A Posteriori Error Analysis of Crank-Nicolson Finite Element Method for Parabolic Integro-Differential Equations

A POSTERIORI ERROR ANALYSIS OF CRANK-NICOLSON FINITE ELEMENT METHOD FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

G MURALI MOHAN REDDY∗ AND RAJEN K. SINHA†

Abstract. We study a posteriori error analysis for the space-time finite element discretizations of linear parabolic integro-differential equations in a bounded convex polygonal or polyhedral domain. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the Crank-Nicolson method. A novel Ritz-Volterra reconstruction operator [2013, IMA J. Numer. Anal., doi:10.1093/imanum/drt059, pp. 1-31], a generalization of elliptic reconstruction operator [2003, SIAM J. Numer. Anal., 41, pp. 1585-1594], is used in a crucial way to obtain optimal rate of convergence in space. In addition, a quadratic (in time) space- time reconstruction operator is introduced to establish second order convergence in time. We use energy method to derive optimal order error estimator in the L∞(L2)-norm. Numerical experiment is performed to validate the optimality of the derived error estimators.

Key words. Parabolic integro-diﬀerential equations, ﬁnite element method, Crank-Nicolson method, a posteriori error estimate

AMS subject classiﬁcations. 65M15, 65M60

1. Introduction. The main objective of this article is to study a posteriori error analysis by means of Crank-Nicolson ﬁnite element method for the linear parabolic integro-diﬀerential equations (PIDE) of the form

t (1.1) u (x, t)+ u(x, t)= (t,s)u(x, s)ds + f(x, t), (x, t) Ω (0,T ], t A B ∈ × Z0 u(x, t)=0, (x, t) ∂Ω (0,T ], ∈ × u(x, 0) = u (x), x Ω. 0 ∈ Here, Ω Rd, d 1 is a bounded convex polygonal or polyhedral domain with bound- ⊂ ≥ ∂u ary ∂Ω and ut(x, t) = ∂t (x, t) with T < . Further, is a self-adjoint, uniformly positive deﬁnite second-order linear elliptic∞ partial diﬀerentialA operator of the form

u = (A u) A −∇ · ∇ and the operator (t,s) is of the form B (t,s)u = (B(t,s) u), B −∇ · ∇ where “ ” denotes the spatial gradient and and A = a (x) and B(t,s)= b (x; t,s) ∇ { ij } { ij } are two d d matrices assumed to be in L∞(Ω)d×d in space variable. Moreover, the el- × ements of B(t,s) are assumed to be smooth in both t and s. The initial function u0(x) and the nonhomogeneous term f(x, t) are assumed to be smooth for our purpose. Such problems and variants of them arise in various applications, such as heat conduction in material with memory [8], the compression of poro-viscoelasticity media [9], nuclear reactor dynamics [13, 14] and the epidemic phenomena in biology [6]. For

∗Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India ([email protected]). †Corresponding author: Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India ([email protected]). 1 2 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

the existence, uniqueness and regularity results of such problems, one may refer to [18] and references therein. In the recent years, a posteriori error analysis for the finite element methods for partial differential equations has been thoroughly studied by several researchers [1, 2, 3, 4, 7, 10, 11, 12, 17]. While much of interest has focussed on elliptic and parabolic problems, relatively less progress has been made in the direction of a posteriori error analysis of PIDE [15]. In order to put the results of the paper into proper prospective, we give a brief account of the relevant literature and motivation for the present investigation. In the absence of the memory term i.e., when (t,s) = 0, a posteriori error analysis for linear parabolic problems have been investigatedB by several authors in [2, 3, 4, 7, 10, 11, 12, 17]. In particular, for the fully discrete Crank-Nicolson method for the heat equation, a continuous, piecewise linear approximation in time is used to obtain suboptimal (with respect to time step) a posteriori error bounds using standard energy techniques in [17]. A continuous, piecewise quadratic polynomial so-called Crank-Nicolson reconstruction is then introduced in [2] to restore the second order of convergence for semidiscrete time discretization of a general parabolic problem. Subsequently, the authors of [11] have introduced the reconstruction based on approximations on one time level (two-point reconstruction) as in [2] as well as the reconstructions based on approximations on two time levels (three-point reconstruction) in the L2(H1)-norm. Recently in [3], for parabolic problems, the elliptic reconstruction in combination with energy techniques are used to derive optimal order a posteriori error estimate for Crank-Nicolson method in the L∞(L2)-norm. Since PIDE (1.1) may be thought of as a perturbation of parabolic equation, an attempt has been made to carry over a posteriori error analysis of parabolic problems [3] to PIDE (1.1). We wish to remark that such an extension is not straightforward due to the presence of the Volterra integral term in (1.1). To the best of our knowledge no article is available in the literature concerning a posteriori error analysis of Crank- Nicolson method for PIDE in L∞(L2)-norm. We derive a posteriori bounds for PIDE in the L∞(L2)-norm of the error for the fully discrete Crank-Nicolson scheme. The optimality in space hinges essentially on this Ritz-Volterra reconstruction operator [15] whereas a quadratic (in time) space-time reconstruction operator is introduced to establish second order a posteriori error estimator in time. Here, we emphasis the fact that choice of such a quadratic space-time reconstruction operator is non-trivial and heavily problem dependent. We allow only nested refinement of the space meshes to avoid further complications arise because the Volterra integral term memorizes the jumps over all element edges in all previous space meshes. The rest of the paper is organized as follows. In Section 2, we introduce some standard notations and preliminary materials to be used in the subsequent sections. Section 3 is devoted to introduce quadratic space-time reconstructions for PIDE. In Section 4, a posteriori error estimate for the fully discrete Crank-Nicolson scheme in L∞(L2)-norm is derived. Numerical results are presented in Section 5.

2. Notations and preliminaries. Given a Lebesgue measurable set ω Rd, we denote by Lp(ω), 1 p , the Lebesgue spaces with corresponding norms⊂ ≤ ≤ ∞2 p . When p = 2, the space L (ω) is equipped with inner product , and the k·kL (ω) h· ·iω induced norm L2(ω). Whenever ω = Ω, we remove the subscripts of . L2(ω) and , ω. Further,k·k we shall use the standard notation for Sobolev spaces Wkm,pk (ω) with h· ·i A POSTERIORI ERROR ANALYSIS FOR PIDE 3

1 p . The norm on W m,p(ω) is defined by ≤ ≤ ∞ 1/p α p u m,p,ω = D u dx , 1 p< k k ω | | ! ≤ ∞ Z |αX|≤m with the standard modification for p = . When p = 2, we write W m,2(Ω) by Hm(Ω) ∞ 1 and denote the norm by m. In particular, H0 (Ω) signifies the space of functions in H1(Ω) that vanish onk·k the boundary of Ω (boundary values are taken in the sense of traces). 1 1 R Let a( , ): H0 (Ω) H0 (Ω) be the bilinear form corresponding to the elliptic operator · ·defined by × → A a(φ, ψ) := A φ, ψ , φ, ψ H1(Ω). h ∇ ∇ i ∀ ∈ 0 Similarly, let b(t,s; , ) be the bilinear form corresponding to the operator (t,s) defined on H1(Ω) ·H·1(Ω) by B 0 × 0 b(t,s; v(s), ψ) := B(t,s) v(s), ψ , v(s), ψ H1(Ω). h ∇ ∇ i ∀ ∈ 0

Let bs(t,s; , ) and bss(t,s; , ) be the bilinear forms corresponding to the operators (t,s) and · (·t,s) deﬁned on· ·H1(Ω) H1(Ω) by Bs Bss 0 × 0 b (t,s; φ(s), ψ) := B (t,s) φ(s), ψ , φ(s), ψ H1(Ω) s h s ∇ ∇ i ∀ ∈ 0 and

b (t,s; φ(s), ψ) := B (t,s) φ(s), ψ , φ(s), ψ H1(Ω), ss h ss ∇ ∇ i ∀ ∈ 0 where Bs(t,s) and Bss(t,s) are obtained by diﬀerentiating B(t,s) partially with respect to s once and twice respectively. We assume that the bilinear form a( , ) is 1 · · coercive and continuous on H0 (Ω) i.e., (2.1) a(φ, φ) α φ 2 and a(φ, ψ) β φ ψ , φ, ψ H1(Ω) ≥ k k1 | |≤ k k1k k1 ∀ ∈ 0 with α, β R+. ∈ Further, we assume that the bilinear forms b(t,s; , ), bs(t,s; , ) and bss(t,s; , ) 1 · · · · · · are continuous on H0 (Ω) i.e., (2.2) b(t,s; φ(s), ψ) γ φ(s) ψ , φ(s), ψ H1(Ω), | |≤ k k1k k1 ∀ ∈ 0

(2.3) b (t,s; φ(s), ψ) γ′ φ(s) ψ , φ(s), ψ H1(Ω) | s |≤ k k1k k1 ∀ ∈ 0 and

(2.4) b (t,s; φ(s), ψ) γ′′ φ(s) ψ , φ(s), ψ H1(Ω) | ss |≤ k k1k k1 ∀ ∈ 0 with γ,γ′,γ′′ R+. The weak∈ formulation of the problem (1.1) may be stated as follows: Find u : [0,T ] H1(Ω) such that → 0 t (2.5) u φdx + a(u, φ)= b(t,s; u(s), φ)ds + fφdx, φ H1(Ω), t (0,T ], t ∀ ∈ 0 ∈ ZΩ Z0 ZΩ u(., 0) = u0. 4 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

Let 0 = t0 < t1 <...

Vn := φ H1(Ω) : φ P , K , { ∈ 0 |K ∈ l ∀ ∈ Tn} + where Pl is the space of polynomials in d variables of degree atmost l Z . Let σn be the quadrature rule used to discretize the Volterra integral∈ term. To be consistent with the Crank-Nicolson scheme, we use the trapezoidal rule given by

n−2 τ τ (2.6) σn(y) := j+1 y(t )+ y(t ) + n y(t )+ y(t ) 2 j j+1 4 n−1 n−1/2 j=0 X tn−1/2 y(s)ds. ≈ Z0 Throughout this paper the following notation will be used for n =1, 2, ,N ··· n n−1 n n n−1 n v v ¯ n n n v P0 v ∂v := − , ∂v = P0 ∂v := − , τn τn t + t vn + vn−1 t := n n−1 and vn−1/2 := . n−1/2 2 2 Representation of the bilinear forms. For a function v Vn, we can represent the bilinear form a( , ) as ∈ · · 1 (2.7) a(v, φ)= elv, φ + J1[v], φ , φ H (Ω), hA i h i n ∀ ∈ 0 where P

v, φ = div(A v), φ , φ H1(Ω) hAel i h− ∇ i ∀ ∈ 0 KX∈Tn is the regular part of the distribution div(A v) and − ∇

(2.8) J1[v] E(x) = [A v]E (x) := lim(A v(x + ενE) A v(x ενE)) νE | ∇ ε→0 ∇ − ∇ − ·

is the spatial jump of the ﬁeld A v across an element side E n, where νE is a unit normal vector to E at the point∇ x. ∈ S A POSTERIORI ERROR ANALYSIS FOR PIDE 5

Similarly, for all φ H1(Ω), we represent the bilinear form b(t ,s; , ) as ∈ 0 n · · tn tn tn (2.9) b(t ,s; v(s), φ)ds = (t ,s)v(s)ds, φ + J [v(s)]ds, φ , n el n 2 n 0 h 0 B i h 0 i Z Z Z P where el(tn,s)v(s) is the regular part of the distribution div(B(tn,s) v(s)) and is deﬁnedB as − ∇

(t ,s)v(s), φ := div(B(t ,s) v(s)), φ , φ H1(Ω), hBel n i h− n ∇ i ∀ ∈ 0 KX∈Tn and J [v(s)] is the spatial jump of the field div(B(t ,s) v(s)) across an element 2 − n ∇ side E n as defined in (2.8) with B(tn,s) replacing A. ∈ S n 1 Vn n−r 1 We define the fully discrete operators : H0 (Ω) and (s): H0 (Ω) Vn, 0 r< 1 by A → B → ≤ nw,χ = a(w,χ ) χ Vn hA ni n ∀ n ∈ and

n−r(s)w(s),χ = b(t ,s; w(s)χ ) χ Vn, s I . hB ni n−r n ∀ n ∈ ∈ n Moreover, let P n : L2(Ω) Vn denotes the L2 projection operator and is given by 0 → P nw,χ = w,χ , χ Vn. h 0 ni h ni ∀ n ∈ The fully discrete Crank-Nicolson scheme may be stated as follows: Given U 0 = P 0u(0), find U n Vn,n [1 : N] such that 0 ∈ ∈ n n−1/2 (2.10) ∂Uh φndx + a(U , φn) ZΩ = σn(b(t ; U, φ )) + f n−1/2φ dx, φ Vn. n−1/2 n n ∀ n ∈ ZΩ Here, the quadrature rule σn is as defined in (2.6). Let U be a continuous, piecewise linear approximation in time defined for all t I by ∈ n n n−1 (2.11) U(t) := ln(t)U + ln−1(t)U , where

(t tn−1) (tn t) (2.12) ln(t) := − and ln−1(t) := − . τn τn Following [15], we recall the deﬁnition of Ritz-Volterra reconstruction operator below. Definition 2.1 (Ritz-Volterra reconstruction). We deﬁne the Ritz-Volterra re- n 1 1 construction H0 (Ω) of v H0 (Ω) to be a solution of the following elliptic Volterra integralW equation∈ in the weak∈ form

tn (2.13) a( n,χ)= gn,χ + b(t ,s; (s),χ)ds, χ H1, W h i n W ∀ ∈ 0 Z0 where gn is given by

tn gn = nv n(s)v(s)ds, v H1(Ω). A − B ∈ 0 Z0 6 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

1 We denote the Ritz-Volterra reconstruction of a function v H0 (Ω) by wv. Remark. The Galerkin orthogonality property holds for the∈ Ritz-VolterraR reconstruction

tn (2.14) a( n v, φ ) b(t ,s; ( v)(s), φ )=0, φ Vn. W − n − n W− n ∀ n ∈ Z0

We use the following deﬁnitions in the subsequent error analysis. For t In, we deﬁne the Ritz-Volterra reconstructions of U(t) by ∈

(2.15) U(t) := l (t) n−1U n−1 + l (t) n U n, Rw n−1 Rw n Rw where ln−1(t) and ln(t) are given by (2.12). Now, set

t (2.16)ω ˆ(t) := B(t,s) U(s)ds. ∇Rw Z0 For t I , deﬁneω ˆ (t) to be the linear interpolant associated with the integral vectors ∈ n I ωˆ(tn−1) andω ˆ(tn) and is given by

(2.17)ω ˆI (t) := ln−1(t)ˆω(tn−1)+ ln(t)ˆω(tn).

Further, let

t (2.18) ˆ(t) := (t,s)U(s)ds, t I . U B ∈ n Z0 For t I , we deﬁne ˆ (t) to be the linear interpolant associate with the integrals ∈ n UI,1 ˆ(t ), ˆ(t ) and ˆ (t) to be the linear interpolant associate with that of the U n U n−1 UI,2 integral ˆ(t ) as follows: U n−1/2 (2.19) ˆ (t) := l (t) ˆ(t )+ l (t) ˆ(t ) UI,1 n−1 U n−1 n U n and

ˆ ˆ d ˆ (2.20) I,2(t) := (tn−1/2) + (t tn−1/2) (t) , U U − dtU t=tn := ˆ(t ) + (t t ) , U n−1/2 − n−1/2 Yn where d (2.21) n = ˆ(t) . Y dtU t=tn

We deﬁne the jump residual, for n [0 : N], as ∈

tn n n (2.22) J [U] := J1[U ] J2[U(s)]ds, − 0 0 0 Z J [U] := J1[U ].

Modified Crank-Nicolson scheme: In the case of parabolic problem, the authors in [3] have observed that during refinements the discrete Laplace operator on the finer A POSTERIORI ERROR ANALYSIS FOR PIDE 7

mesh when applied to coarse grid function leads to the oscillatory behaviour. Since PIDE may be thought of as the perturbation to the parabolic problem, the same oscillatory behaviour is expected concerning the classical Crank-Nicolson scheme for the PIDE (1.1). Therefore, we consider the following modified Crank-Nicolson scheme instead of the classical Crank-Nicolson scheme. For n, 1 n N, find U n Vn such that ≤ ≤ ∈ 1 1 (2.23) ∂U¯ n + nU n + P n n−1U n−1 P n(σn( n−1/2U)) P nf n−1/2 =0. 2A 2 0 A − 0 B − 0 3. Quadratic (in time) space-time reconstructions. In this section, we start with some notations necessary to introduce quadratic space-time reconstruction. Moreover, these notations will be fruitful in the error analysis in Section 4. Let Θ : [0,T ] H1(Ω) be defined by → 0 (3.1) Θ(t) := l (t)P n n−1U n−1 + l (t) nU n P n(σn( n−1/2U)), t I . n−1 0 A n A − 0 B ∈ n Define Fˆ : [0,T ] H1(Ω) by → 0 (3.2) Fˆ(t) := Θ(t) P nϕ(t), t I , − 0 ∈ n where ϕ(t) := Ifˆ (t). Here, Iˆ is a piecewise linear interpolant chosen such that (3.3) Iˆ(φ) P (I ), Iˆ(φ)(t )= φn−1 and Iˆ(φ)(t )= φn−1/2. ∈ 1 n n−1 n−1/2 We now define the quadratic space-time reconstruction Uˆ : [0,T ] H1(Ω) as follows → 0 t (3.4) Uˆ(t) := n−1U n−1 n Fˆ(s)ds Rw − Rw Ztn−1 n n n−1 n−1 n−1 wP0 U w U +(t tn−1)R − R , t In. − τn ∈ This definition is motivated by the fact that Uˆ(t) satisfies the following relation:

n n n−1 n−1 n−1 ˆ n ˆ wP0 U w U (3.5) Ut(t)+ wF (t)= R − R . R τn

Observe that Uˆ(t )= n−1U n−1 n−1 Rw and P n n−1U n−1 + nU n Uˆ(t )= n P nU n−1 n τ 0 A A P nf n−1/2 n Rw 0 − Rw n 2 − 0 h P nσn( n−1/2U)) = n U n, − 0 B Rw i where we have used (2.23) and the integral is evaluated using the mid-point rule. Equivalently, the modiﬁed Crank-Nicolson scheme can be rewritten as ¯ n n−1/2 n n−1/2 (3.6) ∂U +Θ = P0 f . In view of (3.2) (3.7) ∂U¯ n + F n−1/2 =0, ˆ n−1/2 where F (tn−1/2) := F . 8 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

4. Error analysis. In this section, we shall derive a posteriori error estimate for the error e(t) := u(t) U(t), where u is the exact solution of the PIDE (1.1) and U is defined in (2.11).− We now decompose the total error e(t) as e(t) :=ρ ˆ(t) ε(t), whereρ ˆ(t) := u(t) Uˆ(t) denotes the parabolic error and ε(t) := U(t) Uˆ(−t) denotes the reconstruction− error. Further, we decompose the reconstruction error− as ε(t) := ǫ(t) σ(t), where ǫ(t) := U(t) U(t) is the Ritz-Volterra reconstruction − − Rw error and σ(t) := Uˆ(t) wU(t) is the time reconstruction error. With the above decompositions, the error− Re(t) may be expressed as (4.1) e(t) :=ρ ˆ(t)+ σ(t) ǫ(t). − The idea behind the above error decomposition is as follows: (i) optimal order a posteriori error estimates for Ritz-Volterra reconstruction error ǫ(t) in standard norms like L2 and H1 can be obtained; (ii) the parabolic errorρ ˆ(t) satisfies a variant of the original PIDE (1.1) with a right hand side that can be controlled a posteriori in an optimal way; (iii) the time reconstruction Uˆ is chosen in such a way that the difference Uˆ(t) U(t) can be estimated a posteriori and will be of O(τ 2). − Rw We now recall from [16] the following interpolation error estimates. n 1 Vn Proposition 4.1. Let Π : H0 (Ω) be the Cle´ment-type interpolation operator. Then, for sufficiently smooth ψ→and finite element polynomial space of degree l, there exist constants C1,j and C2,j depending only upon the shape-regularity of the family of triangulations such that for j l +1 ≤ h−j(ψ Πnψ) C ψ , k n − k≤ 1,j k kj and

1/2−j n h (ψ Π ψ) C2,j ψ j . k n − k n ≤ k k Below, we now state the following a posterioriP error estimates for the Ritz-Volterra reconstruction error. For a proof, we refer to Lemma 4.2 of [15]. Lemma 4.2 (Ritz-Volterra reconstruction error estimates). For any v Vn, the following estimates hold: ∈ n v v α (v) kRw − k1 ≤ n and n v v β (v), kRw − k≤ n where

tn (4.2) α (v) := C h nv v n(s)v(s)ds n 1 nkA − Ael − B Z0 tn + (t ,s)v(s)ds + C h1/2 Jn[v] el n 2 n n 0 B k k k Z P and

tn (4.3) β (v) := C h2 nv v n(s)v(s)ds n 3 nkA − Ael − B Z0 tn + (t ,s)v(s)ds + C h3/2 Jn[v] el n 4 n n 0 B k k k Z P A POSTERIORI ERROR ANALYSIS FOR PIDE 9

are the Ritz-Volterra reconstruction error estimators. Moreover, the constants appeared in the estimators are positive constants depend upon the interpolation constants and the ﬁnal time T . We state the main result of this section concerning fully discrete Crank-Nicolson a posteriori error estimate in the L∞(L2)-norm. Theorem 4.3 (L∞(L2) a posteriori error estimate). Let u(t) be the exact solution of (1.1) and U(t) be as deﬁned in (2.11). Then, for each m [1 : N], the following error estimate hold: ∈

1/2 m 1/2 2 2 2 2 max u(t) U(t) ρˆ(t0) + C7 τnΛn + Ξ1,m + Ξ2,m t∈[0,tm] k − k≤ k k n=1 ! h n X i + max βn(U )+ max νn, 0≤n≤m 0≤n≤m

where νn and Λn are the time reconstruction error estimators and are deﬁned by

(4.4) ν := τ 2 β ( )+ n n n Wn kWnk h i and 2 βτn (4.5) Λn := αn( n)+ n 1 . √30α W kW k h i Here, αn( n) and βn( n) are given by (4.2) and (4.3) respectively. n is an a posterioriW quantity givenW by W

n n−1/2 n−1 P0 f f 1 n n − (4.6) n := ∂¯ U . W " 2 A − τn #

2 2 Ξ1,m and Ξ2,m are the total estimators corresponding to parabolic error ρˆ(t) and are deﬁned by

m 2 2 (4.7) Ξ1,m := C7 τn λn + ηn + ζn + ϑn,1 n=1 ! X h i and 4(C )2 2 (4.8) Ξ2 := 7 τ µ + ξ + ϑ . 2,m α n n n n,2 X h i Here, λn is the time estimator which captures quadrature error and linear approximation errors and, is deﬁned by

(4.9) λn := C8 θn + ˆI,1(t) ˆ(t) + ˆ(t) ˆI,2(t) , " kU − U k kU − U k#

where ˆ(t), ˆ (t) and ˆ (t) are given by (2.18), (2.19) and (2.20), respectively and U UI,1 UI,2 θn is given by

n (4.10) θ := τ 2 τ ∆nU j + τ ∆n∂U j . n j j k k j k k j=0 " # X 10 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

tn−1 τˆn 2 −1 n n n−1 n−1 n−1 (4.11) ζn := C9CΩ hn τn U U + (s)U(s)ds τn !" k A − A 0 B Z tn tn−1 n(s)U(s)ds + U n−1 U n + (t ,s)U(s)ds − B Ael − Ael Bel n−1 Z0 Z0 tn n−1 (t ,s)U(s)ds + h3/2 ∂Jn[U] + β (U j ) − Bel n k n k kΣn j 0 j=0 # Z X and τ (4.12) η := n β ( ) n 2 n Wn are the spatial error estimators, where β (U n) and β ( ) are given by (4.3). n n Wn 1/2 1 tn (4.13) ξ := ωˆ(t) ωˆ (t) 2dt , n τ k − I k n Ztn−1 ! is the linear interpolation error estimator for the Volterra integral term where ωˆ(t) and ωˆI (t) are given by (2.16) and (2.17), respectively.

1 n n−1 n−1 n n n−1/2 (4.14) µn := C1,1hn (P0 I) U + (P0 I)σ ( U) "√3k − A k k − B k

−1 n n−1 +τn (P0 I)U , k − k# is the mesh change estimator.

1 tn (4.15) ϑ := f(t) ϕ(t) dt n,1 τ k − k n Ztn−1 and

(4.16) ϑ := 2C h max (I P n)f n−1 , (I P n)f n−1/2 n,2 1,1 n k − 0 k k − 0 k n o are the data approximation error estimators, where ϕ(t) := Ifˆ (t) and Iˆ is given by (3.3). Moreover, the constants appeared in the estimators are positive constants independent of the discretization parameters but depend upon the interpolation constants and the ﬁnal time T . The proof of Theorem 4.3 needs some preparations. We ﬁrst proceed to estimate ρˆ(t) which is a cumbersome task. Lemma 4.4 (A posteriori error estimate for the parabolic error). For each m [1 : N], the following estimate holds for ρˆ(t): ∈

1/2 tm 2 α 2 max ρˆ(t) + ρˆ(t) 1dt t∈[0,tm] k k 4 k k Z0 ! 1/2 m 1/2 ρˆ(t ) 2 + C τ Λ2 + Ξ2 + Ξ2 , ≤ k 0 k 7 n n 1,m 2,m n=1 ! h X i A POSTERIORI ERROR ANALYSIS FOR PIDE 11

2 2 where Λn, Ξ1,m and Ξ2,m are given by (4.5), (4.7) and (4.8), respectively and C7 is a positive constant independent of the discretization parameters but depends upon the interpolation constants and the ﬁnal time T .

The proof of the above lemma in turn hinges essentially on several auxiliary results which we shall discuss in detail below. We shall use the notation ρ(t) for the error u(t) wU(t) in the subsequent error analysis. We begin with the following error equation− R forρ ˆ(t). 1 Lemma 4.5. For t In,n [1 : N] and for each φ H0 (Ω), we have the following error equation for∈ ρˆ(t): ∈ ∈

t (4.17) ρˆ , φ + a(ρ, φ) b(t,s; ρ(s), φ)ds = G, φ , h t i − h i Z0 where G is deﬁned by

G, φ := G , φ + (t t ) , φ h i h 1 i − n−1/2 hYn i with

G , φ := (P n I) l (t) n−1U n−1 σn( n−1/2U) τ −1U n−1 , φ h 1 i h 0 − n−1 A − B − n i + ( n nI)(Fˆ(t) F n−1/2), φ + ˆ (t) ˆ(t), φ + ωˆo(t) ωˆ (t), φ h Rw − − i hUI,1 − U i h − I ∇ i tn−1/2 + ˆ(t) ˆ (t), φ + n−1/2U(s)ds σn( n−1/2U), φ hU − UI,2 i h B − B i Z0 + Fˆ(t) Θ(t)+ f(t), φ τ −1 ( n I)U n ( n−1 I)U n−1 , φ h − i − h n Rw − − Rw − i h i and n is given by (2.21). Y 1 Proof. For t In and φ H0 (Ω), we ﬁrst multiply (3.5) by φ and integrate over Ω. Then, subtract∈ the resulting∀ ∈ equation from (2.5) to obtain

t ρˆ (t), φ + a(u(t), φ) b(t,s; u(s), φ)ds h t i − Z0 = f, φ + n Fˆ(t), φ τ −1 n P nU n−1 n−1U n−1, φ . h i hRw i− n hRw 0 − Rw i Using (2.15)-(2.17) and (2.13), we obtain

t ρˆ (t), φ + a(ρ(t), φ) b(t,s; ρ(s), φ)ds h t i − Z0 tn−1 = l (t) n−1U n−1, φ n−1(s)U(s)ds, φ − n−1 hA i − h B i Z0 h tn i n n n n ˆ ln(t) U , φ (s)U(s)ds, φ + f(t), φ + wF (t), φ − hA i − h 0 B i h i hR i h Z i τ −1 n P nU n−1 n−1U n−1, φ + ωˆ(t) ωˆ (t), φ . − n hRw 0 − Rw i h − I ∇ i Using (3.1) and (2.18)-(2.20), we get

t (4.18) ρˆ (t), φ + a(ρ(t), φ) b(t,s; ρ(s), φ)ds h t i − Z0 12 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

= (P n I) l (t) n−1U n−1 σn( n−1/2U) , φ + ˆ (t) ˆ(t), φ h 0 − n−1 A − B i hUI,1 − U i n tn−1/2 o + ˆ(t) ˆ (t), φ + n−1/2U(s)ds σn( n−1/2U), φ hU − UI,2 i h B − B i Z0 +(t t ) , φ + f(t), φ + ωˆ(t) ωˆ (t), φ − n−1/2 hYn i h i h − I ∇ i + n Fˆ(t) Θ(t), φ τ −1 n P nU n−1 n−1U n−1, φ . hRw − i− n hRw 0 − Rw i For the last two terms on the right hand side of (4.18), an application of (2.23) yields

(4.19) n Fˆ(t) Θ(t), φ τ −1 n P nU n−1 n−1U n−1, φ hRw − i− n hRw 0 − Rw i = ( n I)(Fˆ(t) F n−1/2), φ + Fˆ(t) Θ(t), φ h Rw − − i h − i τ −1 ( n I)U n ( n−1 I)U n−1, φ τ −1 P nU n−1 U n−1, φ . − n h Rw − − Rw − i− n h 0 − i Thus, the error equation (4.17) forρ ˆ(t) now follows from (4.18) and (4.19). The next lemma presents a clear picture of the terms to be estimated in order to obtain a bound onρ ˆ(t). Lemma 4.6. The following estimate holds for ρˆ(t)

tm 2 α 2 2 2 max ρˆ(t) + 2 ρ(t) 1 + ρˆ(t) 1 dt ρˆ(0) + C7 m, t∈[0,tm] k k 2 0 k k k k ≤k k I Z h i where

m := T,1 + T,2 + M,3 + S,4 + S,5 + D,6 Im In In In In In In n=1 X := 1 + 2 + 3 + 4 + 5 + 6 Im Im Im Im Im Im with

tn (4.20) T,1 := ρˆ(t) ρ(t) 2dt, In k − k1 Ztn−1 tn tn−1/2 T,2 n−1/2 n n−1/2 (4.21) n := U(s)ds σ ( U), ρˆ(t) I tn−1 " h 0 B − B i Z Z

+ ˆ (t ) ˆ(t), ρˆ(t) + ˆ(t) ˆ (t), ρˆ(t) hUI,1 − U i hU − UI,2 i

+ ωˆ(t) ωˆI (t), ρˆ(t) dt, |h − ∇ i|#

tn M,3 n n−1 n−1 n n−1/2 (4.22) n := (P0 I) ln−1(t) U σ ( U) I tn−1 " − A − B Z D n

−1 n−1 τn U , ρˆ(t) dt, − # o E tn S,4 n ˆ n−1/2 (4.23) n := ( w I)(F (t) F ), ρˆ(t) dt, I t − R − − Z n 1 D E tn S,5 −1 n n n−1 n−1 (4.24) n := τn ( w I)U ( w I)U , ρˆ(t) dt I t − R − − R − Z n 1 D E

A POSTERIORI ERROR ANALYSIS FOR PIDE 13 and

tn D,6 ˆ (4.25) n := F (t) Θ(t)+ f(t), ρˆ(t) dt. I t − − Z n 1 D E

Proof. Set φ =ρ ˆ(t) in (4.17) to obtain

1 d t ρˆ(t) 2 + a(ρ(t), ρˆ(t)) = b(t,s; ρ(s), ρˆ(t))ds + G, ρˆ(t) . 2 dtk k h i Z0 We integrate from tn−1 to tn and use the fact 1 1 1 a(ρ(t), ρˆ(t)) = a(ρ(t),ρ(t)) + a(ˆρ(t), ρˆ(t)) a(ˆρ(t) ρ(t), ρˆ(t) ρ(t)) 2 2 − 2 − − to obtain

tn tn 1 2 2 1 1 ρˆ(tn) ρˆ(tn−1) + a(ρ(t),ρ(t))dt + a(ˆρ(t), ρˆ(t))dt 2 k k −k k 2 t − 2 t − n o Z n 1 Z n 1 1 tn tn t = a(ˆρ(t) ρ(t), ρˆ(t) ρ(t))dt + b(t,s; ρ(s), ρˆ(t))dsdt 2 − − Ztn−1 Ztn−1 Z0 tn + G, ρˆ(t) dt. h i Ztn−1 Using the coercivity of a( , ), and continuity of a( , ), b(t,s; , ) together with a standard kickback argument,· we· obtain · · · ·

tn tn 1 2 2 α 2 α 2 ρˆ(tn) ρˆ(tn−1) + ρ(t) 1dt + ρˆ(t) 1dt 2 k k −k k 2 t − k k 4 t − k k n o Z n 1 Z n 1 β tn tn t tn ρˆ(t) ρ(t) 2dt + C (T ) ρ(s) 2dsdt + G , ρˆ(t) dt, ≤ 2 k − k1 5 k k1 |h 1 i| Ztn−1 Ztn−1 Z0 Ztn−1 where we have used the fact that

tn (t t )dt =0. − n−1/2 Ztn−1 Summing from n = 1 : m with an application of Gronwall’s lemma gives the desired result with C7 = max βC6(T ), 2C6(T ) , where C6(T ) is a Gronwall’s constant. Now, we proceed{ to estimate the terms} appeared in Lemma 4.6. We start with providing a posteriori error bounds on the time discretization error. Lemma 4.7 (Time error estimators). The following a posteriori bounds hold for the time discretization error terms 1 and 2 : Im Im m (4.26) 1 τ Λ2 Im ≤ n n n=1 X and

m m tn 1/2 2 1/2 2 (4.27) m τn max ρˆ(t) λn + τn ξn ρˆ(t) 1dt , I ≤ [0,tm] k k k k n=1 n=1 tn−1 X X Z 14 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

where Λn, λn and ξn are given by (4.5), (4.9) and (4.13), respectively. Proof. We know that

(4.28) ρˆ(t) ρ(t)= (Uˆ(t) U(t)). − − − Rw T,1 ˆ Thus, to estimate the term In , we have to ﬁrst estimate U(t) wU(t). Using (2.15), (3.5) and (3.7), we have − R

Uˆ (t) ( U) (t)= n F n−1/2 Fˆ(t) . t − Rw t Rw − h i We integrate from t to t and use the fact that Uˆ(t) interpolates with U(t) at n−1 Rw tn−1 to obtain

t ˆ n ˆ n−1/2 (4.29) U(t) wU(t)= w F (s) F ds. − R −R t − − Z n 1 n o Using (3.1), (3.2) and the identity l (t)+ l (t)=1, t I , we have n−1 n ∈ n (4.30) Fˆ(t) F n−1/2 = Θ(t) Θ(t ) P n[ϕ(t) ϕ(t )] − − n−1/2 − 0 − n−1/2 = 2(t t ) , − n−1/2 Wn where n is given by (4.6). SubstitutingW (4.30) in (4.29), we get

(4.31) Uˆ(t) U(t) = (t t)(t t ) n . − Rw n − − n−1 RwWn Using the coercivity and the continuity of the bilinear form a( , ), it follows that · · α ρˆ(t) ρ(t) 2 a(ˆρ(t) ρ(t), ρˆ(t) ρ(t)) k − k1 ≤ − − = (t t)(t t )a( n , ρˆ(t) ρ(t)) n − − n−1 RwWn − (t t)(t t )β n ρˆ(t) ρ(t) , ≤ n − − n−1 kRwWnk1k − k1 where we have used (4.28) and (4.31). Thus, using (4.2) we deduce that β(t t)(t t ) (4.32) ρˆ(t) ρ(t) n − − n−1 α ( )+ . k − k1 ≤ α n Wn kWnk1 h i Finally, with an aid of (4.32), we obtain

tn 2 5 2 T,1 2 β τn (4.33) n := ρˆ(s) ρ(s) 1ds 2 αn( n)+ n 1 . I t − k − k ≤ 30α W kW k Z n 1 h i Thus, the ﬁrst inequality (4.26) follows by taking summation over n and using (4.5). Next, to prove the inequality (4.27), we ﬁrst note that

tn tn−1/2 T,2 n−1/2 n n−1/2 ˆ ˆ n := U(s)ds σ ( U), ρˆ(t) + I,1(t) (t), ρˆ(t) I tn−1 " h 0 B − B i hU − U i Z Z

+ ˆ(t) ˆI,2(t), ρˆ(t) + ωˆ(t) ωˆI (t), ρˆ(t) dt, hU − U i |h − ∇ i|#

:= 1 + 2 + 3 + 4 dt. t − |J | |J | |J | |J | Z n 1 h i A POSTERIORI ERROR ANALYSIS FOR PIDE 15

We start with estimating the term 1. A standard Trapezoidal rule argument for a suﬃciently smooth function g(s) yieldsJ

b (b a) 1 b g(s)ds − (g(a)+ g(b)) = (s a)(s b)g′′(s)ds. − 2 2 − − Za Za If we deﬁne (s t )(s t ) for s [t ,t ] and 1 j n 1, ψ (s) := j−1 j j−1 j 2j (s t − )(s t − ) for s ∈ [t ,t ] and≤ j ≤= n,− − j−1 − j−1/2 ∈ j−1 j−1/2 then tj τ 1 tj (4.34) g(s)ds j [g(t )+ g(t )] = ψ (s)g′′(s)ds − 2 j j−1 2 2j Ztj−1 Ztj−1 and tn−1/2 τ 1 tn−1/2 (4.35) g(s)ds n [g(t )+ g(t )] = ψ (s)g′′(s)ds. − 4 n−1 n−1/2 2 2n Ztn−1 Ztn−1 Using (2.6), (2.11), (4.34) and (4.35), we obtain

tn−1/2 (4.36) n−1/2(s)U(s)ds, ρˆ(t) σn( n−1/2U), ρˆ(t) hB i − h B i Z0 n−1 1 tj d2 = ψ (s) n−1/2(s)U(s) ds 2h 2j ds2 B j=1 Ztj−1 X tn−1/2 d2 + ψ (s) n−1/2(s)U(s) ds, ρˆ(t) 2n ds2 B i Ztn−1 n−1 1 tj d2( n−1/2(s)) d( n−1/2(s)) d(U(s)) = ψ (s) B U(s)+ B ds 2h 2j ds2 ds ds j=1 tj−1 ( ) X Z tn−1/2 d2( n−1/2(s)) d( n−1/2(s)) d(U(s)) + ψ (s) B U(s)+ B ds, ρˆ(t) 2n ds2 ds ds i Ztn−1 ( ) 1 n−1 γ′′ τ 2 τ ∆nU j−1 + ∆nU j + γ′τ ∆n∂U j ≤ 2 j 2 j k k k k j k k j=1 " # X ′′ 2 γ n n−1 n n ′ n n +τn τj ∆ U + ∆ U + γ τj ∆ ∂U ρˆ(t) " 2 k k k k k k#!k k γθ¯ ρˆ(t) , ≤ nk k γ′′ γ′ where θn is given by (4.10) andγ ¯ = max 2 , 2 . Thus, in view of (4.36) we have the following bound on J1 γθ¯ ρˆ(t) . |J1|≤ nk k Moreover, an application of Cauchy-Schwarz inequality gives ˆ (t) ˆ(t) ρˆ(t) , |J2|≤kUI,1 − U kk k ˆ(t) ˆ (t) ρˆ(t) |J3|≤kU − UI,2 kk k 16 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

and

ωˆ(t) ωˆ (t) ρˆ(t) . |J4|≤k − I kk∇ k Combine the bounds on , , and to obtain |J1| |J2| |J3| |J4|

tn T,2 γθ¯ + ˆ (t) ˆ(t) + ˆ(t) ˆ (t) ρˆ(t) dt In ≤ n kUI,1 − U k kU − UI,2 k k k Ztn−1 " # 1/2 tn tn 1/2 + ωˆ(t) ωˆ (t) 2dt ρˆ(t) 2dt . k − I k k k1 Ztn−1 ! Ztn−1 Thus, we obtain

tn 1/2 T,2 1/2 2 n τn max ρˆ(t) λn + τn ξn ρˆ(t) 1dt , I ≤ [0,tm] k k k k Ztn−1

where λn, ξn are given by (4.9), (4.13), respectively and C8 := max γ,¯ 1 . The desired estimate now follows by taking summation over n. { } The next lemma gives information on the a posteriori contributions due to mesh change. Lemma 4.8 (Mesh change estimate). We have the following bound on the mesh change error term 3 : Im 1/2 m tn (4.37) 3 τ 1/2µ ρˆ(t) 2dt , Im ≤ n n k k1 n=1 tn−1 ! X Z where µn is given by (4.14). n Proof. The orthogonality property of P0 now leads to

tn n n−1 n−1 n n−1/2 −1 n−1 (P0 I) ln−1(t) U σ ( U) τn U , ρˆ(t) dt tn−1 " − A − B − # Z D n o E tn n n−1 n−1 n n−1/2 = (P0 I) ln−1(t) U σ ( U) tn−1 " − A − B Z D n −1 n−1 n τn U , ρˆ(t) Π ρˆ(t) dt. − − # o E An application of the Cauchy-Schwarz inequality yields

tn 1/2 M,3 2 n n−1 n−1 n C1,1hn ln−1(t)dt (P0 I) U I ≤ " tn−1 k − A k n Z o tn 1/2 1/2 n n n−1/2 −1/2 n n−1 2 +τn (P0 I)σ ( U) + τn (P0 I)U ρˆ(t) 1dt k − B k k − k# tn−1 k k n Z o tn 1/2 1/2 2 τn µn ρˆ(t) 1dt , ≤ t − k k n Z n 1 o where µn is given by (4.14). Taking summation over n completes the proof. The next lemma captures contributions due to the spatial discretizations. A POSTERIORI ERROR ANALYSIS FOR PIDE 17

Lemma 4.9 (Spatial error estimates). The following a posteriori error bound holds on the spatial discretization error term 4 : Im m 4 (4.38) m τn max ρˆ(t) ηn, I ≤ [0,tm] k k n=1 X 5 where ηn is given by (4.12). Moreover, the error bound holds for m corresponds to the spatial discretization error due to mesh change: I

m 5 (4.39) m max ρˆ(t) τnζn, I ≤ t∈[0,tm] k k n=1 X

where ζn is given by (4.11).

Proof. Using (4.30) and (4.3), we have

tn 2 n ˆ n−1/2 τn ( w I)(F (t) F ), ρˆ(t) dt max ρˆ(t) βn( n). t − R − − ≤ 2 [0,tm] k k W Z n 1 D E

Hence, we obtain

S,4 (4.40) n τn max ρˆ(t) ηn, I ≤ [0,tm] k k where ηn is given by (4.12) and the estimate (4.38) follows by taking the summation over n. S,5 Next, to estimate n as given in (4.24), we exploit the orthogonality property of the Ritz-Volterra reconstructions.I We use the standard duality technique here. 2 1 For t (0,T ), let ψ H (Ω) H0 (Ω) be the solution of the following elliptic problem in∈ the weak form∈ ∩

(4.41) a(χ, ψ(t)) = χ, ρˆ(t) , χ H1(Ω) h i ∀ ∈ 0 satisfying the following regularity estimate:

(4.42) ψ(t) C ρˆ(t) , k k2 ≤ Ωk k where the constant CΩ depends on the domain Ω. n n n−1 n−1 n n−1 Setting χ = wU w U U + U in (4.41) and using (2.13), (2.7) and, (2.9), we obtainR − R −

n U n n−1U n−1 U n + U n−1, ρˆ(t) hRw − Rw − i = a( n U n n−1U n−1 U n + U n−1, ψ(t) Πnψ(t)) Rw − Rw − − tn b(t ,s; ( U U)(s), ψ(t) Πnψ(t))ds − n Rw − − Z0 tn−1 + b(t ,s; ( U U)(s), ψ(t) Πnψ(t))ds n−1 Rw − − Z0 tn tn−1 + b(t ,s; ( U U)(s), ψ(t))ds b(t ,s; ( U U)(s), ψ(t))ds n Rw − − n−1 Rw − Z0 Z0 18 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

tn tn = nU n n(s)U(s)ds U n + (t ,s)U(s)ds, ψ(t) Πnψ(t) hA − B − Ael Bel n − i Z0 Z0 tn−1 n−1U n−1 n−1(s)U(s)ds U n−1 −hA − B − Ael Z0 tn−1 + (t ,s)U(s)ds, ψ(t) Πnψ(t) Bel n−1 − i Z0 tn tn−1 + J [U(s)]ds J [U n] J [U(s)]ds + J [U n−1], ψ(t) Πnψ(t) h 2 − 1 − 2 1 − iΣn Z0 Z0 tn tn−1 + b(t ,s; ( U U)(s), ψ(t))ds b(t ,s; ( U U)(s), ψ(t))ds. n Rw − − n−1 Rw − Z0 Z0 We now use (2.22) together with Jn[U] Jn−1[U]= τ ∂Jn[U] to obtain − n (4.43) n U n n−1U n−1 U n + U n−1, ρˆ(t) |hRw − Rw − i| tn tn nU n n(s)U(s)ds U n + (t ,s)U(s)ds ≤ kA − B − Ael Bel n Z0 Z0 tn−1 tn−1 n−1U n−1 + n−1(s)U(s)ds + U n−1 (t ,s)U(s)ds −A B Ael − Bel n−1 k Z0 Z0 ψ(t) Πnψ(t) + τ ∂Jn[U] ψ(t) Πnψ(t) k − k nk kΣn k − kΣn tn tn−1 + b(t ,s; ( U U)(s), ψ(t))ds b(t ,s; ( U U)(s), ψ(t))ds . | n Rw − − n−1 Rw − | Z0 Z0 To handle the last term above, we use the fact

(4.44) b(t ,s; ( U U)(s), ψ(t)) := ( U U)(s), ∗(t ,s)ψ(t) , n Rw − h Rw − B n i ∗ where (tn,s) is the formal adjoint of the operator (tn,s). Now, we apply Cauchy- B ∗ B Schwarz inequality together with (tn,s)ψ(t) CB∗ ψ(t) 2 to obtain kB k≤ 1 k k

tn tn−1 (4.45) b(t ,s; ( U U)(s), ψ(t))ds b(t ,s; ( U U)(s), ψ(t))ds | n Rw − − n−1 Rw − | Z0 Z0 tn tn−1 ( U U)(s), ∗(t ,s)ψ(t) ds ( U U)(s), ∗(t ,s)ψ(t) ds ≤ | h Rw − B n i − h Rw − B n−1 i | Z0 Z0 n tj ( U U)(s) ∗(t ,s)ψ(t) ds ≤k Rw − kkB n k j=1 tj−1 X Z n−1 tj + ( U U)(s) ∗(t ,s)ψ(t) ds k Rw − kkB n−1 k j=1 tj−1 X Z n tj j−1 j−1 j−1 j j j 2CB∗ lj−1(s)( U U )+ lj(s)( U U ) ds ψ(t) 2 ≤ 1 k Rw − Rw − k k k j=1 tj−1 # X Z n n−1 n CB∗ τˆn βj [U]+ˆτn−1 βj [U] ψ(t) 2 2CB∗ τˆn βj [U] ψ(t) 2. ≤ 1 k k ≤ 1 k k j=1 j=0 j=0 h X X i h X i

∗ Using (4.45) in (4.43) and applying Proposition 4.1 with C9 = max 2CB1 , 1 , we A POSTERIORI ERROR ANALYSIS FOR PIDE 19 obtain

(4.46) n U n n−1U n−1 U n + U n−1, ρˆ(t) |hRw − Rw − i| tn C ψ τˆ h2 τ −1( nU n n(s)U(s)ds U n ≤ 9k k2 n nk n A − B − Ael Z0 tn tn−1 + (t ,s)U(s)ds n−1U n−1 + n−1(s)U(s)ds Bel n − A B Z0 Z0 tn−1 n + U n−1 (t ,s)U(s)ds) + h3/2 ∂Jn[U] + β [U] . Ael − Bel n−1 k n k kΣn j 0 j=0 ! Z X Combining (4.24) and (4.46), we arrive at

tn tn S,5 C τ −1 ψ(t) dt τˆ h2 τ −1( nU n n(s)U(s)ds U n In ≤ 9 n k k2 n nk n A − B − Ael Ztn−1 Z0 tn tn−1 + (t ,s)U(s)ds n−1U n−1 + n−1(s)U(s)ds Bel n − A B Z0 Z0 tn−1 n + U n−1 (t ,s)U(s)ds) + h3/2 ∂Jn[U] + β [U] Ael − Bel n−1 k n k kΣn j 0 j=0 ! Z X max ρˆ(t) τnζn, ≤ t∈In k k where we have used (4.11) and the regularity result (4.42). Summing from n = 1 : m, the desired result is obtained. The data approximation error is estimated in the following lemma. Lemma 4.10 (Data approximation error estimate). The following bound holds on the data approximation error term 6 Im

1/2 m tn 6 1/2 2 (4.47) m τnϑn,1 max ρˆ(t) + τn ϑn,2 ρˆ(t) 1dt , I ≤ t∈[0,tm] k k k k n=1 " tn−1 ! # X Z where ϑn,1 and ϑn,2 are given by (4.15) and (4.16). Proof. Using (3.1) and (3.2), we have

tn tn D,6 ˆ n n := F (t) Θ(t)+ f(t), ρˆ(t) dt = f(t) P0 ϕ(t), ρˆ(t) dt I t − − t − − Z n 1 D E Z n 1 D E tn tn n f(t) ϕ(t), ρˆ(t) dt + (I P0 )ϕ(t), ρˆ(t) dt ≤ t − − t − − Z n 1 D E Z n 1 D E := J + J . 1 2

Using Cauchy-Schwarz inequality, we obtain

tn J1 max ρˆ(t) f(t) ϕ(t) dt. ≤ t∈[0,tm] k k k − k Ztn−1 20 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

n For J2, we use orthogonality property of P0 to have

tn tn n n n J2 = (I P0 )ϕ(t), ρˆ(t) dt = (I P0 )ϕ(t), ρˆ(t) Π ρˆ(t) dt t − − t − − − Z n 1 D E Z n 1 D E tn C h (I P n)ϕ(t) ρˆ(t) dt ≤ 1,1 n k − 0 kk k1 Ztn−1 1/2 tn 1/2 n n−1 n n−1/2 2 2C1,1hnτn max (I P0 )f , (I P0 )f ρˆ(t) 1dt . ≤ k − k k − k tn−1 k k ! n o Z Therefore,

1/2 tn D,6 ˆ 1/2 2 n τnϑn,1 max ρ(t) + τn ϑn,2 ρˆ(t) 1dt , I ≤ t∈[0,tm] k k k k Ztn−1 ! where ϑn,1 and ϑn,2 are given by (4.15) and (4.16), respectively. Now, taking summation over n, we obtain the desired result. Proof of Lemma 4.4. Application of Lemmas 4.7-4.10 in Lemma 4.6 yields

tm 2 α 2 2 max ρˆ(t) + 2 ρ(t) 1 + ρˆ(t) 1 dt t∈[0,tm] k k 2 0 k k k k Z h m i m 2 2 ρˆ(0) + C7 τnΛn + max ρˆ(t) τn λn + ηn + ζn + ϑn,1 ≤k k t∈[0,tm] k k h n=1 n=1 h i m X X tn 1/2 + τ 1/2 µ + ϑ + ξ ρˆ(t) 2dt , n n n,2 n k k1 n=1 tn−1 X h i Z i We now use the following elementary fact to complete the proof. For a = (a0,a1,...,am), m+1 2 2 b = (b0,b1,...,bm) R and c R, if a c + a.b, then a c + b . In particular for∈n = [1 : m], taking∈ | | ≤ | | ≤ | | | |

m tn 1/2 1/2 α 2 2 2 a0 = max ρˆ(t) , an = ρˆ(t) 1dt , c = ρˆ(t0) + C7 τnΛn , t∈[0,tm] k k 2 k k k k tn−1 n=1 Z h X i

m 1/2 b0 = C7 τn λn + ηn + ζn + ϑn,1 , bn = C7(2τn/α) µn + ξn + ϑn,2 , n=1 X h i h i we obtain the desired result. Proof of Theorem 4.3. In view of (4.1), we apply triangle inequality to have

(4.48) u(t) U(t) ρˆ(t) + σ(t) + ǫ(t) . k − k≤k k k k k k For t I , ∈ n ǫ(t) = l (t)ǫn−1 + l (t)ǫn max ǫn−1 , ǫn . k k k n−1 n k≤ k k k k Therefore, for t [0,t ], using Lemma 4.2, we have ∈ m n−1 n (4.49) ǫ(t) max ǫ , ǫ max βn[U]. k k≤ n∈[0,m] k k k k ≤ n∈[0,m] A POSTERIORI ERROR ANALYSIS FOR PIDE 21

Also, (4.50) Uˆ(t) U(t) (t t )(t t) n k − Rw k≤ − n−1 n − kRwWnk (t t )(t t) ( n I) + ν , ≤ − n−1 n − k Rw − Wnk kWnk ≤ n h i where νn is given by (4.4). Finally, we use (4.48)-(4.50) and Lemma 4.4 to obtain the desired result. Remarks. (i) The estimator appeared in Theorem 4.3 is formally of optimal order. Moreover, in the absence of the memory term (i.e., (t,s) = 0), the error estimator obtained in Theorem 4.3 is similar to that for the parabolicB problems [3]. Further, we note that the estimator λn, the contribution to the error from the approximation of the integral term, is of O(τ 2). Thus, the a posteriori error bound in Theorem 4.3 generalizes the results of [3] to PIDE. (ii) The mesh change error term 3 can alternatively be estimated as Im m 3 ′ m τn max ρˆ(t) µn, I ≤ t∈[0,tm] k k n=1 X ′ where µn is given by 1 µ′ = C h (P n I) n−1U n−1 + (P n I)σn( n−1/2U) n 1,1 n 2k 0 − A k k 0 − B k h +τ −1 (P n I)U n−1 . n k 0 − k This estimate for mesh change error willi lead to an alternative a posteriori error 2 2 estimate for the main error e. In particular, the terms Ξ1,m and Ξ2,m in Lemma 4.4 take the form m 2 2 ′ Ξ1,m := C7 τn λn + µn + ηn + ζn + ϑn,1 n=1 ! X h i and 4C2 Ξ2 := 7 τ (ξ + ϑ )2. 2,m α n n n,2 The corresponding changes take place inX the Theorem 4.3. (iii) The term 1/2 1 tn (4.51) ωˆ(t) ωˆ (t) 2dt τ k − I k n Ztn−1 ! appeared in Theorem 4.3 (see (4.13)) is not a traditional a posteriori quantity, where ωˆ(t) andω ˆI (t) are given by (2.16) and (2.17), respectively. Since, the error in linear interpolation is bounded as 2 2 d ωˆ(t) ωˆI (t) Cτn max 2 (ˆω(t)) , t In, k − k≤ t∈In kdt k ∈ d2 where dt2 (ˆω(t)) depends upon the quantities ωt(t) and ω(t). The term ωt(t) can be estimated as ∇ ∇ k∇ k ω (t) = ǫ (t) U (t) ǫ (t) + U (t) k∇ t k k∇ t − ∇ t k≤k∇ t k k∇ t k 1 ǫn + ǫn−1 + ∂U n , ≤ τn k∇ k k∇ k k∇ k 22 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

and for the term ω(t) , we have k∇ k ω(t) ǫ(t) + U(t) k∇ k≤k∇ k k∇ k l (t) ǫn−1 + l (t) ǫn + l (t) U n−1 + l (t) U n ≤k n−1 ∇ n ∇ k k n−1 ∇ n ∇ k max ǫn−1 , ǫn + max U n−1 , U n . ≤ k∇ k k∇ k k∇ k k∇ k This shows that (4.51) is now a meaningful a posteriori quantity by noting the fact that ǫn is bounded and is of O(h) (see Lemma 4.2). Taking τ h, it is easy to see thatk∇ thek term (4.51) is of optimal order. ≈ (iv) The a posteriori error analysis of the classical Crank-Nicolson scheme leads 1 n n−1 n−1 to one additional term 2 (P0 I) U in the error bounds. However, it does not aﬀect the optimalityk of the− mainA estimator.k 5. Numerical assessment. In this section, we study the behaviour of the a posteriori error estimator presented in Theorem 4.3 for a two dimensional test problem. We solve the PIDE (1.1) in a square domain Ω = (0, π)2 R2 with homogeneous Dirichlet boundary conditions. We select the coeﬃcient matrices⊂ to be A = I and B(t,s)= I, where I denotes the identity matrix. Then, the forcing term f is chosen such that the exact solution is given by

u(x,y,t) = exp( t/2) sin(x) sin(y). − Our main emphasis here is to understand the asymptotic behaviour of the estimators following which we perform numerical test on uniform meshes with uniform time- step. All computations have been carried out using MATLAB-7.8. We choose a sequence of mesh-sizes (h(i): i [1 : l]), to which we couple a sequence of step-sizes ∈ 0.4 (τ(i): i [1 : l]) i.e., τ(i) = c0h(i), where c0 is taken to be π . Here, l denotes the number∈ of runs. The initial mesh size h(1) and the time step τ(1) are chosen to be π 8 and 0.05. Bisection algorithm is used to generate the refined triangulations. For each run i [1 : l], we compute the following quantities of interest: ∈ the Ritz-Volterra reconstruction error estimator: maxn∈[0:m] βn • m the space error estimator: n=1 τnζn • m 2 1/2 the time reconstruction error estimators: ( τnΛ ) and maxn∈[0:m] νn • P n=1 n for each time tm [0 = t0 : τ(i): tN = .1]. We dropped the linear interpolation estimator, mesh change∈ estimator and data approximationP estimator from study. The experiment is carried out with P1 elements. For each quantities of interest we observe their experimental order of convergence (EOC). The EOC is defined as follows: For a given finite sequence of successive runs (indexed by i), the EOC of the corresponding sequence of quantities of interest E(i) (estimator or part of an estimator) itself is a sequence defined by

log(E(i + 1)/E(i)) EOC(E(i)) = , log(h(i + 1)/h(i))

where h(i) denotes the mesh size of the run i. In order to measure the quality of our estimator the estimated error is compared to the true error so-called effectivity index (EI). The effectivity index is defined by

m m 1/2 2 max βn + max νn + τnζn + τnΛn max e(tn) . n∈[0:m] n∈[0:m] n∈[0:m] k k n=1 n=1 !, X X A POSTERIORI ERROR ANALYSIS FOR PIDE 23

0.9

1 0.8

0.8 0.7

0.6 0.6

0.4 0.5

0.2 0.4 0 0.3 3 0.2 2 3 0.1 1 2 1 0 0 0

Fig. 5.1. The ﬁrst plot shows the exact solution and the second one corresponds to Crank-Nicolson FEM solution. Crank-Nicolson FEM solution is computed using P1 elements with 33025 free nodes at T = 0.1 corresponding to τ = .003125

All the constants involved in the estimators are taken to be equal to 1 except Gron- wall’s constant which is taken to be exp(T ). The effectivity index is to be understood only qualitatively in this paper as the main emphasis is on observing asymptotic behaviour of the estimator. In FIG. 5.2, the abscissa represents time which ranges in [0,0.1]. Each curve in each of the plots in first and third rows of FIG. 5.2 shows the estimator behaviour corresponds to a given run. The most coarse grid corresponds to curve with the largest error value and the finest grid corresponds to curve with the smallest error value. The value of EOC of a given estimator indicates its order. From the FIG. 5.2, it is apparent that the estimators have the optimal rate of convergence. Concluding remarks. In this paper, we have derived optimal order residual based a posteriori error estimator for PIDE (1.1) in the L∞(L2)-norm for the fully discrete Crank-Nicolson method. Moreover, computational results are provided to illustrate that the estimator exhibit optimal rate of convergence which support our theoretical findings. Despite the importance of PIDE, and their variants in the modeling of several physical phenomena, the topic of a posteriori analysis for such kind of equations re- mains unexplored. We believe the work presented here could be a first step towards the development of various space-time adaptive algorithms for PIDE. The Ritz-Volterra reconstruction operator [15] unifies a posteriori approach from parabolic problems to PIDE. Moreover, for the optimality of the estimator, the linear approximation of the Volterra integral term is found to be crucial. It is challenging to study the problem of obtaining a posteriori error estimates with the constants appeared in the bounds be independent of the final time T and hence they can serve as long-time estimates. Moreover, a posteriori error analysis for hyperbolic integro-differential equations in the L∞(L2)-norm is an interesting research problem which will be reported elsewhere. 24 G MURALI MOHAN REDDY AND RAJEN. K. SINHA

1 1 1 10 10 10 h(1) = π/8 π 0 0 0 h(1) = /8 10 10 h(2) = π/16 10 h(2) = π/16 −1 −1 −1 h(3) = π/32 π 10 10 10 h(3) = /32 h(4) = π/64 −2 h(4) = π/64 −2 −2 10 10 10 −3 −3 −3 10 10 10 −4 10 −4 h(1) = π/8 −4 10 −5 h(2) = π/16 10 10 −5 π −5 −6 10 h(3) = /32 10 10 h(4) = π/64 −6 −6 −7 10 10 10 0 0.05 0.1 0 0.05 0.1 0 0.05 0.1 1/2 n m 2 maxn βn(U ) n τnζn n=1 τnΛn P P 3 2.1 4 EOC(max β (Un)(1)) n n 2.5 EOC(max β (Un)(2)) 2.05 n n 3 EOC(max β (Un)(3)) 2 n n 2 1.5 2 EOC(Σ τ ζ (1)) EOC((Σ τ Λ2)1/2(1)) 1 n n n n n n 1.95 EOC(Σ τ ζ (2)) 1 EOC((Σ τ Λ2)1/2(2)) n n n n n n 0.5 EOC(Σ τ ζ (3)) 2 1/2 n n n EOC((Σ τ Λ ) (3)) n n n 1.9 0 0 0 0.05 0.1 0 0.05 0.1 0 0.05 0.1 1/2 n m 2 EOC(maxn βn(U )) EOC( n τnζn) EOC n=1 τnΛn ! 1 1 P P 10 10 π 0 h(1) = /8 0 45 10 h(2) = π/16 10 −1 π −1 10 h(3) = /32 10 h(4) = π/64 40 −2 −2 10 10 −3 −3 35 10 10 π h(1) = π/8 −4 −4 h(1) = /8 10 10 h(2) = π/16 30 h(2) = π/16 −5 −5 π h(3) = π/32 10 10 h(3) = /32 h(4) = π/64 25 h(4) = π/64 −6 −6 10 10 0 0.05 0.1 0 0.05 0.1 0 0.05 0.1

maxn∈[0:m] νn Total Estimator

2.1 2.1 EOC(max ν (1)) EOC(Total Estimator(1)) n n EOC(Total Estimator(2)) EOC(max ν (2)) 2.05 n n 2.05 EOC(Total Estimator(3)) EOC(max ν (3)) n n

2 2

1.95 1.95

1.9 1.9 0 0.05 0.1 0 0.05 0.1

EOC(maxn∈[0:m] νn) EOC(Total Estimator)

Fig. 5.2. Simulation with P1 elements for the PIDE with exact solution u(x,y,t) = exp(−t/2) sin(x) sin(y). On first and third rows we show the behaviour of the different estimators and below them the corresponding EOC. At the end of third row, we plot the effectivity index of the total estimator. We observe that the estimators decrease with at least second order with respect to the mesh n parameters. Moreover, the Ritz-Volterra reconstruction estimator i.e. maxn βn(U ) dominates all other error estimators. A POSTERIORI ERROR ANALYSIS FOR PIDE 25

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