An easily computable error estimator in space and time for the wave equation

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for the wave equation

Olga Gorynina, Alexei Lozinski, Marco Picasso

To cite this version:

Olga Gorynina, Alexei Lozinski, Marco Picasso. An easily computable error estimator in space and

time for the wave equation. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences,

2019, 53 (3), pp.729-747. �10.1051/m2an/2018049�. �hal-02922758�

https://doi.org/10.1051/m2an/2018049 www.esaim-m2an.org

AN EASILY COMPUTABLE ERROR ESTIMATOR IN SPACE AND TIME FOR THE WAVE EQUATION

O. Gorynina

, A. Lozinski

and M. Picasso

Abstract.We propose a cheaper version ofa posteriorierror estimator from Goryninaet al.(Numer.

Anal.(2017)) for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.

Mathematics Subject Classification. 65M15, 65M50, 65M60.

Received October 25, 2017. Accepted August 25, 2018.

1. Introduction

In this paper, we are interested ina posteriori time-space error estimates for finite element discretizations of the wave equation. Such lestimates were designed, for instance, in [9,12] for the case of implicit Euler discretization in time, in [13] for the case of the second order discretization in time by Cosine (or, equivalently, Newmark) scheme, and in [15] for a particular variant of the Newmark schemeβ= 1/4,γ= 1/2 (the advantage of the approach from [15] being its suitability for non constant time steps while the estimator from [13] is restricted to uniform meshes in time). In both [13,15], the error is measured in a physically natural norm:H¹ in space,L^∞ in time. Another common feature of these two papers is that the time error estimators proposed there contain the Laplacian of the discrete solution which should be computedvia an auxiliary finite element problem at each time step. This requires thus a non-negligible extra work in comparison with computing the discrete solution itself. In the present paper, we propose an alternative time error estimator for the particular Newmark scheme considered in [15] that avoids these additional computations.

Note that we have cited above only the articles on explicit residual based error bounds in space and time.

The litterature on the error control of finite element methods for second order hyperbolic problems, although much less abundant than that for eliptic and parabolic problems, also contains different approaches. We can cite the goal-oriented error estimators by Bangerth et al. [6–8] where the error is measured with respect to some functional of the solution, and the work by Adjerid et al. [1–4] proposing asymptotically accurate error estimates at the expense of a workload which is generally bigger than that of explicit estimators. We also note

Keywords and phrases. a posteriorierror bounds in time and space, wave equation, Newmark scheme.

1 Laboratoire de Math´ematiques de Besan¸con, CNRS UMR 6623, Univ. Bourgogne Franche-Comt´e, 16 route de Gray, 25030 Besan¸con Cedex, France.

2 Institute of Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, Station 8, CH 1015 Lausanne, Switzerland.

∗Corresponding author:alexei.lozinski@univ-fcomte.fr

c

The authors. Published by EDP Sciences, SMAI 2019

This is an Open Access article distributed under the terms of theCreative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

that the work of Adjeridet al. is related to some space-time Galerkin discretizations, which are different from the time marching schemes considered in the present work as well as in other papers cited above.

In deriving our a posteriori estimates, we follow first the approach of [15]. First of all, we recognize that the Newmark method can be reinterpreted as the Crank–Nicolson discretization of the reformulation of the governing equation as the first-order system, as in [5]. We then use the techniques stemming from a posteriori error analysis for the Crank–Nicolson discretization of the heat equation in [17], based on a piecewise quadratic polynomial in time reconstruction of the numerical solution. Finally, in a departure from [15], we replace the second derivatives in space (Laplacian of the discrete solution) in the error estimate with the forth derivatives in time by reusing the governing equation. This leads to the newa posteriori error estimate in time and also allows us to easily recover the error estimates in space that turn out to be the same as those of [15]. The resulting estimate is referred to as the 5-point estimator since it contains the fourth order finite differences in time and thus involves the discrete solution at 5 points in time at each time step. On the other hand, the estimate [15]

involves only 3 points in time at each time step and will be thus referred to as the 3-point estimator.

Like in the case of the 3-point estimator, we are able to prove that the new 5-point estimator is reliable on general regular meshes in space and non-uniform meshes in time (with constants depending on the regularity of meshes in both space and time). Moreover, the 5-point estimator is proved to be of optimal order at least on sufficiently smooth solutions, quasi-uniform meshes in space and uniform meshes in time, again reproducing the results known for the 3-point estimator. Numerical experiments demonstrate that 3-point and 5-point error estimators produce very similar results in the majority of test cases. Both turn out to be of optimal order in space and time, even in situations not accessible to the current theory (non quasi-uniform meshes, not constant time steps). It should be therefore possible to use the new estimator for mesh adaptation in space and time. In fact, the best strategy in practice may be to combine both estimators to take benefit from the strengths of each of them: the relative cheapness of the 5-point one, and the better numerical behavior of the 3-point estimator under abrupt changes of the mesh.

The outline of the paper is as follows. We present the governing equations and the discretization in Section2.

Since our work is based on techniques from [15], Section3 recalls the a posteriori bounds in time and space from there. In Section4, the 5-pointa posteriori error estimator for the fully discrete wave problem is derived.

Numerical experiments on several test cases are presented in Section5.

2. The Newmark scheme for the wave equation

We consider initial-boundary-value problem for the wave equation. Let Ω be a bounded domain inR² with boundary∂Ω andT >0 be a given final time. Letu=u(x, t) : Ω×[0, T]→Rbe the solution to



















∂²u

∂t² −∆u=f, in Ω×]0, T], u= 0, on∂Ω×]0, T], u(·,0) =u0, in Ω,

∂u

∂t(·,0) =v₀, in Ω,

(2.1)

where f, u₀, v₀ are given functions. Note that if we introduce the auxiliary unknownv = ^∂u_∂t then model (2.1) can be rewritten as the following first-order in time system















∂u

∂t−v= 0, in Ω×]0, T],

∂v

∂t−∆u=f, in Ω×]0, T], u=v= 0, on∂Ω×]0, T], u(·,0) =u0, v(·,0) =v0, in Ω.

(2.2)

The above problem (2.1) has the following weak formulation [11]: for given f ∈L²(0, T;L²(Ω)), u0∈H₀¹(Ω) andv0∈L²(Ω) find a function

u∈L² 0, T;H₀¹(Ω) , ∂u

∂t∈L² 0, T;L²(Ω) , ∂²u

∂t² ∈L² 0, T;H⁻¹(Ω)

, (2.3)

such thatu(x,0) =u₀ in H₀¹(Ω), ∂u

∂t(x,0) =v₀ inL²(Ω) and ∂²u

∂t², ϕ

+ (∇u,∇ϕ) = (f, ϕ), ∀ϕ∈H₀¹(Ω), (2.4) whereh·,·idenotes the duality pairing betweenH⁻¹(Ω) andH₀¹(Ω) and the parentheses (·,·) stand for the inner product inL²(Ω). Following Chapter 7, Section 2, Theorem 5 of [11], we observe that in fact

u∈C⁰ 0, T;H₀¹(Ω) , ∂u

∂t ∈C⁰ 0, T;L²(Ω) , ∂²u

∂t² ∈C⁰ 0, T;H⁻¹(Ω) . Higher regularity results with more regular data are also available in [11].

Let us now discretize (2.1) or, equivalently, (2.2) in space using the finite element method and in time using an appropriate marching scheme. We thus introduce a regular meshThon Ω with triangles K, diamK=hK, h= max_K∈ThhK, internal edgesE∈ Eh, whereEhrepresents the internal edges of the meshThand the standard finite element spaceVh⊂H₀¹(Ω) of piecewise polynomials of degreek≥1:

V_h=

v_h∈C( ¯Ω) :v_h|K∈Pk ∀K∈ Th andv_h|∂Ω= 0 . Let us also introduce a subdivision of the time interval [0, T]

0 =t0< t1<· · ·< tN =T,

with non-uniform time stepsτ_k =t_n+1−t_k forn= 0, . . . , N −1 andτ= max

0≤n≤N−1τ_k.

The Newmark scheme [18,19] with coefficientsβ= 1/4, γ= 1/2 as applied to the wave equation (2.1): given approximationsu⁰_h, v⁰_h∈V_hofu₀, v₀ computeu¹_h∈V_h from

u¹_h−u⁰_h τ0

, ϕh

+

∇τ₀(u¹_h+u⁰_h) 4 ,∇ϕh

= v_h⁰+τ₀

4(f¹+f⁰), ϕh

, ∀ϕh∈Vh, (2.5) and then computeuⁿ⁺¹_h ∈Vhforn= 1, . . . , N−1 from equation

uⁿ⁺¹_h −uⁿ_h

τ_k −uⁿ_h−uⁿ⁻¹_h τ_n−1 , ϕh

+

∇τk(uⁿ⁺¹_h +uⁿ_h) +τ_n−1(uⁿ_h+uⁿ⁻¹_h )

4 ,∇ϕh

=

τ_k(fⁿ⁺¹+fⁿ) +τ_n−1(fⁿ+fⁿ⁻¹)

4 , ϕh

, ∀ϕh∈Vh. (2.6) wherefⁿ is an abbreviation for f(·, tk).

Following [5,15], we observe that this scheme is equivalent to the Crank–Nicolson discretization of the govern- ing equation written in the form (2.2): takingu⁰_h, v_h⁰∈Vhas some approximations tou0, v0computeuⁿ_h, v_hⁿ∈Vh

forn= 0, . . . , N−1 from the system

uⁿ⁺¹_h −uⁿ_h τk

−v_hⁿ+v_hⁿ⁺¹

2 = 0, (2.7)

v_hⁿ⁺¹−v_hⁿ τ_k , ϕ_h

+

∇uⁿ⁺¹_h +uⁿ_h 2 ,∇ϕ_h

=

fⁿ⁺¹+fⁿ 2 , ϕ_h

, ∀ϕ_h∈V_h. (2.8)

Note that the additional unknowns v_k^h are the approximations are not present in the Newmark scheme (2.5)–(2.6). If needed, they can be recovered on each time step by the following easy computation

v_hⁿ⁺¹= 2uⁿ⁺¹_h −uⁿ_h

τ_k −v_hⁿ. (2.9)

From now on, we shall use the following notations u^n+1/2_h :=uⁿ⁺¹_h +uⁿ_h

2 , ∂n+1/2uh:= uⁿ⁺¹_h −uⁿ_h τk

, ∂nuh:=uⁿ⁺¹_h −uⁿ⁻¹_h

τk+τ_n−1 , (2.10)

∂_n²uh:= 1 τ_n−1/2

uⁿ⁺¹_h −uⁿ_h τk

−uⁿ_h−uⁿ⁻¹_h τ_n−1

withτ_n−1/2:=τk+τ_n−1

2 ·

We apply this notations to all quantities indexed by a superscript, so that, for example, f^n+1/2 = (fⁿ⁺¹+fⁿ)/2. We also denote u(x, tk), v(x, tk) by uⁿ, vⁿ so that, for example, u^n+1/2 = uⁿ⁺¹+uⁿ

/2 = (u(x, t_n+1) +u(x, t_k))/2.

We shall measure the error in the following norm u7→ max

t∈[0,T]

∂u

∂t(t)

2

L²(Ω)

+|u(t)|²_H1(Ω)

!1/2

. (2.11)

Here and in what follows, we use the notations u(t) and ∂u

∂t(t) as a shorthand for, respectively, u(·, t) and

∂u

∂t(·, t). The norms and semi-norms in Sobolev spacesH^k(Ω) are denoted, respectively, byk·k_Hk(Ω)and|·|_Hk(Ω). We call (2.11) the energy norm referring to the underlying physics of the studied phenomenon. Indeed, the first term in (2.11) may be assimilated to the kinetic energy and the second one to the potential energy.

3. The 3-point time error estimator

The aim of this section is to recalla posteriori bounds in time and space from [15] for the error measured in the norm (2.11). Their derivation is based on the following piecewise quadratic (in time) 3-point reconstruction of the discrete solution.

Definition 3.1. Let uⁿ_h be the discrete solution given by the scheme (2.6). Then, the piecewise quadratic reconstruction ˜u_hτ(t) : [0, T]→V_h is constructed as the continuous in time function that is equal on [t_k, t_n+1], n≥1, to the quadratic polynomial intthat coincides withuⁿ⁺¹_h (respectively uⁿ_h, uⁿ⁻¹_h ) at timet_n+1 (respec- tively tk, tn−1). Moreover, ˜uhτ(t) is defined on [t0, t1] as the quadratic polynomial in t that coincides withu²_h (respectivelyu¹_h,u⁰_h) at timet2 (respectivelyt1,t0). Similarly, we introduce piecewise quadratic reconstruction

˜

vhτ(t) : [0, T]→Vh based onvⁿ_h defined by (2.9) and ˜fτ(t) : [0, T]→L²(Ω) based onf(tk,·).

The quadratic reconstructions ˜uhτ, ˜vhτ are thus based on three points in time (normally looking backwards in time, with the exemption of the initial time slab [t0, t1]). This is also the case for the time error estimator (3.3), recalled in the following Theorem and therefore referred to as the 3-point estimator.

Theorem 3.2. The following a posteriori error estimate holds between the solutionuof the wave equation (2.1) and the discrete solutionuⁿ_h given by (2.5) and (2.6) for all tk, 0≤n≤N withvⁿ_h given by (2.9):

vⁿ_h−∂u

∂t(t_n)

2

L²(Ω)

+|uⁿ_h−u(t_n)|²_H1(Ω)

!^1/2

≤

v_h⁰−v₀

2 L²(Ω)+

u⁰_h−u₀

2 H¹(Ω)

^1/2

+ηS(tn) +

n−1

X

k=0

τkηT(tk) + Z t_n

0

kf−f˜τk_L2(Ω)dt, (3.1)

where the space indicator is defined by

ηS(tn) =C1 max

06t6tn

"

X

K∈T_h

h²_K

∂v˜hτ

∂t −∆˜uhτ−f

2

L²(K)

+ X

E∈E_h

hE|[n· ∇˜uhτ]|²_L2(E)

#1/2

(3.2)

+C₂

n−1

X

m=0

Z tm+1

t_m

"

X

K∈Th

h²_K

∂²v˜_hτ

∂t² −∆∂u˜_hτ

∂t −∂f

∂t

2

L²(K)

+ X

E∈Eh

h_E

n· ∇∂˜u_hτ

∂t

2

L²(E)

#^1/2 dt, hereC1, C2 are constants depending only on the mesh regularity,[·]stands for a jump on an edge E∈ Eh, and

˜

uhτ,v˜hτ are given by Definition3.1.

The error indicator in time fork= 1, . . . , N−1 is η_T(t_k) =

1 12τ_k²+1

8τ_k−1τ_k

∂_k²v_h

2

H¹(Ω)+

∂_k²f_h−z_h^k

2 L²(Ω)

^1/2

, (3.3)

wherez_h^k is such that

z_h^k, ϕh

= (∇∂_k²uh,∇ϕh), ∀ϕh∈Vh, (3.4) and

ηT(t0) = 5

12τ₀²+1 2τ1τ0

∂²₁vh

2

H¹(Ω)+

∂²₁fh−z_h¹

2 L²(Ω)

^1/2

. (3.5)

We also recall an optimality result for the 3-point time error estimator. We introduce to this end the H₀¹- orthogonal projectionΠh:H₀¹(Ω)→Vh so that

(∇Π_hv,∇ϕ_h) = (∇v,∇ϕ_h), ∀v∈H₀¹(Ω), ∀ϕ_h∈V_h, (3.6) Theorem 3.3. Let u be the solution of wave equation (2.1) and ∂³u

∂t³(0)∈H¹(Ω), ∂²u

∂t²(0)∈H²(Ω), ∂²f

∂t²(t)∈ L^∞(0, T;L²(Ω)), ∂³f

∂t³(t)∈L²(0, T;L²(Ω)). Suppose that meshTh is quasi-uniform, the mesh in time is uniform (t_k=kτ), and the initial approximations are chosen as

u⁰_h=Πhu0, v⁰_h=Πhv0. (3.7)

Then, the 3-point time error estimator ηT(tk)defined by (3.3,3.5) is of orderτ², i.e.

ηT(tk)≤Cτ².

with a positive constantC depending only onu,f, and the regularity of meshTh.

Remark 3.4. Note that the particular choice for the approximation of initial conditions in (3.7) using the H₀¹-orthogonal projection (3.6) is crucial to obtain the optimal order of the 3-point time error estimator, as confirmed both theoretically and numerically in [15].

4. The 5-point A P

error estimator

As already mentioned in the Introduction, the time error estimator (3.3) contains a finite element approxi- mation to the Laplacian of u^k_h, i.e.z_h^k given by (3.4). This is unfortunate because z^k_h should be computed by solving an additional finite element problem that implies additional computational effort. Having in mind that the term∂_n²f_h−z_hⁿ in (3.3) is a discretization of∂²f /∂t²+ ∆u=∂⁴u/∂t⁴ at timet_n our goal now is to avoid the second derivatives in space in the error estimates and replace them with the forth derivatives in time.

We introduce a “fourth order finite difference in time”∂⁴_n defined by

∂_n⁴w_h= 8

τ_n+τ_n−1+τ_n−2+τ_n−3

∂_n²w_h−∂_n−1² w_h

τ_n+τ_n−2 −∂_n−1² w_h−∂_n−2² w_h τ_n−1+τ_n−3

!

(4.1) on any sequence{wⁿ_h}n=0,1,...∈Vh. This can be rewritten as a composition of two second order finite difference operators

∂_n⁴w_h= ˆ∂²_n∂²w_h, (4.2)

where ∂²w_h is the standard finite difference (2.10) applied to w_h, and ˆ∂_n² is a modified second order finite difference defined by

∂ˆ²_nωh= 2 (ˆt_n−ˆt_n−2)

ω_hⁿ−ω_hⁿ⁻¹

ˆt_n−ˆt_n−1 −ω_hⁿ⁻¹−ωⁿ⁻²_h ˆt_n−1−ˆt_n−2

, (4.3)

ˆtn=t_n+1+t_n−1 2

on any sequence {ω_hⁿ}n=0,1,... ∈ Vh. Note that a lower subscript “n” is lacking from ∂²wh in (4.2) consistent with the fact that ˆ∂_n² is applied there to the sequence{∂_n²wh}n=0,1,... rather than to a single instance of∂_n²wh. In full detail, (4.2) should be interpreted as∂_n⁴wh= ˆ∂²_nωh withω_hⁿ=∂²_nwh.

Remark 4.1. In the case of constant time stepsτn=τ, (4.1) is reduced to

∂_n⁴wh=wⁿ⁺¹_h −4wⁿ_h+ 6wⁿ⁻¹_h −4w_hⁿ⁻²+w_hⁿ⁻³

τ⁴ ·

It is thus indeed a standard finite difference approximation to the fourth derivative. In particular, it is exact on polynomials (in time) of degree up to 4. However, a standard fourth order finite difference in the general case of non constant time steps would be given by the divided differences

∂˜_n⁴wh= 4![w_hⁿ⁻³, . . . , wⁿ⁺¹_h ]

= 12

τ_n+τ_n−1+τ_n−2+τ_n−3

∂²_nw_h−∂_n−1² w_h

τ_n+τ_n−1+τ_n−2 − ∂²_n−1w_h−∂²_n−2w_h τ_n−1+τ_n−2+τ_n−3

! .

Clearly, the formulas for∂_n⁴whand ˜∂_n⁴wh, although similar, do not coincide in general, and consequently∂_n⁴wh

is not necessarily consistent with the fourth derivative in time ofw_h. Definition (4.1) may seem thus artificial and counter-intuitive. We shall see however that it arises naturally in the analysis of Newmark scheme, cf.

forthcoming Lemma4.2. Indeed, in order to “differentiate” in time the averaged quantities ¯wⁿ_h defined by (4.4) and present in the scheme (2.6),cf.also (4.13), one needs to employ the modified second order finite difference

∂ˆ_n², which shall be composed further with∂_n² to give rise to∂_n⁴. For any sequence{w_hⁿ}n=0,1,...∈Vh, we denote

¯

wⁿ_h= τn(w_hⁿ⁺¹+wⁿ_h) +τ_n−1(wⁿ_h+w_hⁿ⁻¹)

4τ_n−1/2 · (4.4)

Consistently with the conventions above, ¯w_h will stand for the collection of any sequence {w¯ⁿ_h}_n=0,1,.... The following technical lemma establishes a connection between second order discrete derivatives ˆ∂²_n and∂_n².

Lemma 4.2. For all integer n= 3, . . . N−1 there exist coefficients α_k, k =n−2, n−1, nsuch that for all {w_hⁿ}n=0,1,...

∂ˆ_n²w¯_h=

n

X

k=n−2

α_k∂²_kw_h, (4.5)

Moreover

|αk| ≤c, fork=n−2, n−1, n, and

n

X

k=n−2

α_k≥C,

where c and C are positive constants depending only on the mesh regularity in time, i.e. on max_k≥0τ

k+1

τ_k +_τ^τk

k+1

.

Proof. We first note that relation (4.5) does not contain any derivatives in space and thus it should hold at any pointx∈Ω. Consequently, it is sufficient to prove this Lemma assuming thatw_hⁿ,∂²_kwh, etc. are real numbers, i.e. replacing V_h by R. This is the assumption adopted in this proof. We shall thus drop the sub-indexes h everywhere. Furthermore, it will be convenient to reinterpretwⁿ in (4.2), (4.3) and (4.4) as the values of a real valued functionw(t) att=tn. We shall also use the notations like ¯wⁿ,∂_n²w, and so on, wherewis a continuous function onR, always assumingwⁿ=w(tn).

Observe that ˆ∂_n²w¯is a linear combination of 5 numbers{wⁿ⁻³, . . . , wⁿ⁺¹}. Thus, it is enough to check equality (4.5) on any 5 continuous functionsφ_(k)(t),k=n−3, . . . , n+ 1, such that the vector of values ofφ_(k)at times tl,l=n−3, . . . , n+ 1, form a basis ofR⁵. For fixedn, let us choose these functions as

φ_(k)(t) =











t−t_k−1

τ_k−1 , ift < tk, t_k+1−t

τk

, ift≥t_k,

k=n−3, . . . , n+ 1. (4.6)

First we notice that for every linear function u(t) on [t_n−3, tn+1] we have ˆ∂_n²u¯ = ∂_n²u = 0. Thus, we get immediately ˆ∂_n²φ¯_(n−3)=∂_n²φ_(n−3)= 0 and ˆ∂_k²φ¯_(n+1)=∂_k²φ_(n+1)= 0 so that (4.5) is fulfilled on functionsφ_(n−3), φ_(n+1)with any coefficientsα_k,k=n−2, n−1, n. Now we want to provide coefficients α_k,k=n−2, n−1, n for which (4.5) is fulfilled on functionsφ_(n−2), φ_(n−1) and φ(n). For brevity, we demonstrate the idea only for functionφ_(n)(t). Function φ_(n)(t) is linear on [t_n−3, tn] and thus

∂_n−2² φ_(n)= 0, ∂_n−1² φ_(n)= 0.

From direct computations it is easy to show that

∂_n²φ_(n)∼ 1

τ_k², φ¯_(n)∼1, ∂ˆ_k²φ¯_(n)∼ 1 τ_k²,

where ∼ hides some factors that can be bounded by constants depending only on the mesh regularity. Thus we are able to establish expression for coefficient α_n =

∂ˆ²_kφ¯_(n)

∂_n²φ_(n)≤C. Similar reasoning for functionφ_(n−1) and φ_(n−2) shows thatα_n−1=

∂ˆ_n²φ¯_(n−1)

∂²_n−1φ_(n−1)≤Candα_n−2=

∂ˆ_n²φ¯_(n−2)

∂_n−2² φ_(n−2)≤C.

The next step is to show boundedness from below of

n

X

k=n−2

α_k. We will show it by applying equality (4.5) to second order polynomial functions(t) = t²

2. Using a Taylor expansion ofs(t) around ˆtn in the definition of ¯sⁿ

gives

¯ sⁿ =

τn(ˆt²_n+ ˆtnτn−1+1

4(τ_n²+τ_n−1² )) +τn−1(ˆt²_n−ˆtnτn+1

4(τ_n²+τ_n−1² )) 2(τn+τ_n−1)

=ˆt²_n 2 +1

8 τ_n²+τ_n−1² .

Substituting this into the definition of ˆ∂_n²s¯we obtain

∂ˆ²_n¯s=

¯

sⁿ−¯sⁿ⁻¹

ˆt_n−ˆt_n−1 −¯sⁿ⁻¹−¯sⁿ⁻² ˆt_n−1−ˆt_n−2 ˆtn−tˆ_n−2

/2 = 1 +1

8









2τ_n²−τ_n−2²

τ_n+τ_n−2 −2τ_n−1² −τ_n−3² τ_n−1+τ_n−3

1

4(τn+τn−1+τn−2+τn−3)









= 1 +τ_n−τ_n−1−τ_n−2+τ_n−3 τn+τn−1+τn−2+τn−3

. Using (4.5) and the fact that∂_n²s= 1 fork=n−2, n−1, nwe note that

1 + τn−τ_n−1−τ_n−2+τ_n−3 τn+τ_n−1+τ_n−2+τ_n−3=

n

X

k=n−2

αk.

This implies

n

X

k=n−2

αk ≥C.

Lemma 4.3. Let wⁿ_h, sⁿ_h∈Vh be such that w_hⁿ⁺¹−wⁿ_h

τk

−sⁿ_h+sⁿ⁺¹_h

2 = 0, ∀n≥0. (4.7)

For alln≥3 there exist coefficientsβk,k=n−2, n−1, nsuch that

n

X

k=n−2

αk∂_n²wh=

n

X

k=n−2

αk

!

∂_k²wh−τk n

X

k=n−2

βk∂_k²sh, (4.8)

where coefficientsα_k,k=n−2, n−1, nare introduced in Lemma4.2. Moreover

|βk| ≤C, k=n−2, n−1, n,

whereC is a positive constant depending only on the mesh regularity in time, i.e. on max_k≥0τ

k+1

τk +_τ^τk

k+1

. Proof. As in proof of Lemma4.2, we assumeV_h=R, drop the sub-indexeshand interpretwⁿ, sⁿ as the values of continuous real valued functions w(t), s(t) at t =tn. Using (4.7) and notations (2.10) implies ∂_k²w=∂ks.

Now, we are able to rewrite (4.8) in terms ofsⁿ only

n

X

k=n−2

α_k∂_ks=

n

X

k=n−2

α_k

!

∂_ns−τ_k

n

X

k=n−2

β_k∂_k²s, (4.9)

As in the proof of Lemma 4.2 we take into account the fact that equation (4.9) should hold for every 5 numbers{sⁿ⁻³, . . . , sⁿ⁺¹}and therefore it’s enough to check equality (4.9) on 5 linearly independent piecewise linear functionsφ_(k)introduced by (4.6). Using the reasoning as in Lemma4.2leads to desired result (4.8).

We can now prove an a posteriori error estimate involving ∂_n⁴u_h. Since the latter is computed through 5 points in time {tn−3, . . . , tn+1}, we shall refer to this approach as the 5-point estimator. For the same reason, this estimator is only applicable from timet4. The error at first 3 time steps should be thus measured differently, for example using the 3-point estimator from Theorem3.2.

Theorem 4.4. The following a posteriori error estimate holds between the solutionuof the wave equation (2.1) and the discrete solutionuⁿ_h given by (2.5)–(2.6) for all t_n, 4≤n≤N withvⁿ_h given by (2.9):

vⁿ_h−∂u

∂t(tn)

2

L²(Ω)

+|uⁿ_h−u(tn)|²_H1(Ω)

!1/2

≤

v_h³−∂u

∂t(t3)

2

L²(Ω)

+

u³_h−u(t3)

2 H¹(Ω)

!1/2

+ηS(tn) +C

n−1

X

k=3

τkηˆT(tk) +C

n−1

X

k=1

τkηˆ_T^h.o.t(tk) +

Z tn

t3

kf−f˜_τk_L2(Ω)dt, (4.10)

where the space error indicator is defined by (3.2) and the time error indicator is ˆ

ηT(tk) = 1

12τ_k²+1 8τk−1τk

∂_k²vh

_H1(Ω)+

∂_k⁴uh

_L2(Ω)

(4.11) with additional higher order terms

ˆ

η^h.o.t_T (tk) =τ_k³

∂_k²f˙h−Ah∂_k²vh

_L2(Ω)

wheref˙_hⁿ satisfy

f_hⁿ⁺¹−f_hⁿ τn

=

f˙_hⁿ+ ˙f_hⁿ⁺¹

2 .

The constant C >0 depends only on the mesh regularity in time, i.e. onmax_k≥0τ

k+1

τ_k +_τ^τk

k+1

.

Proof. We note first of all that it is sufficient to prove the Theorem for the final time,i.e.n=N because the statement for the general casen < Nwill follow by resetting the final timeNton. Introducing theL²-orthogonal projectionP_h:H₀¹(Ω)→V_hand operator A_h:V_h→V_h such that

(A_hw_h, ϕ_h) = (∇wh,∇ϕh), ∀ϕh∈V_h, (4.12) we can rewrite scheme (2.6) as

∂_n²uh+Ahu¯ⁿ_h= ¯f_hⁿ, (4.13) forn= 0, . . . , N−1 where ¯f_hⁿ is defined through averaging (4.4) from f_hⁿ=Phf(tn,·). Taking a linear combi- nation of instances of (4.13) at stepsn, n−1, n−2 with appropriate coefficients gives

∂_n⁴uh+Ah∂ˆ_n²¯uh= ˆ∂_n²f¯h. (4.14) Using the definition of operator ˆ∂_n² and re-introducingv_hⁿ by (2.7) leads to

∂ˆ_n²u¯_h=

n

X

k=n−2

α_k∂_k²u_h=

n

X

k=n−2

α_k

!

∂_k²u_h−τ_n

n

X

k=n−2

β_k∂_k²v_h,

with coefficients α_k, β_k introduced in Lemmas 4.2 and 4.3. Moreover, by Lemma 4.2 γ = Pn

k=n−2α_k−1

is positive and bounded so that

∂_n²uh=γ∂ˆ²_nu¯h+τn n

X

k=n−2

γk∂_k²vh,

withγ_k=γβ_k that are all uniformly bounded on regular meshes in time. Similarly,

∂_n²fh= ˆ∂_n²f¯h+τn n

X

k=n−2

γk∂_k²f˙h.

Thus,

∂_n²fh−Ah∂_n²uh= ˆ∂_n²f¯h−Ah∂ˆ_n²u¯h+τn n

X

k=n−2

γk

∂_k²f˙h−Ah∂_k²vh

=∂_n⁴u_h+τ_n

n

X

k=n−2

γ_k

∂²_kf˙_h−A_h∂_k²v_h .

The rest of the proof follows closely that of Theorem 3.2, cf.[15]. We adopt the vector notationU(t, x) = u(t, x)

v(t, x)

wherev=∂u/∂t. Note that the first equation in (2.2) implies that

∇∂u

∂t,∇ϕ

−(∇v,∇ϕ) = 0, ∀ϕ∈H₀¹(Ω),

by taking its gradient, multiplying it by∇ϕand integrating over Ω. Thus, system (2.2) can be rewritten in the vector notations as

b ∂U

∂t,Φ

+ (A∇U,∇Φ) =b(F,Φ), ∀Φ∈(H₀¹(Ω))², (4.15)

whereA= 0−1

1 0

,F = 0

f

and

b(U,Φ) =b u

v

, ϕ

ψ

:= (∇u,∇ϕ) + (v, ψ).

Similarly, Newmark scheme (2.7) and (2.8) can be rewritten as b

U_hⁿ⁺¹−U_hⁿ τn

,Φ_h

+

A∇U_hⁿ⁺¹+U_hⁿ 2 ,∇Φh

=b

F^n+1/2,Φ_h

, ∀Φh∈V_h², (4.16)

whereU_hⁿ= uⁿ_h

vⁿ_h

andF^n+1/2= 0

f^n+1/2

.

Thea posteriori analysis relies on an appropriate residual equation for the quadratic reconstruction ˜Uhτ = u˜hτ

˜ vhτ

. We have thus fort∈[tn, tn+1],n= 1, . . . , N −1 U˜hτ(t) =U_hⁿ⁺¹+ (t−tn+1)∂n+1/2Uh+1

2(t−tn+1)(t−tn)∂_n²Uh, (4.17)

so that, after some simplifications, b ∂U˜hτ

∂t ,Φ_h

!

+ (A∇U˜_hτ,∇Φ_h) =b

(t−t_n+1/2)∂_n²U_h+F^n+1/2,Φ_h +

(t−tn+1/2)A∇∂n+1/2Uh+1

2(t−tn+1)(t−tn)A∇∂²_nUh,∇Φh

. (4.18) Consider now (4.16) at time stepsnandn−1. Subtracting one from another and dividing byτ_n−1/2 yields

b ∂_n²U_h,Φ_h

+ (A∇∂_nU_h,∇Φ_h) =b(∂_nF,Φ_h), or

b ∂_n²U_h,Φ_h +

A∇

∂_n+1/2U_h−τ_n−1 2 ∂_n²U_h

,∇Φh

=b(∂_nF,Φ_h), so that (4.18) simplifies to

b ∂U˜_hτ

∂t ,Φ_h

! +

A∇U˜_hτ,Φ_h

= p_nA∇∂_n²U_h,∇Φh

+b

t−t_n+1/2

∂_nF+F^n+1/2,Φ_h

= p_nA∇∂_n²U_h,∇Φ_h +b

F˜_τ−p_n∂_n²F,Φ_h

, (4.19)

where

pn= τn−1

2 (t−tn+1/2) +1

2(t−tn+1)(t−tn), F˜τ(t) =F_hⁿ⁺¹+ (t−tn+1)∂n+1/2F+1

2(t−tn+1)(t−tn)∂_n²F.

Introduce the error between reconstruction ˜Uhτ and solutionU to problem (4.15):

E= ˜Uhτ−U, (4.20)

or, component-wise

E= E_u

Ev

=

˜u_hτ−u

˜ vhτ−v

.

Taking the difference between (4.19) and (4.15) we obtain the residual differential equation for the error valid fort∈[tn, tn+1],n= 1, . . . , N−1

b(∂tE,Φ) + (A∇E,∇Φ) =b ∂U˜τ h

∂t −F,Φ−Φh

! +

A∇U˜τ h,∇(Φ−Φh) + p_nA∇∂_n²U_h,∇Φh

+b

F˜_τ−F−p_n∂_n²F,Φ_h

, ∀Φh∈V_h². (4.21) Now we take Φ =E, Φh=

ΠhEu

I˜hEv

whereΠh:H₀¹(Ω)→Vhis theH₀¹-orthogonal projection operator (3.6) and ˜Ih :H₀¹(Ω)→Vh is a Cl´ement-type interpolation operator which is also a projection [10,20]. Noting that (A∇E,∇E) = 0 and

∇∂u˜hτ

∂t ,∇(Eu−ΠhEu)

= (∇˜vhτ,∇(Eu−ΠhEu)) = 0.

we get ∂Ev

∂t , E_v

+

∇Eu,∇∂Eu

∂t

= ∂v˜τ h

∂t −f, E_v−Π_hE_v

+

∇˜u_{τ h},∇

E_v−I˜_hE_v +

pn Ah∂²_nuh−∂_n²fh

,I˜hEv

− pn∇∂_n²vh,∇Eu

+

f˜τ−f,I˜hEv

. Integrating (4.21) in time fromt₃to somet^∗≥t₃yields

1

2 |Eu|²_H1(Ω)+kEvk²_L2(Ω)

(t^∗) = 1

2 |Eu|²_H1(Ω)+kEvk²_L2(Ω)

(t3)

+ Z _t∗

t₃

∂v˜τ h

∂t −f, Ev−I˜hEv

! dt

| {z }

I

+ Z _t∗

t₃

∇˜uτ h,∇(Ev−I˜hEv)

dt

| {z }

II

+ Z _t∗

t₃

h

pn Ah∂_n²uh−∂_n²fh

,I˜hEv

− pn∇∂n²vh,∇Eu

+

f˜τ−f,I˜hEv

i dt

| {z }

III

.

(4.22) Let

Z(t) =q

|Eu|²_H1(Ω)+kEvk²_L2(Ω),

and assume thatt^∗∈[t3, tN] is the point in time whereZ attains its maximum andt^∗∈(tn, tn+1] for somen.

We have for the first and second terms in (4.22)

I+II≤C1

"

X

K∈T_h

h²_K

∂˜vhτ

∂t −∆˜uhτ−f

2

L²(K)

+ X

E∈E_h

hEk[n·∇˜uhτ]k²_L2(E)

#1/2

(t^∗)|Eu|_H1(Ω)(t^∗)

+C2 n

X

m=1

τ_m−1 2

"

X

K∈Th

h²_K

∂_m²vh−∂_m−1² vh

2 L²(K)

#1/2

|Eu|_H1(Ω)(tm)

+C₃

n

X

m=0

Z min(tm+1,t^∗) tm

"

X

K∈Th

h²_K

∂²˜v_hτ

∂t² −∆∂u˜_{τ h}

∂t −∂f

∂t

2

L²(K)

+ X

E∈Eh

h_E

n·∇∂˜uτ h

∂t

2

L²(E)

#^1/2

(t)|Eu|H¹(Ω)(t)dt.

This follows from an integration by parts with respect to time and the estimates on operatorsΠhand ˜Ih,cf.

[15], and gives rise to the space part of the error estimate (4.10). Indeed, we can summarize the bounds above as

I+II ≤η_s(t_N)Z(t^∗).

The third term in (4.22) is responsible for the time estimator. It can be written as III=

n−1

X

m=3

Z min(t_m+1,t^∗) tm

"

pm ∂_m⁴uh+τm m

X

k=m−2

γk

∂_k²f˙h−Ah∂_k²vh

! ,I˜hEv

!

−(pm∇∂²_mvh,∇Eu) + ( ˜fτ−f,I˜hEv)

#

dt. (4.23)

Recalling thatZ(t^∗) is the maximum ofZ(t) and using the estimatekI˜_hE_vkL²(Ω)≤CkEvkL²(Ω)we continue as

III≤Z(t^∗)

n−1

X

m=3

Z min(tm+1,t^∗) tm

|pm|dt

× Ck∂_m⁴u_hk_L2(Ω)+|∇∂_m²v_h|_H1(Ω)+Cτ_m

m

X

k=m−2

γ_kk∂²_kf˙_h−A_h∂_k²v_hk_L2(Ω)

!

+Z(t^∗) Z t_n

t3

kf−f˜τk_L2(Ω)dt.

Noting

Z t_m+1 t_m

|pm|dt≤ 1

12τ_m³ +1

8τm−1τ_m², we can finally boundIII as

III ≤ C

N−1

X

k=3

τ_kηˆ_T(t_k) +C

N−1

X

k=1

τ_kηˆ_T^h.o.t(t_k) + Z tN

t3

kf−f˜_τk_L2(Ω)dt

! Z(t^∗).

Summing together the estimates on the terms I, II, III, and recalling Z(t^∗)≥Z(tN) yields (4.10) at the

final timet_N.

Remark 4.5. The terms ˆη_T^h.o.t(tk) in (4.10) are (at least formally) of higher order than ˆηT(tk). We propose therefore to ignore ˆη^h.o.t_T (tk) in practice together with the integral off−f˜τ, and to use ˆηT(tk) as the indicator of error due to the discretization in time. The following Theorem shows that the latter is indeed of optimal order τ², at least for sufficiently smooth solutions, on quasi-uniform meshes in space and uniform meshes in time.

Theorem 4.6. Let u be the solution of wave equation (2.1) and ∂³u

∂t³(0)∈H¹(Ω), ∂²u

∂t²(0)∈H²(Ω), ∂²f

∂t²(t)∈ L^∞(0, T;L²(Ω)), ∂³f

∂t³(t)∈L²(0, T;L²(Ω)). Suppose that meshTh is quasi-uniform, the mesh in time is uniform (t_k=kτ), and the initial approximations are chosen as in (3.7). Then, the 5-point time error estimator ηˆ_T(t_k) defined by (4.11) is of orderτ², i.e.

ˆ

ηT(tk)≤Cτ².

with a positive constantC depending only onu,f, and the mesh regularity.

Proof. The result follows from Theorem3.3by using (4.14) and Lemma4.2 ∂_k⁴uh

_L2(Ω)=

∂ˆ_k²f¯h−Ah∂ˆ_k²u¯h

L²(Ω)

=

n

X

k=n−2

αk ∂_k²fh−Ah∂_k²uh L²(Ω)

.

Remark 4.7. Note, that as in the case for 3-point error estimator, the approximation of initial conditions is crucial for the optimal rate of our time error estimator.