish. Similar remarks apply to the result of Lemma 4.12 below.

For each n = 1,...,N, we denote byˆTnthe finest common coarsening of Tn

and Tn−1, and byˆVn

with the orthogonal L2-projection operatorˆΠn: L2(Ω) →ˆVn

5.3. Lemma (estimation of the time derivative of the elliptic error). With the

notation introduced in §4.1 we have

?tN

0

h:= Vn

h∩Vn−1

h

, the corresponding finite element space, along

h.

(5.11)?ǫt? ≤ δ2(tN),

where

(5.12)

δ2(tN) :=2

3

N

?

j=0

(2kj+ kj+1)

?

CelEj

∂+ CΩα−1

min?∂fj− ∂¯fj?

?

,

with

(5.13)Ej

∂:= E(∂Uj,∂(AjUj) − ∂(Πjfj) + ∂fj,ˆTj),j = 0,1,...,N.

Page 15

L∞(L2)-NORM A POSTERIORI BOUNDS FOR WAVE EQUATION15

Proof. For t ∈ (tj−1,tj], j = 1,...,N, we have

(5.14)ǫt= ∂wj− ∂Uj− k−1

from which, we deduce

?tj

noting that

?tj

Combining (5.15) for j = 1,...,N, we arrive to

j(tj− t)(tj− 2tj−1+ t)(∂2wj− ∂2Uj),

(5.15)

tj−1?ǫt? ≤4kj

3

?∂wj− ∂Uj? +2kj

3

?∂wj−1− ∂Uj−1?,

(5.16)

tj−1k−2

j(tj− t)(tj− 2tj−1+ t) =2kj

3

.

(5.17)

?tN

0

?ǫt? ≤2

3

N

?

j=0

(2kj+ kj+1)?∂wj− ∂Uj?,

with k0= 0 and kN+1= 0.

It remains to estimate the terms ?∂wj− ∂Uj?. To this end, recalling the def-

inition of the functions wj

0(Ω) from the proof of Lemma 5.1 and, since

ˆVj

h

, we have a(wj

a(Uj−1,V ) for all V ∈ˆVj

(5.18)a(∂wj

∗ ∈ H1

h:= Vj

h∩ Vj−1

∗,V ) = a(Uj,V ) for all V ∈ˆVj

h, for j = 1,...,N. Therefore, we deduce

∗,V ) = a(∂Uj,V )

hand a(wj−1

∗

,V ) =

for allV ∈ˆVj

h,

for j = 1,...,N, i.e., ∂Ujis the finite element solution inˆVj

problem

a(∂wj

hof the boundary-value

(5.19)

∗,V ) = ?∂(AjUj) − ∂(Πjfj) + ∂fj,v?

In view of Theorem 3.7, this implies that

∀v ∈ H1

0(Ω).

(5.20)?∂wj

∗− ∂Uj? ≤ CelEj

∂,

for j = 1,...,N.

a(V0,V ) for all V ∈ V0

Moreover, since

We also recall that, by construction, we have a(∂w0,V ) =

h. Hence, (5.8) also holds.

(5.21)a(∂wj,V ) = ?∂(AjUj) − ∂(Πjfj) + ∂¯fj,v?

j = 1,...,N, (cf. Definition 4.1). As in (5.9), elliptic stability implies

?∂wj− ∂wj

for j = 1,...,N and, using the triangle inequality

?∂wj− ∂Uj? ≤ ?∂wj− ∂wj

along with the bounds (5.22), (5.8) and (5.20), already implies the result.

∀v ∈ H1

0(Ω),

(5.22)

∗? ≤ CΩα−1

min?∂¯fj− ∂fj?,

(5.23)

∗? + ?∂wj

∗− ∂Uj?,

?

5.4. Theorem (fully-discrete residual-type a posteriori bound). With the same

hypotheses and notation as in Theorems 4.6 and 3.7, we have the bound

?e?L∞(0,tN;L2(Ω))≤δ1(tN) +

(5.24)

√2CelE0+

√2?u0− U(0)?

+ 2δ2(tN) + 2

4

?

i=1

ηi(tN) + Ca,N?u1− V0?,

where δ1,E0are defined in Lemma 5.1, δ2 is defined in Lemma 5.3, and ηi, i =

1,2,3,4 after (41) respectively.