1. Introduction. Consider a semilinear parabolic equation in the form

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1 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS USING ELLIPTIC RECONSTRUCTIONS. I: BACKWARD-EULER AND CRANK-NICOLSON METHODS NATALIA KOPTEVA AND TORSTEN LINSS Abstract. A semilinear second-order parabolic equation is considered in a regular and a singularly-perturbed regime. For tis equation, we give computable a posteriori error estimates in te maximum norm. Fully discrete Bakward-Euler and Crank-Nicolson metods are addressed, for wic we employ elliptic reconstructions tat are, respectively, piecewise-constant and piecewiselinear in time. We also use certain bounds for te Green s function of te parabolic operator. Key words. a posteriori error estimate, maximum norm, singular perturbation, elliptic reconstruction, Backward-Euler, Crank-Nicolson, parabolic equations, reaction-diffusion. AMS subject classifications. 65M15, 65M6. 1. Introduction. Consider a semilinear parabolic equation in te form Mu := t u + Lu + f(x, t, u) = for (x, t) Q := Ω (, T ], (1.1a) wit a second-order linear elliptic operator L = L(t) in a spatial domain Ω R n wit Lipscitz boundary, subject to te initial and Diriclet boundary conditions u(x, ) = ϕ(x) for x Ω, u(x, t) = for (x, t) Ω [, T ]. (1.1b) We assume tat f is differentiable in te tird argument and, for some positive constants γ and γ, satisfies γ 2 z f(x, t, z) γ 2 for (x, t, z) Ω [, T ] R. (1.2) Te purpose of tis paper is to obtain computable a posteriori error estimates for fully discrete metods applied to problem (1.1). We consider te first-order Backward- Euler and te second-order Crank-Nicolson discretizations in time. Furtermore, in a fortcoming paper [9], a similar approac will be used to analyze te tird-order Discontinuous Galerkin metod dg(1). Tese results are applied to te model equation wit L := ε 2 = ε 2 n i=1 2 x i : Mu := t u ε 2 u + f(x, t, u) = (1.3) posed in a bounded polyedral spatial domain Ω R n, wit n = 1, 2, 3. Tis equation will be considered in te two regimes: (i) ε = 1, γ ; (ii) ε 1, γ >. Note tat regime (ii) yields a singularly perturbed reaction-diffusion equation, wose solutions may exibit sarp layer penomena. So it is important in tis regime tat a posteriori error estimates are robust in te sense tat any dependence on te small perturbation parameter ε sould be sown explicitly [13, 16]. Tis publication as emanated from researc conducted wit te financial support of Science Foundation Ireland under te Researc Frontiers Programme 28; Grant 8/RFP/MTH1536. Department of Matematics and Statistics, University of Limerick, Limerick, Ireland (natalia.kopteva@ul.ie). Institut für Numerisce Matematik, Tecnisce Universität Dresden, D-162 Dresden, Germany (torsten.linss@tu-dresden.de). 1

2 2 NATALIA KOPTEVA AND TORSTEN LINSS We will give error estimates in te maximum norm, wic is sufficiently strong to capture sarp layers and singularities tat may occur, in particular, if problem (1.1) is of singularly-perturbed type. Our estimates will be of interpolation type in te sense tat tey will include certain terms tat may be interpreted as approximating τ p j p t u, were p and are te discretization order and local step size in time, respectively. We employ te elliptic reconstruction tecnique, wic was introduced in te recent papers [14, 11, 3] as a counterpart of te Ritz-projection in te a posteriori error estimation for parabolic problems. We also use certain bounds for te Green s function of te continuous parabolic operator in a manner similar to [3], only for a more general semilinear parabolic operator of (1.3) (compared to t in [3]). One distinctive feature of our analysis in tis paper (compared, e.g., to [1, 3]) is tat we use computed solutions and elliptic reconstructions tat are piecewise-constant in time wen dealing wit te Backward-Euler metod, and piecewise-linear in time wen dealing wit te Crank-Nicolson metod. Consequently, we allow te residuals of computed solutions, as well as oter functions, to be understood as distributions; tis inclusion plays a crucial role in our analysis. Te paper is organized as follows. In Section 2, we introduce te Green s function and obtain a certain stability lemma, wic is ten used in Sections 3 4 to obtain a posteriori error estimates for semidiscrete Backward-Euler and Crank-Nicolson metods (wit no spatial discretization). Next, in Section 5 we cite some elliptic a posteriori error estimates, wic are used in Sections 6 7 to derive a posteriori error estimates for fully discrete Backward-Euler and Crank-Nicolson metods. Te final Section 8 gives a proof of certain Green s function bounds deferred from Section 2. Notation. Trougout te paper, C, as well as c, denotes a generic positive constant tat may take different values in different formulas, but is independent of te diffusion coefficient ε and any mes sizes. We use x for te Euclidian norm of x R n. Te usual spaces C( Ω) and H 1 (Ω) are used, as well as te spaces L p, 1 p, wit te norm p,ω, wile φ, ψ = Ω φ(x)ψ(x) dx denotes te inner product in L 2(Ω). Distributions and left-continuity convention. Certain functions will be understood as distributions [7], wic will in most cases be indicated. By contrast, if a certain function is Lebesgue-integrable in Ω (, T ), we sall refer to it as a regular function. Wenever we deal wit a regular function, it will be understood rigt-continuous at t = and left-continuous for all t (, T ]. In particular, tis convention will be applied to all piecewise-continuous temporal derivatives. 2. Te Green s function of te parabolic operator. In tis section we consider te Green s function G associated wit te operator M of (1.1). Our interest in te Green s function is in tat it will be used to express te error of a numerical approximation in terms of its residual. For definitions and properties of fundamental solutions and Green s functions of parabolic operators wit variable coefficients, we refer te reader to [6, Cap. 1 and 7 of Cap. 3]. In particular, for fixed (x, t) Q, te Green s function G(x, t; ξ, s) =: Γ(ξ, s) solves te adjoint terminal-value problem [ s L + a(ξ, s) ] Γ(ξ, s) = for (ξ, s) Ω [, t), (2.1a) Γ(ξ, t) = δ(ξ x) for ξ Ω, (2.1b) Γ(ξ, s) = for (ξ, s) Ω [, t]. (2.1c) Here δ( ) is te Dirac δ-distribution in R n [7], and L is te adjoint operator to te linear operator L. We set a(x, t) := z f(x, t, z) if f is linear in te tird argument, i.e.

3 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 3 f(x, t, z) = a(x, t)z + b(x, t). If f is nonlinear, we associate a(x, t) := 1 zf(x, t, w + z[v w]) dz wit a pair of bounded functions v and w tat vanis on Ω. As te standard linearization now yields Mv Mw = [ t + L + a(x, t)](v w), so wit te elp of tis Green s function, te difference v w is represented as [v w](x, t) = G(x, t; ξ, ) [v w](ξ, ) dξ Ω t + G(x, t; ξ, s) [Mv Mw](ξ, s) dξ ds. (2.2) Ω Te analysis in tis paper will be carried out under te following condition. Condition 2.1. Tere are constants κ, κ 1 > and κ 2 suc tat te Green s function G of (2.1), (1.2) satisfies G(x, t;, s) 1,Ω κ e γ2 (t s), t τ s G(x, t;, s) 1,Ω ds κ 1 l(τ, t) + κ 2, were x Ω, τ (, t], t (, T ], and l(τ, t) := t τ s 1 e 1 2 γ2s ds ln(t/τ). Note tat our model problem satisfies tis condition as follows. Lemma 2.2. Let ε (, 1] and γ. Under assumption (1.2), te model problem (1.3) satisfies Condition 2.1 wit κ := 1, κ 1 := 3n and an ε-independent 2 n/2+1 constant κ 2. If f(x, t, z) = a(x)z + b(x, t), ten κ 2 =. In general, κ 2 = ( γ 2 γ 2 ) ˆκ 2, were ˆκ 2 = ˆκ 2 (γ) if γ >, and ˆκ 2 = ˆκ 2 (T ) if γ =. Proof. We defer te proof to Section 8. Remark 2.3. Te constants κ and κ 1 given by Lemma 2.2 are reasonably sarp. E.g., for te constant-coefficient version t u ε 2 xu+γ 2 2 u = b(x, t) of (1.3) in te spatial domain Ω := R, a calculation yields G(x, t;, s) 1,Ω = 1 and s G(x, t;, s) 1,Ω ( 2 π e (t s) 1 + γ 2) e γ2 (t s) so Condition 2.1 is satisfied wit κ = 1 (as in 2 Lemma 2.2), κ 1 = π e.48, κ 2 = 1, wile Lemma 2.2 gives κ 1 = /2 Te above Condition 2.1 will be employed by means of te following lemma, wic plays a crucial role in our analysis ere and in te fortcoming paper [9]. Te lemma is formulated in te context of an arbitrary nonuniform mes in te time direction = t < t 1 < t 2 <... < t M = T, wit = t j t j 1 for j = 1,..., M. (2.3) Lemma 2.4. Suppose te parabolic operator M of (1.1) satisfies (1.2) and Condition 2.1, and v, w are bounded in Ω [, T ]. Furtermore, let v(, t), w(, t) H 1 (Ω) C( Ω) for t [, T ], and Mv Mw = t µ + ϑ in Q, (2.4) were te function µ is continuous and bounded on [t, t 1 ] and eac (t j 1, t j ], wile t µ is continuous and bounded on (t m 1, t m ] for some 1 m M, and ϑ(, s) is integrable on (, t m ) (possibly, in te sense of distributions). Ten [v w](, t m ) κ e γ2 t m [v w µ](, ) + (κ 1 l m + κ 2 ) sup µ(, s) + κ sup s (t m 1,t m] µ(, s) + κ τ m tm + κ e γ2 (t m s) ϑ(, s) ds, s [,t m 1 ] sup s µ(, s) s (t m 1,t m]

4 4 NATALIA KOPTEVA AND TORSTEN LINSS were l m = l m (γ) := t m τ m s 1 e 1 2 γ2s ds ln(t m /τ m ). Remark 2.5. Te term t µ in te rigt-and side of (2.4) is understood in te sense of distributions. Remark 2.6. One can easily ceck tat if γ =, ten l m = ln(t m /τ m ). Oterwise, if γ >, one as l m (γ) = E 1 ( 1 2 γ2 τ m ) E 1 ( 1 2 γ2 t m ), were E 1 (t) = s 1 e s ds; so l t m (γ) ln( 1 2 γ2 τ m ) provided tat 1 2 γ2 τ m.67 (tis is easily cecked by finding te only root.67 of te equation E 1 (s) = ln s on (, 1)). Note also tat l 1 = for any γ. Proof. Combining representation (2.2) wit te notation Γ(ξ, s) := G(x, t m ; ξ, s) for te Green s function of (2.1), one gets [v w](x, t m ) = [v w](, ), Γ(, ) + tm [Mv Mw](, s), Γ(, s) ds. Here, in view of (2.4), te integral on te rigt-and side involves µ and ϑ, so can be represented as a sum J µ + J ϑ of te corresponding integrals, wic we consider separately. We use te notation b + b+β := lim β + and so split Jµ as t + J µ = J µ (1) + J µ (2) := m 1 Here, for J (1) µ, an integration by parts yields s µ, Γ(, s) ds + tm J (1) µ = µ(, t + m 1 ), Γ(, t m 1) µ(, ), Γ(, ) Consequently, we arrive at [v w](x, t m ) = [v w µ](, ), Γ(, ) + µ(, t + m 1 ), Γ(, t m 1) + + tm ϑ(, s), Γ(, s) ds, t + m 1 tm 1 tm 1 tm t + m 1 s µ, Γ(, s) ds. µ(, s), s Γ(, s) ds. µ(, s), s Γ(, s) ds s µ, Γ(, s) ds were te last term represents J ϑ. Finally, Condition 2.1 implies tat Γ(, s) 1,Ω κ e γ2 (t m s) κ, so we get te desired result. tm 1 s Γ(, s) 1,Ω ds κ 1 l m + κ 2, 3. Semidiscrete Backward Euler metod (no spatial discretization). Consider an arbitrary nonuniform mes (2.3) in te time direction and discretize te abstract parabolic problem (1.1) in time using te first-order Backward Euler metod (also referred to as te implicit Euler metod or te Discontinuous Galerkin metod dg()) as follows. We associate an approximate solution U j H 1 (Ω) C( Ω) wit te time level t j and require it to satisfy δ t U j + L j U j + f j = in Ω, j = 1,..., M; U = ϕ, (3.1a)

5 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 5 were δ t U j := U j U j 1, L j := L(t j ) and f j := f(, t j, U j ). (3.1b) For tis discretization, we give te following a posteriori error estimate. Teorem 3.1. Let u solve te problem (1.1) wit te parabolic operator M satisfying (1.2) and Condition 2.1, and U j solve te corresponding semidiscrete problem (3.1). Ten, for m = 1,..., M, one as U m u(, t m ) (κ 1 l m + κ 2 ) max U j U j 1 + 2κ U m U m 1 + κ m j=1,...,m 1 tj e γ j=1 t j 1 2 (t m s) [ϑ L + ϑ f ](, s) ds, (3.2) were ϑ L and ϑ f are regular functions defined, for t (t j 1, t j ], j = 1,..., M, by ϑ L (, t) := [L(t) L j ] U j, ϑ f (, t) := f(, t, U j ) f(, t j, U j ). (3.3) Proof. Let I t U be te standard piecewise-linear interpolant of U j in time: I t U(, t) := tj t U j 1 + t tj 1 U j for t [t j 1, t j ], j = 1,..., M. (3.4) Furtermore, we define a piecewise-constant interpolant Ũ of U j by Ũ(, t) := U j for t (t j 1, t j ], j = 1..., M; Ũ(, ) := U 1, (3.5) (so Ũ is continuous on [t, t 1 ]). Note tat te temporal derivative t Ũ is understood as a distribution, wile t (I t U) is a regular function, equal to δ t U j for t (t j 1, t j ] (in agreement wit our left-continuity convention). Consequently, (3.1a) implies tat t (I t U) + L(t)Ũ + f(x, t, Ũ) = ϑ L + ϑ f for (x, t) Ω (, T ]. (3.6) Here we also used te observation tat, by (3.5), te regular functions ϑ L and ϑ f of (3.3) can be rewritten for t (t j 1, t j ] as ϑ L = L(t)Ũ Lj U j and ϑ f = f(, t, Ũ) fj. Next, combining (3.6) wit (1.1a) yields MŨ Mu = tũ + L(t)Ũ + f(x, t, Ũ) = t[ũ I tu] + [ϑ L + ϑ f ] in Q. Now te desired bound for U m u(, t m ) = [Ũ u](, t m) is obtained by an application of Lemma 2.4 wit µ := Ũ I tu and ϑ := ϑ L + ϑ f, using te following two observations. First, we note tat [Ũ u µ](, ) = U 1 ϕ (U 1 ϕ) =. Second, for t (t j 1, t j ], one as µ = t j t (U j U j 1 ) = µ U j U j 1, t µ = U j U j 1. Tis completes te proof. Corollary 3.2. Under assumption (1.2), te a posteriori error estimate (3.2) applies to te model problem (1.3) wit ϑ f from (3.3), ϑ L =, and te constants κ, κ 1, κ 2 from Lemma 2.2.

6 6 NATALIA KOPTEVA AND TORSTEN LINSS 4. Semidiscrete Crank-Nicolson metod (no spatial discretization). Consider an arbitrary nonuniform mes (2.3) in te time direction and discretize te abstract parabolic problem (1.1) in time using te second-order Crank-Nicolson metod as follows. We associate an approximate solution U j H 1 (Ω) C( Ω) wit te time level t j and require it to satisfy δ t U j + 1 2( L j 1 U j 1 + L j U j) (f j 1 + f j ) = in Ω, j = 1,..., M, (4.1a) were we again let U = ϕ, δ t U j := U j U j 1, L j := L(t j ) and f j := f(, t j, U j ). (4.1b) To give an a posteriori error estimate for tis discretization, we will use te standard piecewise linear interpolation I t, wic, for any continuous function w = w(t), is defined by I t w(t) := t j t w(t j 1 ) + t t j 1 w(t j ) for t [t j 1, t j ], j = 1,..., M, (4.2) (tis definition is almost identical wit (3.4)). Teorem 4.1. Let u solve te problem (1.1) wit te parabolic operator M satisfying (1.2) and Condition 2.1, and U j solve te corresponding semidiscrete problem (4.1). Ten for m = 1,..., M, one as U m u(, t m ) 1 8 (κ 1 l m + κ 2 ) max τ j 2 δ t (L j U j + f j ) j=1,...,m κ τm 2 δ t (L m U m + f m ) + κ m j=1 tj t j 1 e γ 2 (t m s) [ϑl + ϑ f ](, s) ds. (4.3) Here ϑ L and ϑ f are regular functions defined, for t (t j 1, t j ], j = 1,..., M, by ϑ L := L(t) U I t [L(t) U], ϑ f := f(, t, U) I t [f(, t, U)], (4.4) were we use U(, t) := I t U(, t) of (3.4) and I t of (4.2). Remark 4.2. As te exact solution u of (1.1) satisfies 2 t u = t (Lu+f(x, t, u)), so te terms τ 2 j δ t(l j U j + f j ) in (4.3) can be considered discrete analogues of τ 2 j 2 t u. We also note tat tese terms can be bounded using τ 2 j δ tw j ( w j 1 + w j ) wit w j := L j U j + f j. Proof. Consider t [t j 1, t j ]. In view of (4.2), for any function w = w(t) wit te notation w j := w(t j ), one as I t w(t) = 1 2 (wj 1 + w j ) + (t t j 1/2 ) δ t w j. So using (4.4) and ten tis property, for t [t j 1, t j ], one easily gets f(, t, U) 1 2( f j 1 + f j) = { I t [f(, t, U)] 1 2( f j 1 + f j)} + ϑ f and a similar relation = (t t j 1/2 ) δ t f j + ϑ f, L(t) U 1 2( L j 1 U j 1 + L j U j) = { I t [L(t) U] 1 2( L j 1 U j 1 + L j U j)} + ϑ L = (t t j 1/2 ) δ t [L j U j ] + ϑ L.

7 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 7 Note also tat U(, t) = I t U(, t) implies tat δ t U j = t U for t (t j 1, t j ]. Combining tese tree observations wit (4.1a), one deduces tat t U + L(t) U + f(, t, U) = (t t j 1/2 ) δ t [L j U j + f j ] + [ϑ L + ϑ f ], (4.5) for t (t j 1, t j ] (ere te left-and side is a regular function). Next, combining (4.5) wit (1.1a) yields MU Mu = t U + L(t) U + f(x, t, U) = t µ + [ϑ L + ϑ f ] in Q, (4.6) were µ = µ(x, t) is a continuous function defined by µ(, t) := 1 2 (t j t)(t t j 1 ) δ t [L j U j + f j ] for t [t j 1, t j ]. (4.7) Tis is easily cecked by using te relation d dt [ 1 2 (t j t)(t t j 1 )] = t t j 1/2 to evaluate t µ. Now te desired bound (4.3) for U m u(, t m ) is obtained by an application of Lemma 2.4 to te equation (4.6) wit µ defined by (4.7), and ϑ := ϑ L + ϑ f, using te following two observations. First, note tat [U u µ](, ) = U 1 ϕ =. Second, for t (t j 1, t j ], one as µ δt (L j U j + f j ) and t µ δt (L j U j + f j ). Tis completes te proof. Corollary 4.3. Under assumption (1.2), te a posteriori error estimate (4.3) applies to te model problem (1.3) wit ϑ f from (4.4), ϑ L =, and te constants κ, κ 1, κ 2 from Lemma 2.2. Remark 4.4. If te term 1 2 (f j 1 + f j ) in te Crank-Nicolson discretization (4.1) is replaced by f j 1 tj := τj t j 1 f(, t, U) dt, ten te proof of Teorem 4.1 remains applicable wit te only modification tat te rigt-and side in (4.6) involves, for t (t j 1, t j ], an additional term ϑ f := 1 2 (f j 1 + f j ) f j. Consequently, te statement of Teorem 4.1 remains valid wit [ϑ L +ϑ f ] in te final line of (4.3) replaced by [ϑ L + ϑ f ϑ f ], were, as one can easily deduce, ϑ f = τ 1 tj j t j 1 ϑ f dt is te average value of ϑ f on [t j 1, t j ]. Remark 4.5. Te a posteriori error estimates given by Teorem 4.1 and Remark 4.4 resemble (but are not identical wit) error estimates of [1]. Our analysis of te semidiscrete Crank-Nicolson metod seems more straigtforward as we work wit te standard piecewise linear interpolant of te computed solution, wile te analysis in [1] involves a construction of a certain piecewise-quadratic polynomial of te computed solution in time. Furtermore, in Section 7, we derive a posteriori error estimates for fully discrete Crank-Nicolson metods, wic were not considered in [1]. 5. Elliptic a posteriori error estimators. In tis section, we consider a steady-state version of te abstract parabolic problem (1.1): Lv + g(x, v) = for x Ω, v = for x Ω, (5.1) and its discretizations in te form Find v V : L v + P [g(, v )] =, were V := V H 1 (Ω). (5.2a)

8 8 NATALIA KOPTEVA AND TORSTEN LINSS Here V C( Ω) is some finite-element space, and for te related Lagrange interpolation operator I onto V, we use some linear operators L and P suc tat L : H 1 (Ω) V I [g(, )], P v V + I v v C( Ω), P v = v v V. (5.2b) Note tat as any v V vanises on Ω, so V I [g(, )] = V I [g(, v )], so te definition (5.2) is consistent. Assumptions. We assume, for any admissible g, tat (i) tere exist unique solutions v and v of problems (5.1) and (5.2), respectively; (ii) an a posteriori error estimate is available for tese solutions in te form v v η ( V, v, g(, v ) ). (5.3) Note tat te availability of elliptic a posteriori error estimates, suc as (5.3), enables one to employ elliptic reconstructions in te a posteriori error estimation of te related parabolic problems. Moreover, L and P are not necessarily needed to be evaluated explicitly to compute te a posteriori estimator eiter for te elliptic problem or te parabolic problem Elliptic model problem. Many standard finite element discretizations of elliptic equations (including tose wit quadrature) allow a representation of type (5.2). For example, consider a steady-state elliptic version of our model problem (1.3) posed in a bounded polyedral domain Ω R n : ε 2 v + g(x, v) = for x Ω, v = for x Ω, z g(x, z) γ 2 >. (5.4) Wit a finite-element space V C( Ω) and V := V H 1 (Ω), a standard Galerkin finite element metod for tis problem can be described by Find v V : ε 2 v, χ + g(, v ), χ = χ V, (5.5) were, is eiter exactly te inner product, in L 2 (Ω), or some quadrature formula for,. Remark 5.1. Te discretization (5.5) is of type (5.2). Suppose, for example, tat ψ, χ = ψ, χ for all ψ, χ V. Ten L ϕ, χ = ε 2 ϕ, χ and P ψ, χ = ψ, χ, subject to (5.2b), for all ϕ H 1 (Ω), ψ C( Ω) and χ V. In particular, (i) if, :=, (i.e. no quadrature is used), ten P is te L 2 projection; (ii) if a quadrature of type ψ, χ := I ψ, χ is used, were I is te Lagrange interpolation operator associated wit V, ten P := I. Remark 5.2. Suppose tat one employs a quadrature of lumped-mass type defined by ψ, χ i = I (ψχ i ), 1 = ψ i χ i, 1 for all basis functions χ i of V, were ψ C( Ω) and ψ i χ i = I ψ. Ten again P := I, but L ϕ := a i χ i wit a i := ε 2 ϕ, χ i χ i,1 for interior mes nodes, and a i := [g(, )] i for boundary mes nodes. Consequently, L v is easily computable for any v V by applying te normalized stiffness matrix to te column vector of nodal values {v,i }. We now cite elliptic estimators of type (5.3) for particular cases of (5.4) and (5.5).

9 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS Elliptic model problem: regular regime. We first consider te steadystate version (5.4) of our model problem (1.3) in te regular regime of ε := 1. Let v solve te problem (5.4) wit ε = 1, γ, posed in a bounded polyedral domain Ω R n, n = 2, 3, and v solve te discrete problem (5.5) wit V and, defined as follows. Given a conforming and sape-regular triangulation T of Ω made of elements T, we let V be te space of continuous piecewise polynomial finite element functions of degree l 1, and V := V H 1 (Ω). We employ ϕ, ψ := T T Q T (ϕψ), were Q T is a quadrature formula for te integral over T wit positive weigts, and quadrature points contained in T, suc tat Q T is exact for te polynomials of degree q wit q max{2l 2, 1}. In [15, Teorem 4.2], an a posteriori error estimate of type (5.3) is given wit η = η defined by ( η V, v, g(, v ) ) [ := c max { 2 T ( v g(, v )) T T,T + T [ n u ]] }, T \ Ω + c 1 ν q n/2,t ln/2 + c 2 T ν q 1 ] n,t ln ln min 2, (5.6) were min is te smallest mes size, T is te diameter of T, [[ n u ] is te jump of te normal derivatives across an inter-element side, lp is te l p norm, and te quantity ν q n,t := T 1/n g(, v ) I,q [g(, v )],T is defined using te Lagrange interpolation operator I,q onto te space of piecewise polynomials of degree q Elliptic model problem: singularly-perturbed regime in one dimension. We now consider te steady-state version (5.4) of our model problem (1.3) in te singularly-perturbed regime of ε 1. Let v solve te problem (5.4) wit ε (, 1] and γ >, posed in te domain Ω := (, 1), and v solve te discrete problem (5.5) using te space V of continuous piecewise-linear finite element functions on an arbitrary nonuniform mes {x i } N i=1 wit = x < x 1 < < x N = 1 and i := x i x i 1. Note tat ere we make absolutely no mes regularity assumptions (as solutions of our problem typically exibit sarp layers so a suitable mes is expected to be igly-nonuniform; see, e.g., [13]). Consider two coices of,, wic are defined using te standard piecewiselinear Lagrange polynomial I onto V : ϕ, ψ := I ϕ, ψ, (quadrature) (5.7a) ϕ, ψ := I [ϕψ], 1. (lumped-mass quadrature) (5.7b) Remark 5.3. To illustrate Remarks 5.1 and 5.2, note tat te described two discretizations using eiter (5.7a) or (5.7b) are of type (5.2). In particular, for (5.7a), we get L := ε 2 [ x] 2 and P := I. Here te operator [ x] 2 : H 1 (Ω) V + ε 2 I [g(, )] is defined by [ x] 2 ϕ, χ = ϕ, χ for all ϕ H 1 (Ω), χ V. Consequently, te discrete problem using (5.7a) may be represented as ε 2 [ 2 x] v + I [g(, v )] =. (5.8a) By contrast, (5.7b) can be rewritten as a difference [ sceme: ε 2 δxv 2,i +g(x i, v,i ) =, for i = 1,..., N 1, were δxv 2 2,i := 1 i + i+1 i+1 (v,i+1 v,i ) 1 i (v,i v,i 1 ) ]

10 1 NATALIA KOPTEVA AND TORSTEN LINSS is te standard finite-difference operator. Letting δxv 2,i := ε 2 g(x i, v,i ) for i =, N and applying te linear interpolation I to {δxv 2,i } N i=, we can represent te discrete problem using (5.7b) as ε 2 I [δ 2 xv ] + I [g(, v )] =, (5.8b) were te values δxv 2,i are easily explicitly computable. ( We cite a posteriori error bounds [8, 12, 13] of type (5.3) wit η := η ε V, g(, v ) ) ( for (5.7a) and η := η ε; l.m. V, g(, v ) ) for (5.7b), respectively, defined by η ε ( V, g ) := max i=1,...,n η ε; l.m. ( V, g ) := ηε + were g := g(, v ). { 2 i 4ε 2 I g,(xi 1,x i) max i=1,...,n { 2 i 6γ ε x(i g ),(xi 1,x i ) } + γ 2 g I g,(,1), (5.9a) }, (5.9b) Remark 5.4. Te error estimators (5.9a) and (5.9b) are robust altoug tey involve negative powers of te small parameter ε. Indeed, an inspection of representations (5.8a) and (5.8b) for te two considered numerical metods sows tat ε 2 2 i I g = ε 2 2 i I [g(, v )] becomes 2 i [ 2 x] v or 2 i δ2 xv, so it approximates 2 i 2 xv, were v is te exact solution of our equation ε 2 xv 2 + g(, v) =. Similarly, te term ε 1 2 i x(i g ) approximates ε xv, 3 wic as similar magnitude to 2 i 2 xv in te layer regions. By contrast, if, :=, (i.e. no quadrature is used), ten one can obtain a { 2 i simpler-looking error estimate of type (5.3) wit η := max i=1,...,n 4ε g 2,(xi 1,x i )}. However, tis estimate is not robust. To see tis, split g = P g + (g P g ) using te standard L 2 projection P. Ten, instead of (5.8a), we ave te representation ε 2 [ x] 2 v + P [g ] = for our numerical metod. Te component ε 2 2 i P g approximates 2 i 2 xv so it yields a robust part of te estimator. But te oter component ε 2 2 i g P g may be as large as O(ε 2 4 i ), wic may become quite large if ε is small compared to te local mes size. For tis numerical metod one can, in fact, obtain a robust error estimator, wic is almost identical wit (5.9a), only I in η ε sould be replaced by P (but tis latter estimator is less practical, as it requires te L 2 projection P g to be explicitly computed). 6. Fully discrete Backward Euler metod. To fully discretize te abstract parabolic problem (1.1), we now apply a spatial discretization of type (5.2) to te semidiscrete problem (3.1) as follows. We associate a finite-element space V j C( Ω) and a computed solution u j V j := V j H1 (Ω) wit te time level t j and require, for j = 1,..., M, tat L j uj + Pj [f(, t j, u j ) + δ t u j ] =, (6.1a) wit some u ϕ in V. Here Lj and Pj are also associated wit te time level t j and, in agreement wit (5.2b), for te Lagrange interpolation operator I j onto V j, we let L j : H1 (Ω) V j Ij [f(, t j, )], P j v V j + Ij v v C( Ω), P j v = v v V j. (6.1b) Note tat as bot u j and δ t u j vanis on Ω, so V j Ij [f(, t j, )] coincides wit V j Ij [f(, t j, u j ) + δ t u j ], so te definition (6.1) is consistent.

11 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 11 Te term δ t u j in (6.1a) approximates tu and is defined by δt u j := uj ûj 1, were û := u τ. (6.1c) j Te operator δt is identical wit δ t of (3.1b) for j = 1, wile for j > 1 it involves te intermediate computed solution û j 1 H 1 (Ω) tat we associate wit te time level t + j 1 (tis is indicated by te at notation). We do not presently specify û j 1, but note two particular cases of interest: Case A: Case B: û j 1 û j 1 := u j 1 δt u j j 1 span{ V, V j }, (6.2a) := I j u j 1, I j : V j 1 V j δ t u j V j. (6.2b) Here I j is some interpolation operator suc tat I j u j 1 Note tat if V j 1 := u j 1 if V j 1 V j. V j (wic includes te case of V j = V being fixed for all j =,..., M), ten Cases A and B are identical. Note also tat to define I j in Case B, one may employ, e.g., te standard Lagrange interpolation or te L 2 projection A posteriori error estimate using piecewise-constant elliptic reconstructions. To estimate te error of te fully discrete Backward-Euler metod (6.1), we sall employ te elliptic reconstruction, wic was introduced in te recent papers [14, 11, 3] as a counterpart of te Ritz-projection in te a posteriori error estimation for parabolic problems. We associate an elliptic reconstruction R j H 1 (Ω) C( Ω) wit te time level t j and require, for x Ω, j = 1,..., M, tat L j R j + g j (x, R j ) =, were g j (x, v) := f(x, t j, v) + δ t u j, (6.3) so tis relation may be considered somewat between (3.1a) and (6.1a). On te oter and, (6.3) is a version of te elliptic problem (5.1) wit L := L j and g := g j, wile te numerical metod (5.2) applied to tis problem is identical wit (6.1) and yields te computed solution u j. Furtermore, applying te elliptic a posteriori error estimate (5.3) to te exact solution R j and te corresponding computed solution u j, one gets η j := R j u j η ( V j, uj, gj (, u j )) for j = 1,..., M. (6.4) We now give an a posteriori error estimate for te fully discrete metod (6.1). Teorem 6.1. Let u solve te problem (1.1), (1.2) wit te parabolic operator M satisfying Condition 2.1, u j solve te discrete problem (6.1), and Rj be te elliptic reconstruction defined by (6.3) and satisfying (6.4). Ten for m = 1,..., M, one as u m u(, t m ) κ e γ2 t m u ϕ { + (κ 1 l m + κ 2 ) max u j j=1,...,m 1 uj 1 + η j} + 2κ u m u m 1 + (κ + 1) η m m tj + κ 2 (t m s) [ϑ L,R + ϑ f,r ](, s) ds j=1 + κ m j=2 t j 1 e γ e γ2 (t m t j) û j 1 u j 1, (6.5)

12 12 NATALIA KOPTEVA AND TORSTEN LINSS were ϑ L,R and ϑ f,r are regular functions defined, for t (t j 1, t j ], j = 1,..., M, by ϑ L,R (, t) := [L(t) L j ] R j, ϑ f,r (, t) := f(, t, R j ) f(, t j, R j ). (6.6) Remark 6.2. Te final term in (6.5) vanises in Case A of (6.2); in particular, wen one as V j 1 V j for all j = 1,..., M. Proof. In view of (6.4), to get te desired bound (6.5) for u m u(, t m), it suffices to obtain a bound of type (6.5) for R m u(, t m ) only wit (κ + 1) replaced by κ, and ten apply te triangle inequality. So we focus on estimating R m u(, t m ). We partially imitate te proof of Teorem 3.1. Let I t u be a standard piecewiselinear interpolant of u j in time: I t u (, t) := t j t u j 1 + t t j 1 u j for t [t j 1, t j ], j = 1,..., M. (6.7) Furtermore, we define a piecewise-constant interpolant R of R j in time by R(, t) := R j for t (t j 1, t j ], j = 1..., M; R(, ) := R 1, (6.8) (so R is continuous on [t, t 1 ]; compare wit Ũ of (3.5)). Note tat te temporal derivative t R is understood in te sense of distributions, wile t (I t u ) is a regular function. Now, in view of (6.3), one gets t (I t u ) + L(t) R + f(, t, R) = ϑ L,R + ϑ f,r + ϑ in Q, (6.9) were ϑ is a regular function defined by ϑ (, t) := t (I t u ) δ t u j for t (t j 1, t j ]. (6.1) To get (6.9), we also used te observation tat, by (6.8), te regular functions ϑ L,R and ϑ f,r of (6.6) can be rewritten for t (t j 1, t j ] as ϑ L,R = L(t) R L j R j and ϑ f,r = f(, t, R) f(, t j, R j ). Next, combining (6.9) wit (1.1a) yields M R Mu = t R + L(t) R + f(, t, R) = t [ R I t u ] + [ϑ L,R + ϑ f,r + ϑ ]. Now te desired bound of type (6.5) for R m u(, t m ) = [ R u](, t m ), only wit (κ +1) replaced by κ, is obtained by an application of Lemma 2.4 wit µ := R I t u and ϑ := ϑ L,R + ϑ f,r + ϑ, using te following tree observations. First, note tat [ R u µ](, ) = R 1 ϕ (R 1 u ) = u ϕ. (6.11) Next, for t (t j 1, t j ], we ave µ = R j u j + tj t (u j uj 1 ). Tus, µ R j u j + uj uj 1 and t µ = u j uj 1, (6.12) were R j u j = η j. Finally, (6.1) combined wit (6.1c), (6.7) implies tat ϑ (, t) = 1 (û j 1 u j 1 ) for t (t j 1, t j ]. Terefore, tj t j 1 e γ 2 (t m s) ϑ (, s) ds e γ2 (t m t j ) û j 1 u j 1, (6.13)

13 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 13 were û u =. Te tree observations (6.11), (6.12), (6.13) yield te required bound for R m u(, t m ). Teorem 6.1. Te statement of Teorem 6.1 remains valid wit te terms u j uj 1 and u m um 1 in (6.5) respectively replaced by u j ûj 1 and u m ûm 1. Proof. We imitate te proof of Teorem 6.1, but wit I t u of (6.7) replaced by te piecewise-continuous interpolant I t u (, t) := t j t û j 1 + t t j 1 u j for t (t j 1, t j ], j = 1,..., M, (6.14) wit I t u (, ) := û = u. Furtermore, ϑ is defined not by (6.1), but by ϑ (, t) := t (I t u ) δ t u j = [ûj 1 u j 1 ] δ(t t + j 1 ) for t (t j 1, t j ], (6.15) were δ( ) is te one-dimensional Dirac δ-distribution. (Note tat û = u and te rigt-continuity convention at t = imply tat ϑ = on [, t 1 ].) So instead of (6.13) we get tm e γ2 (t m s) ϑ (, s) ds m j=2 e γ2 (t m t j 1) ûj 1 u j 1. (6.16) Te required bound for R m u(, t m ) = [ R u](, t m ) is again obtained by an application of Lemma 2.4 only wit µ := R It u, for wic we ave a version of (6.12) wit u j 1 replaced by û j 1. Remark 6.3. Te terms ϑ L,R and ϑ f,r in (6.5) involve te elliptic reconstruction R j. In view of te bound (6.4), teir discrepancy from ϑ L,u and ϑ f,u, respectively, can be easily estimated. E.g., for ϑ f,r wit t (t j 1, t j ], we ave [ϑ f,r ϑ f,u ](, t) η j η j sup z f(, t, z) z f(, t j, z) (t j 1,t j ] R sup (t j 1,t j] R t z f(, t, z), were η j is estimated using (6.4). In fact, if t z f C, ten te discrepancy [ϑ f,r ϑ f,u ](, t) between ϑ f,r and ϑ f,u becomes O( η j ), i.e. negligible compared wit te terms η j tat explicitly appear in (6.5) Applications to te model problem (1.3). Consider a fully discrete Backward-Euler metod for (1.3), obtained by applying te spatial discretization (5.5) to a version of te semidiscrete Backward-Euler metod (3.1): Find u j V j : ε2 u j, χ + f(, t j, u j ) + δ t u j, χ = χ V j, (6.17) were, is eiter exactly te inner product, in L 2 (Ω), or some quadrature formula for,, and δt u j is defined by (6.1c), (6.2). Note tat te full discretization (6.17) is of type (6.1). For some particular cases of,, te operators L j and Pj are defined as in Remarks 5.1 and 5.2 only using V j instead of V.

14 14 NATALIA KOPTEVA AND TORSTEN LINSS Model problem (1.3): regular regime. Let u solve te problem (1.3) wit ε = 1, γ, posed in a bounded polyedral spatial domain Ω R n, n = 2, 3, and u j solve te discrete problem (6.17) wit V j and, defined, for eac time level t j, as in 5.2. To be more specific, we let T j be a conforming and sape-regular triangulation of Ω made of elements T, V j be te space of continuous piecewise polynomial finite element functions of degree l 1, and V j := V j H1 (Ω). We ten employ a quadrature formula ϕ, ψ := T T j Q T (ϕψ), as described in 5.2. Corollary 6.4. Let te above numerical metod be applied to problem (1.3) wit ε = 1, γ. Ten te a posteriori error estimates of Teorems 6.1 and 6.1 are valid wit ϑ L,R =, ϑ f,r computed as described in Remark 6.3, and η j ( := η V j, uj, f(, t j, u j ) + δ t u j ) for j = 1,..., M, were η is defined in (5.6). Remark 6.5. Te Backward-Euler metod for a linear version of (1.3) wit ε = 1 was considered in [4, 3]. Te a posteriori error estimate (6.5) of Corollary 6.4 resembles (but is not identical wit) te one of [4, (1.13)] in tat it involves terms suc as u j uj 1, tat may be interpreted as approximating t u. (Note also tat [4, (1.13)] is given witout proof, and does not appear to be proved elsewere). By contrast, te a posteriori error estimates of [3] include terms (denoted tere by g j g j 1 ) tat may be interpreted as approximating te quantity t 2 u +..., wic seems less suitable for a first-order metod in time Model problem (1.3): singularly perturbed regime in one dimension. Now, consider ε 1. Let u solve (1.3) wit ε (, 1], γ >, posed in te domain Ω := (, 1). Let u solve te discrete problem (6.17) wit V j and, defined, for eac time level t j, as in 5.3. Tus V j is te space of continuous piecewise-linear finite element functions on an arbitrary nonuniform mes { x j } N i i=1 wit = x j < xj 1 < < xj N = 1 under absolutely no mes regularity assumptions. We consider te two coices (5.7a) and (5.7b) of,, using te piecewise-linear Lagrange interpolant I := I j onto V j. Corollary 6.6. Let te above numerical metod be applied to problem (1.3) wit ε (, 1], γ >, Ω := (, 1). Ten te a posteriori error estimates of Teorems 6.1 and 6.1 are valid wit ϑ L,R =, ϑ f,r computed as described in Remark 6.3, and { ( η j η ε V j :=, f(, t j, u j ) + δ t u j ) for (5.7a), ( η ε; l.m. V j, f(, t j, u j ) + δ t u j ) for (5.7b), for j = 1,..., M, were η ε and η ε; l.m. are defined in (5.9), wit I replaced by I j. Remark 6.7. Te a posteriori error estimators of Corollary 6.6 are robust as te argument of Remark 5.4 applies to η j wit v replaced by u(, t j ), so ε 2 2 i I g approximates 2 i 2 xu(, t j ), wile te term ε 1 2 i x(i g ) approximates ε 3 xu(, t j ), wic as similar magnitude to 2 i 2 xu(, t j ) in te layer regions. Remark 6.8. Consider te term g I j g in te error estimators of Corollary 6.6 for Cases A and B of (6.2). In Case B, one as δt u j Ij [δ t u j ] =, ence g I j g simplifies to f(, t j, u j ) Ij f(, t j, u j ). In Case A, te final term in (6.5) vanises (see Remark 6.2). However, g I j g again involves f(, t j, u j ) Ij f(, t j, u j ) and, furtermore, δ t u j Ij [δ t u j ] = 1 (u j 1 I j [uj 1 ]).

15 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 15 Interestingly, Case A and Case B wit I j := I j are identical, but, in view of te above, yield different error estimators. Note tat one seems to get a sarper estimator wen tis metod is interpreted as Case B wit I j := I j. 7. Fully discrete Crank-Nicolson metod. We now describe a full discretization of Crank-Nicolson type for te abstract parabolic problem (1.1). To tis end, we apply a spatial discretization of type (5.2) to te semidiscrete problem (4.1) as follows. A finite-element space V j C( Ω) and a computed solution u j V j := V j H1 (Ω) are associated wit te time level t j, wile a computed solution û j 1 V j is associated wit te time level t + j 1 (tis is indicated by te at notation). We require, for j = 1,..., M, tat δ t u j ( ˆLj 1 û j 1 + L j ) uj Pj [ f(, tj 1, û j 1 ) + f(, t j, u j )] = (7.1a) wit some u ϕ in V. Here Lj and Pj are associated wit te time level t j, j 1 wile ˆL is associated wit te time level t + j 1. In agreement wit (5.2b), for te Lagrange interpolation operator I j onto V j, we let L j : H1 (Ω) V j Ij [f(, t j, )], ˆLj 1 : H 1 (Ω) V j Ij [f(, t j 1, )], P j v V j + Ij v v C( Ω), P j v = v v V j. (7.1b) Note tat any v V j vanises on Ω, so V j I [f(, t k, )] = V j I [f(, t k, v )] for k = j 1, j, wile we also impose tat δt u j V j. So te definition (7.1) is consistent. Te term δt u j in (7.1a) approximates tu and is identical wit (6.1c): δt u j := uj ûj 1, were û := u τ. (7.1c) j Te operator δt is identical wit δ t of (4.1b) for j = 1, wile for j > 1 it involves defined by û j 1 û j 1 := I j u j 1, I j j 1 : V V j = δt u j V j. (7.1d) Here I j is some linear interpolation operator suc tat I j u j 1 = u j 1 if V j 1 V j. Tus, in tis section we restrict our analysis to Case B of (6.2b). Note tat if V j 1 V j (wic includes te case of V j = V being fixed for all j), ten û j 1 = u j 1. Also, to define I j, one may employ, e.g., te standard Lagrange interpolation or te L 2 projection A posteriori error estimate using piecewise-linear elliptic reconstructions. We associate elliptic reconstructions R j, ˆR j 1 H 1 (Ω) C( Ω), respectively, wit te time levels t j and t + j 1, and require, for x Ω, j = 1,..., M, tat were L j R j + g j (x, R j ) =, L j 1 ˆRj 1 + ĝ j 1 (x, ˆR j 1 ) =, (7.2a) } g j (, v) := f(, t j, v) {L j uj + Pj [f(, t j, u j )], (7.2b) { } ĝ j 1 (, v) := f(, t j 1, v) ˆLj 1 û j 1 + P j [f(, t j 1, û j 1 )]. (7.2c) Combining (7.2) wit (7.1a) immediately yields δ t u j + 1 2[ L j 1 ˆRj 1 + L j R j] + 1 2[ f(, tj 1, ˆR j 1 ) + f(, t j, R j ) ] =, (7.3)

16 16 NATALIA KOPTEVA AND TORSTEN LINSS wic may be considered somewat between (4.1a) and (7.1a). Note tat (7.2a) describes two versions of te elliptic problem (5.1) wit L := L j, g := g j, and wit L := ˆL j 1, g := ĝ j 1, and exact solutions R j and ˆR j 1, respectively. Furtermore, te numerical metod (5.2), using te finite element space V j, applied to tese two problems yields L j Rj + Pj [gj (x, R j )] =, ˆLj 1 ˆR j 1 + P j [ĝj 1 (x, ˆR j 1 ) ] =. (7.4) We ave assumed tat solutions of tese two discrete problems are unique. Tus, R j = uj j 1 and ˆR = û j 1. Tis is easily cecked by combining (7.4) wit te definitions of g j and ĝ j 1 in (7.2), in wic te terms { } V j. Consequently, applying te elliptic a posteriori error estimate (5.3) to te exact solutions R j and ˆR j 1 and te corresponding computed solutions u j and ûj 1, one gets, for j = 1,..., M, η j := R j u j η ( V j, uj, gj (, u j )), ˆη j 1 := ˆR j 1 û j 1 η ( V j, ûj 1 (7.5a), ĝ j 1 (, û j 1 ) ). (7.5b) To formulate our a posteriori error estimate for u u, we generalize te piecewise linear interpolation I t of (4.2) to any left-continuous function w = w(t) by setting I t w(t) := tj t w(t + j 1 ) + t tj 1 w(t j ) for t (t j 1, t j ], j = 1,..., M. (7.6) In a similar manner, we apply I t to te elliptic reconstruction (7.2) and define R(, t) := I t R(, t) := tj t ˆRj 1 + t tj 1 R j for t (t j 1, t j ], j = 1,..., M. (7.7) (In agreement wit te adapted left-/rigt-continuity convention, bot R and I t w are rigt-continuous at t =.) Teorem 7.1. Let u solve (1.1), (1.2) wit a parabolic operator M satisfying Condition 2.1, u j solve te discrete problem (7.1), and Rj, ˆRj 1 be te elliptic reconstructions defined by (7.2) and satisfying (7.5). Ten, for m = 1,..., M, one as u m u(, t m ) κ e γ2 t m u ϕ { 1 δ t Ψ j + η j + ˆη j 1} + (κ 1 l m + κ 2 ) max j=1,...,m κ τm 2 δt Ψ m + (2κ + 1) η m + 2κ ˆη m 1 + κ m j=1 + κ m j=2 tj t j 1 e γ 2 (t m s) [ϑ L,R + ϑ f,r ](, s) ds e γ2 (t m t j 1) û j 1 Here, for j = 1,..., M, we use δ t Ψ j = 1 (Ψ j ˆΨ j 1 ) wit Ψ j := L j uj + Pj [f(, t j, u j )], ˆΨj 1 := u j 1. (7.8) j 1 ˆL û j 1 + P j [f(, t j 1, û j 1 )], (7.9) wile ϑ L and ϑ f are regular functions defined, for t (t j 1, t j ], j = 1,..., M, by ϑ L,R := L(t) R I t [L(t) R], ϑ f,r := f(, t, R) I t [f(, t, R)], (7.1)

17 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS 17 wit I t and R from (7.6) and (7.7). Proof. In view of (7.5a), to get te desired bound (7.8) for u m u(, t m), it suffices to obtain a bound of type (7.8) for R m u(, t m ), wit (2κ + 1) replaced by 2κ, and ten apply te triangle inequality. So we consider R m u(, t m ) only. We partially imitate te proof of Teorem 4.1. Let t [t j 1, t j ]. In view of (7.6), for any left-continuous function w = w(t) wit te notation w j := w(t j ) and ŵ j 1 := w(t + j 1 ), one as I t w(t) = 1 2 (ŵj 1 + w j ) + (t t j 1/2 ) δ t w j. So combining (7.1) and tis property, for t (t j 1, t j ], one easily gets f(, t, R) 1 2( f(, tj 1, ˆR j 1 ) + f(, t j, R j ) ) = (t t j 1/2 ) δ t [f(, t, R)] j + ϑ f,r, and a similar relation L(t) R 1 2( L j 1 ˆRj 1 + L j R j) = (t t j 1/2 ) δ t [L(t) R] j + ϑ L,R. Combining tese two observations wit (7.3), one deduces tat δ t u j + L(t) R + f(, t, R) = (t t j 1/2) δ t [L(t) R + f(, t, R)] j + [ϑ L,R + ϑ f,r ] were δ t u j = 1 (Ψ j ˆΨ j 1 ) wit = (t t j 1/2 ) δ t Ψ j + [ϑ L,R + ϑ f,r ], Ψ j := L j R j + f(, t j, R j ) and ˆΨj 1 := L j 1 ˆRj 1 + f(, t j 1, ˆR j 1 ). In view of (7.2), tese definitions of Ψ j and ˆΨ j 1 are equivalent to (7.9). Finally, δ t u j + L(t) R + f(, t, R) = tµ 1 (, t) + [ϑ L,R + ϑ f,r ] (7.11) for t (t j 1, t j ], were µ 1 = µ 1 (x, t) is a continuous function defined by µ 1 (, t) := 1 2 (t j t)(t t j 1 ) δ t Ψ j for t [t j 1, t j ]. (7.12) Tis is easily cecked by using te relation d dt [ 1 2 (t j t)(t t j 1 )] = t t j 1/2 to evaluate t µ 1. Next, we sall invoke I t u defined by (6.14), for wic we ave (6.15) and (6.16). Combining (7.11) wit (1.1a) and ten (6.15) yields MR Mu = t R + L(t) R + f(, t, R) = t (R I t u ) + t µ 1 (, t) + [ϑ L,R + ϑ f,r + ϑ ] in Q. (7.13) So te desired bound of type (7.8) for R m u(, t m ), only wit (2κ + 1) replaced by 2κ, is obtained by an application of Lemma 2.4 to (7.13) wit µ := µ + µ 1 := (R It u ) + µ 1 and ϑ := ϑ L,R + ϑ f,r + ϑ, using (6.16) and te following two observations. First, [R u µ](, ) = ˆR ϕ [( ˆR u ) + ] = u ϕ. Second, for t (t j 1, t j ], we ave µ R j u j + ˆR j 1 û j 1, t µ R j u j + ˆR j 1 û j 1, µ τ j 2 δ t Ψ j, t µ τ j 2 δ t Ψ j, were we used µ = R I t u = I t (R u ) and (7.12). Finally, recall (7.5) to obtain te required bound for R m u(, t m ).

18 18 NATALIA KOPTEVA AND TORSTEN LINSS Remark 7.2. It is essential for te computability of te error estimator (7.8) tat some explicitly computable bounds are available for η j, ˆη j 1 and δt Ψ j. For tis, it is sufficient (but not necessary) tat one can explicitly compute Ψ j and ˆΨ j 1. Indeed, (7.5) and (7.9) sow tat η j and ˆη j 1 are computable via te functions g j and ĝ j 1 of (7.2b), (7.2c), wile te latter can be represented as g j (, v) = f(, t j, v) Ψ j, ĝ j 1 (, v) = f(, t j 1, v) ˆΨ j 1. (7.14) Te remaining terms ϑ L,R and ϑ f,r in (6.5) can be computed using ϑ L,I t u and ϑ f,i t u as described in Remark 6.3. For example, if z t 2 f C for some constant C, ten ϑ f,r can be effectively replaced by ϑ f,i t u as a calculation sows, for t (t j 1, t j ], tat [ϑ f,r ϑ f,i t u ](, t) Cτj 2(ηj + ˆη j 1 ) Applications to te model problem. Estimator computability. Consider a fully discrete Crank-Nicolson metod for (1.3), obtained by applying te spatial discretization (5.5) to te semidiscrete problem (4.1): Find u j V j suc tat ε (ûj 1 + u j ), χ [f(, t j 1, û j 1 ) + f(, t j, u j )] + δ t u j, χ =, (7.15) χ V j, were, is eiter exactly te inner product, in L 2 (Ω), or some quadrature formula for,, and δt u j is defined by (7.1c), (7.1d). j 1 Note tat te full discretization (7.15) is of type (7.1) wit ˆL := L j. For some particular cases of,, te operators L j and Pj are defined as in Remarks 5.1 and 5.2 only using V j instead of V Model problem (1.3): regular regime. Let u solve problem (1.3) wit ε = 1, γ, posed in a bounded polyedral spatial domain Ω R n, n = 2, 3, and u j solve te discrete problem (7.15) wit V j and, defined, for eac time level t j, as in 5.2. To be more specific, we let T j be a conforming and sape-regular triangulation of Ω made of elements T, V j be te space of continuous piecewise polynomial finite element functions of degree l 1, and V j := V j H1 (Ω). We ten employ a quadrature formula ϕ, ψ := T T j Q T (ϕψ), as described in 5.2. Corollary 7.3. Let te above numerical metod be applied to problem (1.3) wit ε = 1, γ. Ten te a posteriori error estimate of Teorem 7.1 is valid wit ϑ L,R = and ϑ f,r computed as described in Remark 7.2. Te quantities η j and ˆη j 1 satisfy (7.5) combined wit (7.14), were η := η is defined in (5.6) Model problem (1.3): singularly perturbed regime in one dimension. Now consider te regime of ε 1. Let u solve te problem (1.3) wit ε (, 1], γ >, posed in te domain Ω := (, 1), and u solve te discrete problem (7.15) wit V j and, defined, for eac time level t j, as in 5.3. Tus V j is te space of continuous piecewise-linear finite element functions on an arbitrary } Nj i=1 wit = xj < xj 1 < < xj N j = 1 under absolutely no nonuniform mes { x j i mes regularity assumptions. We consider te two coices (5.7a) and (5.7b) of,, using te piecewise-linear Lagrange polynomial I := I j onto V j. Corollary 7.4. Let te above numerical metod be applied to problem (1.3) wit ε (, 1], γ >, Ω := (, 1). Ten te a posteriori error estimate of Teorem 7.1 is valid wit ϑ L,R = and ϑ f,r computed as described in Remark 7.2. Te quantities η j and ˆη j 1 satisfy (7.5) combined wit (7.14), were η := η ε for (5.7a), and η := η ε; l.m. for (5.7b), wile η ε and η ε; l.m. are defined in (5.9a) and (5.9b), respectively, in wic I is now replaced by I j.

19 A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS Computability. In view of Remark 7.2, we now furter discuss te computability of te error estimator (7.8) wen applied to te model problem (1.3). (i) Suppose tat in (7.15), one employs a lumped-mass quadrature ψ, χ i. Ten P j := Ij is te Lagrange interpolation operator onto V j, and Lj v = v is easily computable for any v V j by applying te normalized stiffness matrix to te column vector of nodal values {v,i }; see Remark 5.2. Consequently, η j, ˆη j 1 and δt Ψ j are explicitly computable, as described in Remark 7.2. (ii) Let V j V j 1 for all j. In general, (7.1a) combined wit (7.9) yields δt u j + ( 1 ˆΨj Ψ j ) =. In our case, one as ˆΨj 1 = Ψ j 1 so Ψ j = Ψj 1 2δt u j is explicitly computable for j = 1,..., M provided tat Ψ is computed (as in (iii)). (iii) In te general case, te computation of L j v j 1, and ence of ˆΨ and Ψ j, may be more expensive. Rougly speaking, L j v can be obtained by an application of M 1 j K j to te column vector of nodal values {v,i }, were M j is te mass matrix and K j is te stiffness matrix associated wit te time level t j. Note tat te computation of u j at eac time level by te Crank-Nicolon metod already involves an application of M 1 j. Furtermore, in some cases, an inversion of M j can be entirely avoided by using bounds of te type δ t Ψ j M 1 j δ t (M j Ψ j ), were M 1 j denotes te associated matrix norm (wic may be bounded a priori). ˆL j 1 8. Proof of Lemma 2.2. First, note tat te Green s function G associated wit our problem (1.3) in te spatial domain Ω and te Green s function Ĝ for te related problem ˆMû := t û û + f(x/ε, t, û) = in te spatial domain ˆΩ := Ω/ε satisfy s k G(x, t;, s) 1,Ω = s kĝ(x/ε, t;, s) 1,ˆΩ for k =, 1. Consequently, it suffices to prove Condition 2.1 for te case of ε = 1 wit κ, κ 1 and κ 2 independent of Ω, so trougout te proof we set L = in (2.1a). (i) We start by proving te first bound in Condition 2.1. Te Green s function Ḡ associated wit M := t + γ 2 in te domain Ω := R n can be easily obtained from te fundamental solution of te eat equation (te latter can be found, e.g., in [17, III.3], [5, 2.3.1]. So one gets t ( ) Ḡ(x, t; ξ, s) = g(x ξ, t s), were g(x, t) := e γ2 (4πt) exp x 2. (8.1) n/2 4t Next, note tat, by (1.2), te coefficient a in (2.1a) satisfies a γ 2 so an application of te maximum principle to problem (2.1) yields G Ḡ. Finally, note tat Ḡ(x, t; ξ, s) dξ = e γ2 (t s) ψ(ζ) dζ, were ψ(ζ) := e ζ 2 ξ x, ζ := πn/2 2 t s. (8.2) As ψ(ζ) dζ = 1, we immediately get R Ḡ(x, t;, s) n 1,Ω 1, wic yields te first bound in Condition 2.1 wit κ = 1. (ii) Next, we prove te second bound in Condition 2.1 in te linear case of f(x, t, z) = a(x)z + b(x, t) wit κ 2 =. In tis case, te differential operator in (2.1) does not involve s, so one can invoke [2, Corollary 5] (in using tis result, we imitate te proof of [3, Lemma 2.1]). In view of te above bound G Ḡ, an application of [2, Corollary 5] wit β = 2, γ = 1, c 1 = 1 4, c 2 = 4 9 c 1 and α(t) = e γ2 t (4πt) n/2 yields s G(x, t; ξ, s) 18c 1 c 2 (t s) 1 α( 1 2 [t s]) e (c2/c1) ζ 2, were ζ is cosen as in part (i) of tis proof. Now an observation similar to (8.2) leads to te estimate s G(x, t;, s) 1,Ω κ 1 (t s) 1 e 1 2 γ2 (t s), wic immediately implies te second bound in Condition 2.1 wit κ 2 =.

20 2 NATALIA KOPTEVA AND TORSTEN LINSS (iii) It remains to establis te second bound in Condition 2.1 in te general case of f(x, t, z) satisfying (1.2), wic implies tat for te coefficient a in (2.1a) one as γ 2 a(ξ, s) γ 2. For any fixed (x, t) Ω (, T ], consider te Green s function Ĝ(x, t; ξ, s) =: ˆΓ(ξ, s) associated wit te operator t + γ 2 in te domain Ω so ˆΓ(ξ, s) satisfies a version of (2.1) wit a replaced by γ 2. Comparing tis problem wit te problem (2.1) for Γ and noting tat L = L =, we find tat for any fixed (x, t), te function v(ξ, s) := ˆΓ(ξ, s) Γ(ξ, s) solves te terminal-value problem [ s + γ 2 ] v(ξ, s) = F (ξ, s) for (ξ, s) Ω [, t), (8.3a) v(ξ, t) = for ξ Ω, (8.3b) v(ξ, s) = for (ξ, s) Ω [, t], (8.3c) were F (ξ, s) := [a(ξ, s) γ 2 ] Γ(ξ, s) so, using Γ Ḡ and (8.1), F (ξ, s) ( γ 2 γ 2 ) g(x ξ, t s). (8.3d) Note tat in part (ii) we ave sown tat ˆΓ satisfies te second bound in Condition 2.1 wit κ 2 =. So it remains to sow tat v satisfies te second bound in Condition 2.1 wit κ 1 = and κ 2 = ( γ 2 γ 2 )ˆκ 2. Tis latter bound is immediately obtained by an application of Lemma 8.1 below to te terminal-value problem (8.3). Te next lemma is applied to te terminal-value problem (8.3), but it is convenient to formulate it in te context of an initial-value problem. Lemma 8.1. Let v satisfy [ t + γ 2 ] v = F in Q and vanis for t = and x Ω, were F (x, t) g(x x, t) wit g from (8.1) and some x Ω. Ten T tv(, t) 1,Ω dt ˆκ 2, were ˆκ 2 is independent of Ω, and ˆκ 2 = ˆκ 2 (γ) if γ >, wile ˆκ 2 = ˆκ 2 (T ) if γ =. Proof. Witout loss of generality, assume tat x = Ω so F (x, t) g(x, t). Recall tat Mg = wit M = t + γ 2 ; tis implies tat M[tg] = g, so an application of te maximum principle yields v(x, t) t g(x, t). (8.4) (i) First we establis te desired estimate wit ˆκ 2 tat depends on Ω. Let w(x, t) := ϱ(t) v wit te weigt ϱ := t 1 3 e 1 2 γ2t so ϱ = ( 1 3 t γ2 ) ϱ. Note tat t v 1,Ω [,T ] ϱ 1 2,Ω [,T ] ϱ t v 2,Ω [,T ] ) ˆκ 3 Ω 1 2 ( t w 2,Ω [,T ] + ϱ v 2,Ω [,T ], (8.5) were we used ϱ t v = t w ϱ v and T ϱ 1 2 2,Ω [,T ] = Ω t 2 3 e γ 2t dt =: Ω ˆκ 2 3 (so ˆκ 2 3 3T 1/3 for γ, and t 2 3 e t dt 2.7 implies ˆκ γ 2/3 for γ > ). To estimate t w in (8.5), we note tat Mw = ϱ F + ϱ v ϱ g + ϱ v and so apply an a priori estimate [1, (6.6) of Capter III]: t w 2,Ω [,T ] Mw 2,Ω [,T ] (8.6)