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

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 posteriori 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 discrete scheme. The derivation of these bounds relies crucially on carefully constructed 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 problems mesh-adaptivity plays an important role, in that it drives variable resolution 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 problems (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 finite 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 timestepping finite element approximations to the linear wave equation have been proposed and analyzed (but only in very specific situations) by Adjerid [Adj02]. Also, Bernardi and S¨ uli [BS05] derive rigorous a posteriori L∞ (H 1 )-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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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 setting [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-thanenergy 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. Our approach is based on the one-field formulation of the wave equation and, thus, non-trivial threepoint 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. 2. Preliminaries 2.1. Model problem and notation. We denote by Lp (ω), 1 ≤ p ≤ +∞, ω ⊂ Rd , the Lebesgue spaces, with corresponding norms k · kLp (ω) . The norm of L2 (ω), denoted by k·kω , corresponds to the L2 (ω)-inner product h·, ·iω . We denote by H s (ω), the Hilbertian Sobolev space of order s ≥ 0 of real-valued functions defined on ω ⊂ Rd ; in particular H01 (ω) signifies the space of functions in H 1 (ω) that vanish 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 h·, ·iω is extended to the standard duality pairing between H −1 (ω) and H01 (ω). For 1 ≤ p ≤ +∞, we also define the spaces Lp (0, T, X), with X being a real Banach space with norm k·kX , consisting of all measurable functions v : (0, T ) → X, for which Z T 1/p < +∞, for 1 ≤ p < +∞, kvkLp(0,T ;X) := kv(t)kpX dt (2.1) 0 kvkL∞ (0,T ;X) := ess sup0≤t≤T kv(t)kX < +∞, for p = +∞.

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

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Let Ω ⊂ Rd be a bounded open polygonal domain with Lipschitz boundary ∂Ω. For brevity, the standard inner product on L2 (Ω) will be denoted by h·, ·i and the corresponding norm by k·k. For time t ∈ (0, T ], we consider the linear second order hyperbolic initialboundary value problem of finding u ∈ L2 (0, T ; H01(Ω)), with ut ∈ L2 (0, T ; L2 (Ω)) and utt ∈ L2 (0, T ; H −1 (Ω)) such that (2.2)

utt − ∇ · (a∇u) = f

in (0, T ) × Ω,

¯ with 0 < where f ∈ L2 (0, T ; L2 (Ω)) and a is a scalar-value function in ∈ C(Ω), αmin ≤ a ≤ αmax , such that (2.3)

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 ∈ H01 (Ω) and u1 ∈ L2 (Ω). We identify a function v ∈ Ω × [0, T ] → R with the function v : [0, T ] → H01 (Ω) 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 disjoint open simplicial or quadrilateral elements. Each element κ ∈ T is constructed via mappings Fκ : κ ˆ → κ, where κ ˆ is the reference simplex or reference square, so ¯ = ∪κ∈T κ that Ω ¯ [Cia78]. For a nonnegative integer p, we denote by Pp (ˆ κ) either the set of all polynomials on κ ˆ of degree p or less, when κ ˆ is the simplex, or the set of polynomials of at most degree p in each variable, when κ ˆ is the reference square (or cube). We consider p fixed and use the finite element space (2.4)

Vh := {v ∈ H01 (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) = diam κ, 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)

hUtt , V i + a(U, V ) = hf, V i ∀V ∈ L2 (0, T ; Vh ),

where the bilinear form a is defined for each z, v ∈ H01 (Ω) by Z a∇z · ∇v dx, (2.6) a(z, v) = Ω

and the corresponding energy norm is defined for v ∈ H01 (Ω) by √ (2.7) kvka = k a∇vk. 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 T n , n = 0, . . . , N , be a sequence of meshes which are assumed to be compatible, in the sense that for any two consecutive meshes T n−1 and T n , T n can be obtained from T n−1 by locally coarsening some

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of its elements and then locally refining some (possibly other) elements. The finite element space corresponding to T n will be denoted by Vhn . We consider the fully discrete scheme for the wave problem (2.2), (2.3) for each n = 1, . . . , N , find U n ∈ Vhn such that

(2.8)

h∂ 2 U n , V i + a(U n , V ) = hf n , V i

∀V ∈ Vhn ,

where f n := f (tn , ·), the backward second and first finite differences ∂ 2 U n :=

(2.9)

∂U n − ∂U n−1 , kn

with (2.10)

 n n−1 U − U , n ∂U := k  0 n 0 V := π u1

for n = 1, 2, . . . , N, for n = 0,

where U 0 := π 0 u0 , and π 0 : L2 (Ω) → Vh0 a suitable projection onto the finite element space (e.g., the orthogonal L2 -projection operator). 3. A posteriori error bounds for the semi-discrete problem We derive here a posteriori error bound for the error ku − U kL∞ (0,T ;L2 (Ω)) between 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 (semidiscrete) finite element solution to the problem (2.5). Let also Π : L2 (Ω) → Vh be the orthogonal L2 -projection operator onto the finite element space Vh . We define the elliptic reconstruction w = w(t) ∈ H01 (Ω), t ∈ [0, T ], of U to be the solution of the elliptic problem (3.1)

a(w, v) = hg, vi ∀v ∈ H01 (Ω)

where (3.2)

g := AU − Πf + f,

and A : Vh → Vh is the discrete elliptic operator defined by (3.3)

for q ∈ Vh ,

hAq, χi = a(q, χ) ∀χ ∈ Vh .

We decompose the error as follows: (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)

hett , vi + a(ρ, v) = 0

∀v ∈ H01 (Ω).

Proof. We have, respectively, (3.6)

hett , vi + a(ρ, v) = hUtt , vi + a(w, v) − hf, vi

= hUtt , Πvi + a(w, v) − hf, vi

= −a(U, Πv) + a(w, v) + hΠf − f, vi = 0, observing the identity a(U, Πv) − hΠf − f, vi = a(w, v), due to the construction of w. 

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

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3.3. Theorem (abstract semidiscrete error bound). With the notation introduced in (3.4), the following error bound holds:  √  kekL∞(0,T ;L2 (Ω)) ≤kǫkL∞ (0,T ;L2 (Ω)) + 2 ku0 − U (0)k + kǫ(0)k Z T (3.7) kǫt k + Ca,T ku1 − Ut (0)k, +2 0

p where Ca,T := min{2T, 2CΩ /αmin }, where CΩ is the constant of the Poincar´e– Friedrichs inequality kvk2 ≤ CΩ k∇vk2 , for v ∈ H01 (Ω). Proof. We use a testing procedure due to Baker [Bak76]. Let v˜ : [0, T ] × Ω → R with Z τ (3.8) v˜(t, ·) = ρ(s, ·)ds, t ∈ [0, T ], t

from some fixed τ ∈ [0, T ]. Clearly v˜ ∈ H01 (Ω) as ρ ∈ H01 (Ω). Also, we observe that: (3.9)

v˜(τ, ·) = 0,

∇˜ v (τ, ·) = 0,

and v˜t (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 Z τ Z τ (3.10) − het , v˜t i + het (τ ), v˜(τ )i − het (0), v˜(0)i + a(ρ, v˜) = 0. 0

0

Using (3.9), we have Z τ Z τ Z τ 1 d 1 d (3.11) kρk2 − a(˜ v , v˜) = hǫt , ρi + het (0), v˜(0)i, 0 2 dt 0 2 dt 0 which implies Z τ 1 1 1 2 2 (3.12) kρ(τ )k − kρ(0)k + a(˜ v (0), v˜(0)) = hǫt , ρi + het (0), v˜(0)i. 2 2 2 0 Hence, we deduce (3.13) Z τ 1 1 1 kρ(τ )k2 − kρ(0)k2 + a(˜ v (0), v˜(0)) ≤ max kρ(t)k kǫt k + ket (0)kk˜ v(0)k. 0≤t≤T 2 2 2 0 Now, we select τ = τˆ such that kρ(ˆ τ )k = max0≤t≤T kρ(t)k, and we present two alternative, but complementary, ways to complete the proof. In the first way, we start by observing that k˜ v (0)k ≤ τ kρ(ˆ τ )k, gives Z τ  2 1 1 kρ(τ )k2 − kρ(0)k2 ≤ k∂t ǫk + τ ket (0)k . (3.14) 4 2 0 Using the bound kρ(0)k ≤ ke(0)k + kǫ(0)k, e(0) = U (0)− u0 and et (0) = Ut (0)− u1 , we conclude that  √  kekL∞ (0,T ;L2 (Ω)) ≤kǫkL∞ (0,T ;L2 (Ω)) + 2 ku0 − U (0)k + kǫ(0)k (3.15) Z T  +2 kǫt k + T ku1 − ∂U (0)k . 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 k˜ v(0)k2 ≤ CΩ α−1 v(0)k2a , for min k˜

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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 Z τ 1 1 1 2 kρ(τ )k2 − kρ(0)k2 ≤ max kρ(t)k kǫt k + CΩ α−1 (3.16) min ket (0)k , 0≤t≤T 2 2 2 0 which implies  √  kekL∞ (0,T ;L2 (Ω)) ≤kǫkL∞ (0,T ;L2 (Ω)) + 2 ku0 − U (0)k + kǫ(0)k Z T (3.17) p +2 kǫt k + 2CΩ /αmin ku1 − Ut (0)k. 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 arguments 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 ∈ Vh in (3.1), we have (3.18)

a(w, V ) = hAU − Πf + f, V i = 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 w(x + δne ) − w(x − δne ) · ne , δ→0

for x ∈ e, where ne denotes 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 ∈ H01 (Ω) be the solution to the elliptic problem: (3.21)

− ∇ · (a∇z) = r

r ∈ L2 (Ω) and Ω convex, and let Z ∈ Vh be the finite element approximation of z satisfying (3.22)

a(Z, V ) = hr, V i

∀V ∈ Vh .

Then, there exists a positive constant Cel , independent of T , h, z and Z, so that kz − Zk2 ≤ Cel E(Z, r, T ),

(3.23) where

(3.24)

E(Z, r, T ) :=

X 1/2 X . kh2 (r + ∇ · (a∇Z)k2κ + kh3/2 [ a∇Z]]k2e κ∈T

e⊂Γ

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

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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: Z T kekL∞ (0,T ;L2 (Ω)) ≤Cel kE(U, g, T )kL∞ (0,T ) + 2Cel E(Ut , gt , T ) 0 √ (3.25) + 2Cel E(U (0), g(0), T ) √ + 2ku0 − U (0)k + Ca,T ku1 − Ut (0)k. Proof. Using (3.18), kǫk and kǫt k 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 problems [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 U n , n = 0, . . . , N , be the fully discrete solution computed by the method (2.8), Πn : L2 (Ω) → Vhn be the orthogonal L2 -projection, and An : Vhn → Vhn to be the discrete operator defined by for q ∈ Vhn ,

(4.1)

hAn q, χi = a(q, χ) ∀χ ∈ Vhn .

We define the elliptic reconstruction wn ∈ H01 (Ω), of U n to be the solution of the elliptic problem a(wn , v) = hg n , vi ∀v ∈ H01 (Ω),

(4.2) with

g n := An U n − Πn f n + f¯n , R tn where f¯0 (·) := f (0, ·) and f¯n (·) := kn−1 tn−1 f (t, ·)dt for n = 1, . . . , N . Finally, we need to define the elliptic reconstruction ∂w0 ∈ H01 (Ω), of V 0 to be the solution of the elliptic problem

(4.3)

a(∂w0 , v) = h∂g 0 , vi

(4.4) with

∀v ∈ H01 (Ω),

∂g 0 := A0 V 0 − Π0 f 0 + f 0 .

(4.5)

The time-reconstruction U : [0, T ] × Ω → R of {U n }N n=0 , is defined by (4.6)

U (t) :=

t − tn−1 n tn − t n−1 (t − tn−1 )(tn − t)2 2 n U + U − ∂ U , kn kn kn

ˆ for t ∈ (tn−1 , tn ], n = 1, . . . , N , noting that ∂U 0 is well defined. We note that U 1 n n n n ˆ (t ) = U and U ˆt (t ) = ∂U for , is a C -function in the time variable, with U n = 0, 1, . . . , N . We shall also use the time-continuous elliptic reconstruction w, defined by (4.7)

w(t) :=

t − tn−1 n tn − t n−1 (t − tn−1 )(tn − t)2 2 n w + w − ∂ w , kn kn kn

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

noting that ∂w0 is well defined. By construction, this is also a C 1 -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) hett , vi+a(ρ, v) = h(I−Πn )Utt , vi+µn h∂ 2 U n , Πn vi+a(w−wn , v)+hf¯n −f, vi,

for all v ∈ H01 (Ω), with Πn : L2 (Ω) → Vhn denoting the orthogonal L2 -projection operator onto Vhn , I is the identity mapping in L2 (Ω), and 1

µn (t) := −6kn−1 (t − tn− 2 ),

(4.10) 1

where tn− 2 := 12 (tn + tn−1 ).

Proof. Noting that Utt (t) = (1 + µn (t))∂ 2 U n , for t ∈ (tn−1 , tn ], n = 1, . . . , N , and the identity a(U n , Πn v) − hΠn f n − f¯n , vi = a(wn , v), we deduce hett , vi + a(ρ, v) = hUtt , vi + a(w, v) − hf, vi,

= h(I − Πn )Utt , vi + hUtt , Πn vi + a(w, v) − hf, vi,

(4.11)

= h(I − Πn )Utt , vi + µn (t)h∂ 2 U n , Πn vi

− a(U n , Πn v) + a(w, v) + hΠn f n − f, vi

= h(I − Πn )Utt , vi + µn (t)h∂ 2 U n , Πn vi + a(w − wn , v) + hf¯n − f, vi.  4.4. Remark (vanishing moment property). The particular form of the remainder µn (t) satisfies the vanishing moment property Z tn µn (t) dt = 0, (4.12) tn−1

which appears to be of crucial importance for the optimality of the a posteriori bounds presented below. 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 Z τ m−1 X Z tj j k(I − Πm )Ut k, k(I − Π )Ut k + (4.13) η1,1 (τ ) := tj−1

j=1

tm−1

and (4.14)

η1,2 (τ ) :=

m−1 X j=1

(τ − tj )k(Πj+1 − Πj )∂U j k + τ k(I − Π0 )V 0 (0)k,

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

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evolution error indicator: (4.15)

η2 (τ ) :=

Z

0

τ

kGk,

where G : (0, T ] → R with G|(tj−1 ,tj ] := G j , j = 1, . . . , N and (4.16)

G j (t) :=

(tj − t)3  2 j (tj − t)2 j  (tj − t)4 − ∂g − ∂ g − γj , 2 4kj 3

with g j as in Definition 4.1 and γj := γj−1 + with γ0 = 0; data error indicator:

(4.17)

kj2 j 2 ∂g

k3

+ 12j ∂ 2 g j , j = 1, . . . , N ,

m−1 Z j 1/2  Z τ 1/2 1 X t 3 ¯m + kj3 kf¯j − f k2 η3 (τ ) := ; km kf − f k2 2π j=1 tj−1 tm−1

time reconstruction error indicator:

(4.18)

η4 (τ ) :=

m−1 Z j 1/2  Z τ 1/2 1 X t 3 kj3 kµj ∂ 2 U j k2 + km kµm ∂ 2 U m k2 . 2π j=1 tj−1 tm−1

4.6. Theorem (abstract fully-discrete error bound). Recalling the notation of Definition 4.1 and the indicators of Definition 4.5 we have the bound

(4.19)

 √  kekL∞ (0,tN ;L2 (Ω)) ≤kǫkL∞ (0,tN ;L2 (Ω)) + 2 ku0 − U (0)k + kǫ(0)k 4   Z tN X ηi (tN ) + Ca,N ku1 − V 0 k, +2 kǫt k + 0

where Ca,N := min{2tN , stant.

i=1

p 2CΩ /αmin }, CΩ is Poincar´e–Friedrichs inequality con-

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 ≤ tm for 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)

Z

0

τ

hett , v˜i +

Z

0

τ

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

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

where I1 (τ ) :=

m−1 X Z tj

h(I − Πj )Utt , v˜i +

I2 (τ ) :=

m−1 X Z tj

a(w − wj , v˜) +

I3 (τ ) :=

m−1 X Z tj

hf¯j − f, v˜i +

I4 (τ ) :=

m−1 X Z tj

µj h∂ 2 U j , Πj v˜i +

(4.21)

tj−1

j=1

tj−1

j=1

tj−1

j=1

tj−1

j=1

Z

Z

τ

Z

tm−1

τ tm−1

τ

h(I − Πm )Utt , v˜i,

a(w − wm , v˜)

hf¯m − f, v˜i,

tm−1

Z

τ

tm−1

µm h∂ 2 U m , Πm v˜i.

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 kρ(t)k, 0≤t≤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 kρ(t)k. 0≤t≤T

Proof. Observing that the projections Πj , j = 1, . . . , N , commute with timedifferentiation, we integrate by parts with respect to t, arriving to Z τ m−1 X Z tj h(I − Πm )Ut , ρi h(I − Πj )Ut , ρi + I1 (τ ) = j=1

(4.24)

+

tm−1

tj−1

m−1 X j=1

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

The first two terms on the right-hand side of (4.24) are bounded by  m−1  Z τ X Z tj (4.25) max kρ(t)k k(I − Πm )Ut k . k(I − Πj )Ut k + 0≤t≤T

j=1

tm−1

tj−1

Recalling the definition of v˜ and that U (tj ) = ∂U j , j = 0, 1, . . . , N , we can bound the last two terms on the right-hand side of (4.24) by   m−1 X j j+1 j j 0 0 (τ − t )k(Π − Π )∂U k + τ k(I − Π )V (0)k . (4.26) max kρ(t)k 0≤t≤T

j=1

 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 kρ(t)k. 0≤t≤T

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

11

Proof. First, we observe the identity (4.28)

  w − wj = −(tj − t)∂wj + kj−1 (tj − t)3 − (tj − t)2 ∂ 2 wj ,

on each (tj−1 , tj ], j = 2, . . . , m. Hence, from Definition 4.1, we deduce   (4.29) a(w − wj , v˜) = h−(tj − t)∂g j + kj−1 (tj − t)3 − (tj − t)2 ∂ 2 g j , v˜i

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 Z τ (4.30) I2 (τ ) = hG, ρi, 0

which already implies the result.



4.9. Lemma (data approximation error bound). Under the assumptions of Theorem 4.6 and with the notation (4.21) we have (4.31)

I3 (τ ) ≤ η3 (τ ) max kρ(t)k. 0≤t≤T

Proof. We begin by observing that Z tj (f¯j − f ) = 0, (4.32) tj−1

for all j = 1, . . . , m − 1. Hence, we have m−1 m−1 XZ X Z tj j ¯ hf − f, v˜i = (4.33) j=1

tj−1

tj

tj−1

j=1

hf¯j − f, v˜ − v¯˜j i,

R j −1 t

where v¯ ˜j (·) := kj

v˜(t, ·)dt. Using the inequality Z tj Z tj kj2 (4.34) k˜ v − v¯˜j k2 ≤ k˜ vt k2 , 4π 2 tj−1 tj−1 and recalling that v˜t = ρ, we have, respectively, m−1 m−1 1/2  Z X Z tj X  Z tj j 2 j ¯ ¯ kf − f k hf − f, v˜i ≤ j=1

tj−1

tj−1

(4.35)

tj−1

j=1

1 ≤ 2π 1 2π

m−1 X Z

≤

j=1

k˜ v − v¯˜j k2

tj−1

m−1 X Z j=1

tj

j

t

tj−1

kf¯j − f k2

1/2  Z

j

t

tj−1

kj3 kf¯j − f k2

tj

tj−1

1/2

kj2 kρk2

tm−1

0≤s≤T

t

which implies (4.37)

Z

τ

tm−1

hf¯m − f, v˜i ≤

Z

τ

tm−1

3 ¯m km kf − f k2

1/2

1/2

max kρ(t)k.

0≤t≤T

For the remaining term in I3 , we first observe that Z τ Z τ Z τ 3 (4.36) k˜ v k2 dt ≤ km kρk2 dsdt ≤ km max kρ(t)k2 , tm−1

1/2

max kρ(t)k.

0≤t≤T

12

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

Recalling η3 from 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 g n in the elliptic reconstruction, results to the property (4.32). When f is differentiable, we have η3 (τ ) = O(k 2 ) as k := max1≤j≤m kj → 0, and the convergence is of second order with respect to the maximum time-step. In this case, η3 is, therefore, a higher order term. 4.11. Lemma (time-reconstruction error bound). Under the assumptions of Theorem 4.6 and with the notation (4.21) we have (4.38)

I4 (τ ) ≤ η4 (τ ) max kρ(t)k. 0≤t≤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 ∂ 2 U j is piecewise constant in time, we have (4.39)

m−1 X Z tj j=1

tj−1

µj h∂ 2 U j , Πj v˜i =

m−1 X Z tj j=1

tj−1

µj h∂ 2 U j , Πj (˜ v − v¯˜j )i,

R tj where v¯ ˜j (·) = kj−1 tj−1 v˜(t, ·)dt. Hence, since Πj commutes with time integration, we obtain (4.40) m−1 Z j m−1 1/2  Z tj 1/2 X Z tj 1 X t kµj ∂ 2 U j k2 kj2 kΠj ρk2 µj h∂ 2 U j , Πj (˜ v − v¯ ˜j )i ≤ 2π j=1 j−1 tj−1 tj−1 j=1 t ≤

m−1 Z j 1/2 1 X t kj3 kµj ∂ 2 U j k2 max kρ(t)k. 0≤t≤T 2π j=1 tj−1

For the remaining term in I4 , upon using an argument similar to (4.36), we have Z τ Z τ 1/2 m 2 m m 3 (4.41) hµ ∂ U , Π v˜i ≤ max kρ(t)k. km kµm ∂ 2 U m k2 tm−1

0≤t≤T

tm−1

Recalling the definition of η4 in §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 Z τ Z τ Z τ 4 X 1 d 1 d Ii (τ ), kρk2 − a(˜ v , v˜) = hǫt , ρi + het (0), v˜(0)i + (4.42) 0 2 dt 0 0 2 dt i=1 which implies 1 1 1 (4.43) kρ(τ )k2 − kρ(0)k2 + a(˜ v (0), v˜(0)) = 2 2 2

Z

0

τ

hǫt , ρi+het (0), v˜(0)i+

4 X i=1

Ii (τ ).

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

13

Hence, we deduce (4.44)

1 1 1 kρ(τ )k2 − kρ(0)k2 + a(˜ v (0), v˜(0)) 2 2 2 4  Z τ X ηi (τ ) + ket (0)kk˜ v (0)k. ≤ max kρ(t)k kǫt k + 0≤t≤T

0

i=1

We select τ = τˆ such that kρ(ˆ τ )k = max0≤t≤tN kρ(t)k. First, observing that k˜ v(0)k ≤ τ kρ(ˆ τ )k, gives 4 2 Z τ X 1 1 2 2 ηi (τ ) + τ ket (0)k . kρ(τ )k − kρ(0)k ≤ (4.45) kǫt k + 4 2 0 i=1 ˆ (0) − u(0) = Using the bound kρ(0)k ≤ ke(0)k + kǫ(0)k and observing that e(0) = U 0 0 ˆ U − u0 and that et (0) = Ut (0) − ut (0) = V − u1 , we arrive to  √  kekL∞(0,tN ;L2 (Ω)) ≤kǫkL∞ (0,tN ;L2 (Ω)) + 2 ku0 − U 0 k + kǫ(0)k 4   Z tN (4.46) X ηi (tN ) + tN ku1 − V 0 k . +2 kǫt k + 0

i=1

The second way is completely analogous to the proof of the semidiscrete case. 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 √ √ (5.1) kǫkL∞ (0,tN ;L2 (Ω)) + 2kǫ(0)k ≤ δ1 (tN ) + 2Cel E 0 , where (5.2)

n 8k 1 δ1 (tN ) := max Cel E(V 0 , ∂g 0 , T 0 ), 27  35 31
o kj  ¯j − f j k , max Cel E j + CΩ α−1 k f + max min 27 27 1≤j≤N kj−1 0≤j≤N

with E j := E(U j , Aj U j − Πj f j + f j , T j ), j = 0, 1, . . . , N .

Proof. For t ∈ (tj−1 , tj ], j = 1, . . . , N , we have (5.3) tj − t j−1 (t − tj−1 )(tj − t)2 2 j t − tj−1 j (w − U j ) + (w − U j−1 ) − (∂ w − ∂ 2 U j ), ǫ= kj kj kj from which, we can deduce n 35 31 o 8k1 kj  max kwj − U j k, (5.4) kǫk ≤ max + max k∂w0 − V 0 k , 27 27 1≤j≤N kj−1 0≤j≤N 27 noting that (5.5)

4kj2 (t − tj−1 )(tj − t)2 = . kj 27 ,tj ]

max j−1

t∈(t

14

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

It remains to estimate the terms kwj − U j k and k∂w0 − V 0 k. To this end, recalling the notation of Definition 4.1, we define w∗j ∈ H01 (Ω) to be the solution of the elliptic problem (5.6)

a(w∗j , v) = hAj U j − Πj f j + f j , vi ∀v ∈ H01 (Ω),

for j = 0, 1, . . . , N . Note that, due to the fact that f¯0 = f 0 , we have w∗0 = w0 . By construction, we have a(w∗j , V ) = hAj U j − Πj f j + f j , V i = a(U j , V ) for all V ∈ Vhj , j = 0, 1, . . . , N . Hence, U j is the finite element solution (in Vhj ) of the elliptic boundary-value problem (5.6). In view of Theorem 3.7, this implies that kw∗j − U j k ≤ Cel E j ,

(5.7)

for j = 0, . . . , N . Similarly, by construction, we have a(∂w0 , V ) = hA0 V 0 − Π0 f 0 + f 0 , V i = a(V 0 , V ) for all V ∈ Vh0 . Hence, (5.8)

k∂w0 − ∂U 0 k ≤ Cel E(V 0 , ∂g 0 , T 0 ).

Moreover, since wj − w∗j is the solution of an elliptic problem with right hand-side f¯j − f j , standard elliptic stability results yield j ¯j kwj − w∗j k ≤ CΩ α−1 min kf − f k,

(5.9)

for j = 1, . . . , N . Finally, using the triangle inequality (5.10)

kwj − U j k ≤ kwj − w∗j k + kw∗j − U j k,

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 kf¯j − f j k 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 f j = f¯j (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 by Tˆ n the finest common coarsening of T n and T n−1 , and by Vˆhn := Vhn ∩ Vhn−1 , the corresponding finite element space, along ˆ n : L2 (Ω) → Vˆ n . with the orthogonal L2 -projection operator Π h 5.3. Lemma (estimation of the time derivative of the elliptic error). With the notation introduced in §4.1 we have Z tN kǫt k ≤ δ2 (tN ), (5.11) 0

where (5.12)

δ2 (tN ) :=

N   2X j ¯j k , (2kj + kj+1 ) Cel E∂j + CΩ α−1 k∂f − ∂ f min 3 j=0

with (5.13)

E∂j := E(∂U j , ∂(Aj U j ) − ∂(Πj f j ) + ∂f j , Tˆ j ),

j = 0, 1, . . . , N.

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

15

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

ǫt = ∂wj − ∂U j − kj−1 (tj − t)(tj − 2tj−1 + t)(∂ 2 wj − ∂ 2 U j ),

(5.14)

from which, we deduce Z tj 2kj 4kj k∂wj − ∂U j k + k∂wj−1 − ∂U j−1 k, (5.15) kǫt k ≤ 3 3 j−1 t noting that Z tj 2kj (5.16) kj−2 (tj − t)(tj − 2tj−1 + t) = . 3 tj−1 Combining (5.15) for j = 1, . . . , N , we arrive to Z tN N 2X (5.17) kǫt k ≤ (2kj + kj+1 )k∂wj − ∂U j k, 3 j=0 0

with k0 = 0 and kN +1 = 0. It remains to estimate the terms k∂wj − ∂U j k. To this end, recalling the definition of the functions w∗j ∈ H01 (Ω) from the proof of Lemma 5.1 and, since Vˆhj := Vhj ∩ Vhj−1 , we have a(w∗j , V ) = a(U j , V ) for all V ∈ Vˆhj and a(w∗j−1 , V ) = a(U j−1 , V ) for all V ∈ Vˆhj , for j = 1, . . . , N . Therefore, we deduce a(∂w∗j , V ) = a(∂U j , V ) for all V ∈ Vˆhj ,

(5.18)

for j = 1, . . . , N , i.e., ∂U j is the finite element solution in Vˆhj of the boundary-value problem a(∂w∗j , V ) = h∂(Aj U j ) − ∂(Πj f j ) + ∂f j , vi ∀v ∈ H01 (Ω).

(5.19)

In view of Theorem 3.7, this implies that

k∂w∗j − ∂U j k ≤ Cel E∂j ,

(5.20)

for j = 1, . . . , N . We also recall that, by construction, we have a(∂w0 , V ) = a(V 0 , V ) for all V ∈ Vh0 . Hence, (5.8) also holds. Moreover, since (5.21) a(∂wj , V ) = h∂(Aj U j ) − ∂(Πj f j ) + ∂ f¯j , vi ∀v ∈ H 1 (Ω), 0

j = 1, . . . , N , (cf. Definition 4.1). As in (5.9), elliptic stability implies (5.22) k∂wj − ∂wj k ≤ CΩ α−1 k∂ f¯j − ∂f j k, ∗

min

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

k∂wj − ∂U j k ≤ k∂wj − ∂w∗j k + k∂w∗j − ∂U j k,

(5.23)

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



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 √ √ kekL∞ (0,tN ;L2 (Ω)) ≤δ1 (tN ) + 2Cel E 0 + 2ku0 − U (0)k (5.24)

+ 2δ2 (tN ) + 2

4 X i=1

0

ηi (tN ) + Ca,N ku1 − V 0 k,

where δ1 , E are defined in Lemma 5.1, δ2 is defined in Lemma 5.3, and ηi , i = 1, 2, 3, 4 after (41) respectively.

16

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

Proof. Combining Theorem 4.6 with the bounds derived for ǫ in Lemma 5.1, and ǫt in Lemma 5.3, we arrive to an a posteriori error bound.  6. Final remarks The design and implementation of adaptive algorithms for the wave equation based or rigorous a posteriori error estimators is a largely unexplored subject, despite the importance of these problems in the modeling of a number of physical phenomena. To this end, this work presents rigorous a posteriori error bounds in the L∞ (L2 )-norm for second order linear hyperbolic initial/boundary value problems. The derived bounds are formally of optimal order. The numerical implementation of the proposed bounds in the context of adaptive algorithm design for second order hyperbolic problems remains a challenge that deserves special attention and will be considered elsewhere. References [Adj02]

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Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom. E-mail address: [email protected] Department of Mathematics, University of Sussex, Falmer near Brighton, GB-BN1 9RF, England. http://www.maths.sussex.ac.uk/Staff/OL E-mail address: [email protected] Department of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece; and Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Vasilika Vouton P.O. Box 1527, GR-71110 Heraklion, Greece. E-mail address: [email protected]