c 2017 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vol. 55, No. 6, pp c 2017 Society for Industrial and Applied Matematics GUARANTEED, LOCALLY SPACE-TIME EFFICIENT, AND POLYNOMIAL-DEGREE ROBUST A POSTERIORI ERROR ESTIMATES FOR HIGH-ORDER DISCRETIZATIONS OF PARABOLIC PROBLEMS ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK Abstract. We consider te a posteriori error analysis of approximations of parabolic problems based on arbitrarily ig-order conforming Galerkin spatial discretizations and arbitrarily ig-order discontinuous Galerkin temporal discretizations. Using equilibrated flux reconstructions, we present a posteriori error estimates for a norm composed of te L 2 (H 1 ) H 1 (H 1 )-norm of te error and te temporal jumps of te numerical solution. Te estimators provide guaranteed upper bounds for tis norm witout unknown constants. Furtermore, te efficiency of te estimators wit respect to tis norm is local in bot space and time, wit constants tat are robust wit respect to te mes-size, time-step size, and te spatial and temporal polynomial degrees. We furter sow tat tis norm, wic is key for local space-time efficiency, is globally equivalent to te L 2 (H 1 ) H 1 (H 1 )-norm of te error, wit polynomial-degree robust constants. Te proposed estimators also ave te practical advantage of being robust wit respect to refinement and coarsening between te time steps. Key words. parabolic partial differential equations, a posteriori error estimates, local spacetime efficiency, polynomial-degree robustness, ig-order metods (1.1) AMS subject classifications. 65M15, 65M60 DOI /16M Introduction. We consider te eat equation t u u = f in Ω (0, T ), u = 0 on Ω (0, T ), u(0) = u 0 in Ω, were Ω R d, 1 d 3, is a bounded, connected, polyedral open set wit Lipscitz boundary, and T > 0 is te final time. We assume tat f L 2 (0, T ; L 2 (Ω)), and tat u 0 L 2 (Ω). We are interested ere in developing a posteriori error estimates for a class of ig-order discretizations of (1.1). In particular, we consider a conforming finite element metod (FEM) in space on unstructured sape-regular simplicial meses, and a discontinuous Galerkin discretization in time, were one is free to vary te approximation orders p in space and q in time, as well as te mes size and time-step size τ, leading to wat we call an p-τq metod. Tese metods are igly attractive from te point of view of flexibility, accuracy, and computational efficiency, since it is known from a priori analysis tat judicious local adaptation of te discretization parameters can lead to exponential convergence rates wit respect to te number of degrees of freedom, even for solutions wit singularities near domain corners, edges, and at initial times [37, 39, 47]. In practice, it is desirable to determine te adaptation Received by te editors October 6, 2016; accepted for publication (in revised form) August 7, 2017; publised electronically November 14, ttp:// Funding: Te work of te autors was supported by te European Researc Council (ERC) under te European Union s Horizon 2020 researc and innovation program (grant agreement GATIPOR). Université Paris-Est, CERMICS (ENPC), Marne-la-Vallée 2, France and Inria Paris, 2 rue Simone Iff, Paris, France (alexandre.ern@enpc.fr). Inria Paris, 2 rue Simone Iff, Paris, France and Université Paris-Est, CERMICS (ENPC), Marne-la-Vallée 2, France (iain.smears@inria.fr, martin.voralik@inria.fr). 2811

2 2812 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK algoritmically, wic requires rigorous and ig-quality a posteriori error control in order to exploit te potential for ig accuracy and efficiency of p-τ q discretizations. We recall tat a posteriori error estimates sould ideally give guaranteed upper bounds on te error, i.e., witout unknown constants, sould be locally efficient, meaning tat te local estimators sould be bounded from above by te error measured in a local neigborood, and, moreover, sould be robust, wit all constants in te bounds being independent of te discretization parameters; we refer te reader to [46] for an introduction to tese concepts. In te context of parabolic problems, te a posteriori error analysis for low- and fixed-order metods as received significant attention over te past decade, wit efforts mostly concentrated on fixed-order FEM in space coupled wit an implicit Euler or Crank Nicolson time-stepping sceme, leading to estimates for a wide range of norms. Tese include estimates for te L 2 (H 1 )-norm of te error considered independently by Picasso [35] and Verfürt [44], wit efficiency bounds typically requiring restrictions on te relation between te sizes of te time steps and te meses. Estimates for te L 2 (H 1 ) H 1 (H 1 )-norm were first considered by Verfürt in [45], wo crucially proved local-in-time yet global-in-space efficiency of estimators witout restrictions between time-step and mes sizes; see also Bergam, Bernardi, and Mgazli [1]. Guaranteed upper bounds for a large family of spatial discretizations were later obtained by Ern and Voralík in [15], wit similar efficiency results as in [45]. Tere are also upper bounds in L 2 (L 2 ), L (L 2 ), and L (L ) and iger-order norms, based on eiter duality tecniques as in Eriksson and Jonson [12] or te elliptic reconstruction tecnique originally due to Makridakis and Nocetto [30] and later considered in te fully discrete context by Lakkis and Makridakis [27]; see also [28] and te references terein. Repin [36] studied so-called functional estimates. Finally, a posteriori error estimates developed in te context of te eat equation often serve as a starting point for extensions to diverse applications, including nonlinear problems and spatially nonconforming metods among oters [8, 9, 22, 25, 34]. Adaptive algoritms for parabolic problems are studied in [5, 21, 26]. It is apparent from te literature tat, even for low- and fixed-order metods, tere are remaining outstanding issues, particularly in terms of te efficiency of te estimators. Te efficiency of te estimators is significantly influenced by te coice of norm to be estimated, wit te strongest available results being attained by Y -norm estimates, were, encefort, Y := L 2 (H0 1 ) H 1 (H 1 ). However, even in tis norm, te full space-time local efficiency of te estimators is not known. It is elpful to examine ere more closely tis issue in order to motivate te approac adopted in tis work. For example, let us momentarily consider an implicit Euler discretization in time and a conforming FEM in space, recalling tat te implicit Euler metod corresponds to te lowest-order discontinuous Galerkin time-stepping metod, wic uses piecewise constant approximations wit respect to time. Te resulting numerical solution u τ is discontinuous wit respect to time, so it is not possible to estimate u u τ Y. Terefore, it is usual to consider a reconstruction, denoted by Iu τ Y, obtained by piecewise linear interpolation at te time-step nodes, and it is seemingly natural to seek a posteriori error estimates for u Iu τ Y, were Y is defined in (2.1) below, and were u is te solution of (1.1); for instance, tis corresponds to te approac adopted in [45]. However, te main issue in estimates for u Iu τ Y is tat Iu τ fails to satisfy te Galerkin ortogonality property: instead, Iu τ satisfies (1.2) (f, v τ ) ( t Iu τ, v τ ) ( Iu τ, v τ )dt = ( (u τ Iu τ ), v τ )dt for all discrete test functions v τ V τ ; see section 3 for complete definitions. Te lack

3 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2813 of Galerkin ortogonality for Iu τ is associated wit te discrete residual on te rigtand side of (1.2) tat involves te L 2 (H 1 )-norm of u τ Iu τ, and it is tis discrete residual tat causes te loss of local spatial efficiency in previous analyses. Tis issue is independent of te specific construction of te error estimators, weter tey are residual-type estimators as in [45] or equilibrated flux estimators as considered ere. It turns out tat te discrete residual in (1.2) is related to te temporal jumps in te numerical solution u τ Y, wic is a form of error in itself since it is tied to te nonconformity of te numerical sceme. Tis motivates te introduction of a composite norm u u τ EY tat includes bot u Iu τ Y and te L 2 (H 1 )-norm of u τ Iu τ. Our analysis is ten centered on te error estimation of u u τ EY instead of u Iu τ Y, and tis allows us to recover te fully space-time local efficiency of te estimators: see (1.4) below, and see Teorem 5.2 of section 5. For p-fem discretizations, one of te key issues concerns te robustness of te estimators wit respect to te polynomial degree; tis issue appears already in te context of elliptic problems, were Melenk and Wolmut [33] and Melenk [32] sowed tat te well-known residual estimators fail to be polynomial-degree robust. In a breaktroug work, Braess, Pillwein, and Scöberl [3] establised te polynomialdegree robustness of estimators based on equilibrated fluxes, in te context of elliptic diffusion problems. Tese estimators are based on a globally H(div)-conforming flux computed by solving independent local mixed finite element problems. Te polynomial-degree robustness of tese estimators was recently generalized to nonconforming and mixed metods in [16], to wic we refer te reader for furter references on elliptic problems. For parabolic problems, tere is te additional question of robustness of te estimators wit respect to te temporal polynomial degrees. In comparison to low- and fixed-order metods, tere are comparatively few works on a posteriori error estimates for ig-order discretizations of parabolic problems. Building on te earlier work of Makridakis and Nocetto [31], Scötzau and Wiler [38] studied te effect of te temporal approximation order of a posteriori estimates for a composite norm of L (L 2 ) L 2 (H 1 )-type, in te context of ig-order temporal semidiscretizations of abstract evolution equations. Oterwise, a posteriori error estimates for p-τ q discretizations of parabolic problems remain essentially untouced. In tis work, we present guaranteed, locally space-time efficient, and polynomialdegree robust a posteriori error estimators for p-τ q discretizations of parabolic problems. Tis is by no means simple, as it requires te treatment of te callenges tat ave been outlined above. Our main results are te following. Let te spaces Y := L 2 (H0 1 ) H 1 (H 1 ) and X := L 2 (H0 1 ) be, respectively, equipped wit teir standard norms Y and X defined in (2.1) below. Let Y +V τ be te sum of te continuous and approximate solution spaces, recalling tat V τ X and tat u τ V τ Y due to te temporally discontinuous approximation. Let I : Y + V τ Y be te reconstruction operator defined in section 3.5 below, were we note tat Iv = v if and only if v Y. Let te norm EY be defined by v 2 E Y := Iv 2 Y + v Iv 2 X for all v Y + V τ. In particular, Iu = u, so u u τ 2 E Y = u Iu τ 2 Y + u τ Iu τ 2 X. Hence EY as te natural functional interpretation as an extension of te Y -norm to te discrete approximation space. Guaranteed upper bounds. In Teorem 5.2 of section 5, we sow a posteriori estimates in te norm EY. In te absence of data oscillation, our bound takes te simple form N { } (1.3) u u τ 2 E Y σ τ + Iu τ 2 K + (u τ Iu τ ) 2 K dt, n=1 K T n

4 2814 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK were σ τ is te H(div)-conforming reconstruction; see sections 3 and 4 for full definitions of te notation and construction of te estimators. Polynomial-degree robustness and local space-time efficiency. We establis local space-time efficiency of our estimators wit polynomial-degree robust constants, expressed by te lower bound (1.4) σ τ + Iu τ 2 K + (u τ Iu τ ) 2 Kdt u u τ 2 E a,n + oscillation, Y a V K were K is an element of te mes T n for time step, were V K denotes te set of vertices of K, and were u u τ E a,n is te local component of u u Y τ EY on te patc associated wit te vertex a and time interval. Here, and in te following, te notation a b means tat a Cb wit a constant C tat depends possibly on te sape regularity of te spatial meses, but is oterwise independent of te mes size, time-step size, as well as te spatial and temporal polynomial degrees. We stress tat tis efficiency bound does not require any relation between te sizes of te time step and te mes. Te full bound is stated in Teorem 5.2 below. In addition to te above results, te estimators proposed ere are advantageous in terms of flexibility, since tey do not require restrictions on coarsening or refinement between time steps tat appeared in earlier works, suc as te transition condition used in [45, pp. 196, 201]. Te main tool to avoid tis condition is Lemma 8.1 below. Relation between u u τ EY and u Iu τ Y. Te a posteriori analysis in tis work concerns te estimation of u u τ EY. Independently of te error estimation, we also consider te question of te relation between te new norm u u τ EY and te previously considered norm u Iu τ Y. Specifically, for arbitrary polynomial degrees, we sow te global equivalence result (1.5) u Iu τ Y u u τ EY 3 u Iu τ Y + oscillation. Te oscillation term in (1.5) is te minimum between te source term data oscillation and te coarsening error, as fully detailed in Teorem 5.1 of section 5.1 below. Notice tat te constant in te equivalence is terefore robust wit respect to all parameters. Te proof is based on a simplification and generalization to te iger-order case of a key result of Verfürt [45], namely, tat te jumps in te numerical solution can be controlled locally-in-time and globally-in-space by te Y -norm of u Iu τ. Te key implication of (1.5) is tat te global space-time norms u u τ EY and u Iu τ Y are essentially equivalent, altoug we stress tat teir local (spatial) distributions may differ. Tis paper is organized as follows. First, in section 2 we introduce a functional setting for te a posteriori error analysis. We find it wortwile to provide a complete derivation of te inf-sup analysis of te problem, as we give ere quantitatively sarp results tat are advantageous for te efficiency of te estimators in practice. Section 3 defines te setting in terms of notation, finite element approximation spaces, and te numerical sceme. Ten, in section 4, we define te equilibrated flux reconstruction used in te a posteriori error estimates. In section 5 we gater our main results underlying (1.3), (1.4), and (1.5). Te proofs of te main results are treated in te subsequent sections: section 6 establises te relation between u u τ EY and u Iu τ Y ; te proof of te guaranteed upper bound is given in section 7; and te efficiency of te estimators is te subject of section Inf-sup teory. Recall tat Ω R d, 1 d 3, is a bounded, connected, polyedral open set wit Lipscitz boundary. For an arbitrary open subset ω Ω,

5 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2815 we use (, ) ω to denote te L 2 -inner product for scalar- or vector-valued functions on ω wit associated norm ω. In te special case were ω = Ω, we drop te subscript notation, i.e., := Ω. Following [29, Cap. 3], we consider te function spaces X := L 2 (0, T ; H 1 0 (Ω)) and Y := L 2 (0, T ; H 1 0 (Ω)) H 1 (0, T ; H 1 (Ω)), wit norms (2.1) T ϕ 2 Y := t ϕ 2 H 1 (Ω) + ϕ 2 dt + ϕ(t ) 2 ϕ Y, 0 T v 2 X := v 2 dt v X. Define te bilinear form B Y : Y X R by (2.2) B Y (ϕ, v) := 0 T 0 t ϕ, v + ( ϕ, v) dt, were ϕ Y and v X are arbitrary functions, and, denotes ere te duality pairing between H 1 (Ω) and H 1 0 (Ω). Ten, te problem (1.1) admits te following weak formulation: find u Y suc tat u(0) = u 0 and suc tat (2.3) B Y (u, v) = T 0 (f, v) dt v X. Te well-posedness of (2.3) is well known and can be sown by inf-sup arguments in te above functional setting [13]; it can also be sown by Galerkin s metod [20, 48]. Te inf sup stability result presented ere as te interesting and important property of taking te form of an identity, wic is advantageous for te sarpness of a posteriori error analysis. Te fact tat te constant equals 1 in (2.4) below can also be found in [42, 43]; it can also be seen from [24, p. 249]. Teorem 2.1 (inf sup identity). (2.4) ϕ 2 Y = [ sup v X\{0} For every ϕ Y, we ave ] 2 B Y (ϕ, v) + ϕ(0) 2. v X Proof. For a fixed ϕ Y, let w X be defined by ( w, v) = t ϕ, v for all v H0 1 (Ω), a.e. in (0, T ), wic implies te identity w 2 = t ϕ 2 H 1 (Ω) a.e. in (0, T ). Furtermore, we ave B Y (ϕ, v) = T 0 ( (w +ϕ), v) dt, tus implying tat sup v X\{0} B Y (ϕ, v)/ v X = w + ϕ X. We ten obtain te desired identity (2.4) by expanding te square (2.5) [ sup v X\{0} ] 2 B Y (ϕ, v) = v X = = T 0 T 0 T 0 (w + ϕ) 2 dt w 2 + 2( w, ϕ) + ϕ 2 dt t ϕ 2 H 1 (Ω) + 2 tϕ, ϕ + ϕ 2 dt = ϕ 2 Y ϕ(0) 2, were we note tat we ave used te identity T 0 2 tϕ, ϕ dt = ϕ(t ) 2 ϕ(0) 2.

6 2816 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK In order to estimate te error between te solution u of (1.1) and its approximation, we define te residual functional R Y : Y X by (2.6) R Y (ϕ), v := B Y (u ϕ, v) = T 0 (f, v) t ϕ, v ( ϕ, v) dt, were v X and ϕ Y. Te dual norm of te residual is naturally defined by R Y (ϕ) X := sup v X\{0} R Y (ϕ), v / v X. Teorem 2.1 implies te following equivalence between te error and dual norm of te residual: for all ϕ Y, we ave (2.7) u ϕ 2 Y = R Y (ϕ) 2 X + u 0 ϕ(0) Finite element approximation. Consider a partition of te interval (0, T ) into time-step intervals := (t n 1, t n ) wit 1 n N, were it is assumed tat [0, T ] = N n=1, and tat {t n } N n=0 is strictly increasing wit t 0 = 0 and t N = T. For eac interval, we let τ n := t n t n 1 denote te local time-step size. We will not need any special assumptions about te relative sizes of te time steps to eac oter. We associate a temporal polynomial degree q n 0 wit eac time step, and we gater all te polynomial degrees in te vector q = (q n ) N n=1. For a general vector space V, we sall write Q qn ( ; V ) to denote te space of V -valued univariate polynomials of degree at most q n over te time-step interval Meses. We consider a matcing simplicial mes T n of te domain Ω for eac 0 n N, were we assume sape regularity of te meses uniformly over all time steps. Tis allows us to treat many applications were te meses are obtained by refinement or coarsening between time steps. We consider ere only matcing simplicial meses for simplicity, altoug we indicate tat mixed simplicial parallelepiped meses, possibly containing anging nodes, can be also be treated; see [10] for instance. Te mes T 0 will be used to approximate te initial datum u 0. For eac element K T n, let K := diam K denote te diameter of K. We associate a local spatial polynomial degree p K 1 wit eac K T n, and we gater all spatial polynomial degrees in te vector p n = (p K ) K T n. In order to keep our notation sufficiently simple, te dependence of te local spatial polynomial degrees p K on te time step is kept implicit, altoug we bear in mind tat te polynomial degrees may cange between time steps Approximation spaces. For a general matcing simplicial mes T wit associated vector of polynomial degrees p = (p K ) K T, p K 1 for all K T, te H 1 0 (Ω)-conforming p-finite element space V (T, p) is defined by (3.1) V (T, p) := { v H 1 0 (Ω), v K P pk (K) K T }, were P pk (K) denotes te space of polynomials of total degree at most p K on K. For sortand, we denote V n := V (T n, p n ) for eac 0 n N. Let Π u 0 V 0 denote an approximation to te initial datum u 0, a typical coice being te L 2 - ortogonal projection onto V 0. Given te collection of time steps {} N n=1, te vector q of temporal polynomial degrees, and te p-finite element spaces {V n}n n=1, te spatio-temporal finite element space V τ is defined by (3.2) V τ := { v τ (0,T ) X, v τ In Q qn ( ; V n ) n = 1,..., N, v τ (0) V 0 }. Functions in V τ are generally discontinuous wit respect to te time variable at te partition points, altoug we take tem to be left-continuous: for all 1 n N,

7 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2817 we define v τ (t n ) as te trace at t n of te restriction v τ In. Functions in V τ are tus left-continuous; moreover, tey also ave a well-defined value at t 0 = 0. For all 0 n < N, we denote te rigt-limit of v τ V τ at t n by v τ (t + n ). Ten, te temporal jump operators n, 0 n N 1, are defined on V τ by (3.3) v τ n := v τ (t n ) v τ (t + n ), 0 n N Refinement and coarsening. Similary to oter works, e.g., [45, p. 196], we assume tat we ave at our disposal a common refinement mes T n of T n 1 and T n for eac 1 n N, as well as associated polynomial degrees p n = (p K) K T n, suc tat V n 1 + V n Ṽ n := V ( T n, p n ). For a function v τ V τ, we observe tat v τ n 1 Ṽ n for eac 1 n N since v τ (t n 1 ) V n 1, v τ (t + n 1 ) V n, and V n 1 + V n Ṽ n. It is assumed tat T n as te same sape regularity as T n 1 and T n, and tat every element K T n is wolly contained in a single element K T n 1 and a single element K T n. We empasize tat we do not require any assumptions on te relative coarsening or refinement between successive spaces V n 1 and V n. We note tat in te present context, refinement and coarsening can be obtained by modification of te meses as well as cange in te polynomial degrees. Concerning te polynomial degrees, we may coose, for example, p K = max(p K, p K ). In te case were V n n 1 is obtained from V by refinement witout coarsening, ten we may coose T n := T n and p n := p n so tat Ṽ n = V n. However, we do not need te transition condition assumption from [45, pp. 196, 201], wic requires a uniform bound on te ratio of element sizes between T n and T n Numerical sceme. Te numerical sceme for approximating te solution of te parabolic problem (1.1) consists of finding u τ V τ suc tat u τ (0) = Π u 0, and, for eac time-step interval, (3.4) ( t u τ, v τ ) + ( u τ, v τ ) dt ( u τ n 1, v τ (t + n 1 )) = (f, v τ ) dt v τ Q qn ( ; V n ). Here te time derivative t u τ is understood as te piecewise time-derivative on eac time-step interval. Te numerical solution u τ V τ can tus be obtained by solving te fully discrete problem (3.4) on eac successive time step. At eac time step, tis requires solving a linear system tat is symmetric only in te lowest-order case; tis can be performed efficiently in practice for arbitrary orders; see [40] and te references terein Reconstruction operator. For eac time-step interval and eac nonnegative integer q, let L n q denote te polynomial on obtained by mapping te standard qt Legendre polynomial under an affine transformation of ( 1, 1) to. It follows tat L n q (t n ) = 1 for all q 0, and L n q (t n 1 ) = ( 1) q, and tat te mapped Legendre polynomials {L n q } q 0 are L 2 -ortogonal on, and satisfy L n q 2 dt = τn 2q+1 for all q 0. We introduce te Radau reconstruction operator I defined on V τ by (3.5) (Iv τ ) In := v τ In + ( 1)qn 2 ( L n qn L n ) q n+1 vτ n 1 v τ V τ. It is clear tat I is a linear operator on V τ. It follows from te properties of te Legendre polynomials tat Iv τ In (t n ) = v τ (t n ), and tat Iv τ In (t + n 1 ) = v τ (t n 1 )

8 2818 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK for all 1 n N. Terefore, Iv τ is continuous wit respect to te temporal variable at te interval partition points {t n } N 1 n=0, and tus we ave (3.6) Iv τ H 1 ( 0, T ; H0 1 (Ω) ) Y, Iv τ In Q qn+1( In ; Ṽ ) n v τ V τ, were we recall tat V n 1 + V n Ṽ n. We easily deduce te following property of te reconstruction operator I from integration by parts and te ortogonality of te polynomials L n q n and L n q n+1 to all polynomials of degree strictly less tan q n on te time-step interval : (3.7) t Iv τ φ dt = t v τ φ dt v τ n 1 φ ( t + ) n 1 φ Q qn ( ; R), were equality olds in te above equation in te sense of functions in Ṽ n. We may terefore use (3.7) to rewrite te numerical sceme (3.4) as (3.8) ( t Iu τ, v τ ) + ( u τ, v τ ) dt = (f, v τ ) dt v τ Q qn ( ; V n ). Note also tat Iu τ (0) = Π u 0. Remark 3.1 (alternative equivalent definitions). Te operator I is te Radau reconstruction operator commonly used in te a posteriori error analysis of te discontinuous Galerkin time-stepping metod [31] and in te a priori error analysis of timedependent first-order PDEs [14]. Several equivalent definitions of I ave appeared in te literature, altoug it will be particularly advantageous for our purposes to use te definition (3.5) of I; see [23, 40] for furter details on te equivalence of te various definitions. Remark 3.2 (extensions of I to Y + V τ ). In wat follows, it will be elpful to extend I to a linear operator over Y + V τ. Note tat te definition of te jump operators (3.3) can be naturally extended to Y + V τ and, terefore, te definition (3.5) also extends naturally to Y + V τ. In particular, I : Y + V τ Y, and we ave Iϕ = ϕ if and only if ϕ Y, since te jumps of any ϕ Y vanis identically. Remark 3.3 (a priori analysis in te Y -norm). Altoug we focus ere on te a posteriori analysis of te error, it is elpful to briefly mention some results from a priori analysis in te current functional framework. An important yet peraps nonobvious point is tat te inclusion of te time derivative in te Y -norm does not necessarily decrease te convergence order wit respect to te time-step size in comparison to oter norms, suc as te X-norm. Indeed, since tis is primarily related to te temporal discretization, let us sow tis by momentarily considering a temporal semidiscretization wit solution u τ obtained by replacing all discrete spaces V n by H1 0 (Ω) in (3.2) and in (3.4). Ten, wit f τ as defined in section 4.2, we ave T 0 t(u Iu τ ), v dt = T 0 (f f τ, v) ( (u u τ ), v) dt for all v X. Terefore, we deduce tat T 0 t (u Iu τ ) 2 H 1 (Ω) dt { u u τ X + f f τ X } 2. Tis explains in a nustell wy te time derivative error is not necessarily worse tan te oter terms. Tere are some works on Y -norm-type estimates for te fully discrete case, going back to Dupont [6]. We also mention te recent analysis on quasioptimality in te spatially semidiscrete case by Tantardini [41] and Tantardini and Veeser [42].

9 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS Construction of te equilibrated flux. Te a posteriori error estimates presented in tis paper are based on a discrete and locally computable H(div)- conforming flux σ τ tat satisfies te key equilibration property (4.1) t Iu τ + σ τ = f τ in Ω (0, T ), were Iu τ is defined in section 3.5, and f τ f is a data approximation defined in (4.4) below. We call σ τ an equilibrated flux. We consider ere te natural extension of existing flux reconstructions for elliptic problems [3, 4, 7, 16] to te parabolic setting; see also [11]. In particular, for eac time step, σ τ is obtained as a sum of fluxes computed by solving local mixed finite element problems over te vertex-based patces of te current mes; see Definition 4.1 of section 4.3 below Local mixed finite element spaces. We now define te mixed finite element spaces tat are required for te construction of te equilibrated flux. For eac 1 n N, let V n denote te set of vertices of te mes T n, were we distinguis te set of interior vertices V n int and te set of boundary vertices Vn ext. For eac a V n, let ψ a denote te at function associated wit a, and let ω a denote te interior of te support of ψ a wit associated diameter ωa. Furtermore, let T a,n denote te restriction of te mes T n to ω a. Recalling tat te common refinement spaces Ṽ n were obtained wit a vector of polynomial degrees p n = (p K) K T n, we associate wit eac a V n te fixed polynomial degree (4.2) p a := max K T a,n (p K + 1). Observe tat ψ a t Iu τ K In is a polynomial function wit degree at most q n in time and at most p a in space for eac K T a,n, 1 n N. For a polynomial degree p 0, let te local spaces P p ( T a,n ) and RTN p ( T a,n ) be defined by P p ( T a,n ) := {q L 2 (ω a ), q K P p ( K) K T a,n }, RTN p ( T a,n ) := {v L 2 (ω a ; R d ), v K RTN p ( K) K T a,n }, were RTN p ( K) := P p ( K; R d )+P p ( K)x denotes te Raviart Tomas Nédélec space of order p on K. It is important to notice tat, wereas te patc ω a is subordinate to te vertices of te mes T n, te spaces P p ( T a,n ) and RTN p ( T a,n ) are subordinate to te submes T a,n ; of course, in te absence of coarsening, tis distinction vanises. We now introduce te local spatial mixed finite element spaces V a,n and Q a,n, defined by } {v H(div, ω a ) RTN V a,n pa ( T a,n ), v n = 0 on ω a if a Vint n, := { } v H(div, ω a ) RTN pa ( T a,n ), v n = 0 on ω a \ Ω if a Vext, n { } q Q a,n P pa ( T a,n ), (q, 1) ωa = 0 if a Vint n :=, P pa ( T a,n ) if a Vext. n We ten define te following space-time mixed finite element spaces (4.3) V a,n τ := Q qn ( ; V a,n ), Q a,n τ := Q q n ( ; Q a,n ).

10 2820 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK 4.2. Data approximation. Our a posteriori error estimates given in section 5 involve certain approximations of te source term f appearing in (1.1). It is elpful to define tese approximations ere. First, we define te semidiscrete approximation f τ of f by an L 2 -ortogonal projection in time. In particular, te approximation f τ Q qn ( ; L 2 (Ω)) is defined on eac interval by (f f τ, v) dt = 0 for all v Q qn ( ; L 2 (Ω)). Next, for eac 1 n N and for eac a V n, let Π a,n τ be te L 2 ψ a -ortogonal projection from L 2 ( ; L 2 ψ a (ω a )) onto Q qn ( ; P pa 1( T a,n )), were L 2 ψ a (ω a ) is te space of measurable functions v on ω a suc tat ω a ψ a v 2 dx <. In oter words, te projection operator Π a,n τ is defined by (ψ a Π a,n τ v, q τ ) ωa dt = (ψ a v, q τ ) ωa dt for all q τ Q qn ( ; P pa 1( T a,n )). We adopt te convention tat Π a,n τ v is extended by zero from ω a to Ω (0, T ) for all v L 2 ( ; L 2 ψ a (ω a )). Ten, we define f τ by (4.4) f τ := N n=1 τ f. a V n ψ a Π a,n Remark 4.1 (definition of f τ ). Te somewat tecnical appearance of te definition of f τ is due to te possible variation in polynomial degrees across te mes and te particular requirements of te analysis of efficiency, in particular, te ypoteses of Lemma 8.1 below. Neverteless, f τ as several important approximation properties. First, for any 1 n N, any K T n, and any real-valued polynomial φ of degree at most q n, we ave (4.5) (f f τ, 1) Kφ dt = a V K (ψ a (f Π a,n τ f), φ1) K dt = 0, were V K denotes te set of vertices of K, and were we use te fact tat te at functions {ψ a } a V n form a partition of unity on Ω. Furtermore, using te ortogonality of te projector Π a,n τ and te fact tat 0 ψ a 1 in Ω, it is straigtforward to sow tat (4.6) f f τ L2 (;L 2 ( K)) d + 1 inf f w τ w τ Q qn (;P ( K)) L 2 (;L 2 ( K)). p K Tis sows tat f τ defines an approximation of f tat is at least of te same order as te one associated wit te finite element approximation. Furtermore, te approximations are exact, i.e., f = f τ = f τ, if f is a piecewise polynomial of appropriate degrees wit respect to te time steps and te common refinement meses Flux reconstruction. For eac 1 n N and eac a V n, let te scalar function g a,n τ Q q n ( ; P pa ( T a,n )) and vector field τ a,n τ Q qn ( ; RTN pa ( T a,n )) be defined by (4.7a) (4.7b) τ a,n τ := ψ a u τ ωa, := ψ a (Π a,n τ f tiu τ ) ωa ψ a u τ ωa. g a,n τ We claim tat for all a V n int, (4.8) (g a,n τ (t), 1) ω a = 0 t,

11 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2821 wic is equivalent to sowing tat g a,n τ Q a,n τ for all a V n. Indeed, we first observe tat te construction of te numerical sceme, in particular, identity (3.8), implies tat, for any univariate real-valued polynomial φ of degree at most q n on, (g a,n τ, φ1) ( ) ω a dt = f, φ ψa ( ω t Iu τ, φ ψ a a )ω ( u τ, (φ ψ a ) ) dt = 0, a ω a were we ave used te ortogonality of te projection Π a,n τ and te fact tat φψ a Q qn ( ; V n ) is a valid test function in (3.8). Since te function ga,n τ is polynomial in time wit degree at most q n, i.e., g a,n τ Q q n ( ; P pa ( T a,n )), we deduce (4.8). Definition 4.1 (flux reconstruction). Let u τ V τ be te numerical solution of (3.4). For eac time-step interval and for eac vertex a V n, let te space-time mixed finite element spaces V a,n τ and Q a,n τ be defined by (4.3). Let g a,n τ and τ a,n τ be defined by (4.7). Let σ a,n τ V a,n τ be defined by (4.9) σ a,n τ := argmin v τ τ a,n τ 2 ω a dt. v τ V a,n τ v τ =g a,n τ Ten, after extending σ a,n τ by zero from ω a to Ω (0, T ) for eac a V n and for eac 1 n N, we define (4.10) σ τ := N σ a,n τ. n=1 a V n Note tat σ a,n τ V a,n τ is well-defined for all a V n ; in particular, for interior vertices a Vint n, we use (4.8) to guarantee te compatibility of te datum ga,n τ wit te constraint σ a,n τ = ga,n τ. Te following key result sows tat σ τ from Definition 4.1 leads to an equilibrated flux. Teorem 4.2 (equilibration). Let te flux reconstruction σ τ be defined by (4.10) of Definition 4.1. Ten σ τ L 2 (0, T ; H(div, Ω)) and we ave (4.1), were te discrete approximation f τ is defined in (4.4). Proof. After extending eac σ a,n τ by zero from ω a to Ω (0, T ), we ave σ a,n τ L 2 (0, T ; H(div, Ω)) as a consequence of te boundary conditions included in te definition of te space V a,n. Tis immediately implies tat σ τ L 2 (0, T ; H(div, Ω)). To sow (4.1), te definition of te flux reconstruction σ τ in (4.10) implies tat for any time-step interval and any K T n, (4.11) σ τ K In = σ a,n τ K a V K = a V K = g a,n τ K a V K ( ψa Π a,n τ f ψ a t Iu τ ψ a u τ ) K In = (f τ t Iu τ ) K In, were V K denotes te set of vertices of K, were we use te fact tat te at functions {ψ a } a V n form a partition of unity in order to pass to te last line of (4.11), and were we ave used te definition of f τ in (4.4). Tis yields (4.1) as required.

12 2822 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK For te purposes of practical implementation, it is easily seen tat, for eac timestep interval, te fluxes σ a,n τ can be computed by solving q n + 1 independent spatial mixed finite element problems, provided only tat an ortogonal or ortonormal polynomial basis is used in time over. Moreover, te q n + 1 linear systems eac sare te same matrix, wic elps to simplify te implementation and reduce te computational cost. Lemma 4.3 (decoupling). Let σ a,n τ V a,n τ be defined by (4.9). Ten σ a,n τ is equivalently uniquely defined as te first component of te pair (σ a,n τ, ra,n τ ) V a,n τ tat solves Q a,n τ (4.12a) (σ a,n τ, v τ ) ωa ( v τ, r a,n τ ) ω a dt = (4.12b) ( σ a,n τ, q τ ) ωa dt = (τ a,n τ, v τ ) ωa dt v τ V a,n τ, (g a,n τ, q τ ) ωa dt q τ Q a,n τ. Furtermore, for eac 1 n N, let {φ n j }qn j=0 be an L2 ( )-ortonormal basis for te space of univariate real-valued polynomials of degree at most q n. For eac a V n, define te functions {g a,n,j }qn j=0 (4.13) g a,n,j := a,n and {τ,j }qn j=0 over te patc ω a by := g a,n τ φn j dt, τ a,n,j τ a,n τ φn j dt. Ten, te solution (σ a,n τ, ra,n τ ) of (4.12) can be obtained by solving te following spatial problems: for eac 0 j q n, find σ a,n,j V a,n and r a,n,j in Qa,n suc tat ( ) ( ) ( ) (4.14a) σ a,n,j, v v, r a,n,j = τ a,n,j, v v V a,n, ω a ω a ω ( ) ( ) a (4.14b) σ a,n,j, q g a,n,j, q q Q a,n, ω a = ω a and ten by defining σ a,n τ := q n j=0 σa,n,j φn j and ra,n τ := q n j=0 ra,n,j φn j. Remark 4.2 (implementation). Several tecniques can be used to reduce te computational cost of computing te flux equilibration. First, altoug te flux σ τ is defined on te space-time region Ω (0, T ) from a teoretical viewpoint, in practice it is only its restriction to te current time step wic is required, because values from previous time steps do not need to be reevaluated or even stored. Second, at eac time step and at eac vertex patc, a single matrix is sared by te decoupled local problems in (4.14), so a single factorization is sufficient. Tird, if a patc and its associated polynomial degrees are not canged at te next time step, tis factorization can simply be reused. Terefore tere are ample opportunities for reuse of previous computations to reduce te total cost. Turning to te cases of refined or coarsened patces, te analysis in te subsequent sections sows tat one particular advantage of te equilibrated flux σ τ of Definition 4.1 is tat it leads to estimators tat are robust wit respect to coarsening (and refinement) between time steps. Te price to pay is tat te size of te linear systems in (4.14) grows wit te size of coarsening between two successive time steps, as (4.14) are defined on te patces ω a partitioned by te common refinement mes T n for eac 1 n N. Te analysis in [19, section 6], toug, sows tat tis computational cost can be significantly reduced to te solution of two low-order systems over te patces ω a, followed by local ig-order corrections on te subpatces of T a,n. We refer te reader to [19, section 6] for te full details of tis approac.

13 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS Main results. In tis section, we present te a posteriori error estimate featuring guaranteed upper bounds, local space-time efficiency, and polynomial-degree robustness. Let te norm EY : Y + V τ R 0 be defined by (5.1) v 2 E Y := Iv 2 Y + v Iv 2 X v Y + V τ, were we recall Remark 3.2 on te extension of te linear operator I to Y + V τ. Since te exact solution u Y implies tat Iu = u, we ave te identities (5.2) u u τ 2 E Y = u Iu τ 2 Y + u τ Iu τ 2 X N = u Iu τ 2 τ Y + n(q n+1) (2q n+1) (2q u n+3) τ n 1 2, n=1 τ n(q n+1) were we ave simplified (u τ Iu τ ) 2 dt = (2q n+1) (2q u n+3) τ n 1 2, wic is an identity easily deduced from (3.5) and from L n q 2 dt = τn 2q+1 for all q 0; see also [38]. We also introduce te localized seminorms E a,n for eac Y 1 n N and eac a V n, defined by (5.3) v 2 E := a,n t Iv 2 H Y 1 (ω + a) Iv 2 ω a + (v Iv) 2 ω a dt v Y + V τ. Similarly to (5.2), we find tat (5.4) u u τ 2 E a,n = t (u Iu τ ) 2 H Y 1 (ω + (u Iu a) τ ) 2 ω a dt + τ n(q n+1) (2q n+1) (2q n+3) u τ n 1 2 ω a. Altoug it migt not be immediately obvious tat u u τ EY is equivalent to te Hilbertian sum of te u u τ E a,n, up to data oscillation, tis will come as a Y consequence of te results sown ere and in section 8. Remark 5.1 (role of u τ Iu τ X ). Te error estimators in tis work focus on u u τ EY, wic is based on te inclusion of te additional term u τ Iu τ X. First, tis term allows te extension of te Y -norm to te sum of te continuous and discrete solution spaces Y + V τ. Tus it measures te lack of conformity of u τ / Y coming from te jumps between time steps, as sown by (5.2). Te second reason to consider tis term is tat it is naturally connected to u Iu τ Y. Indeed, recall tat u τ Iu τ appears in te discrete residual from te rigt-and side of (1.2), and tat te X norm of tis discrete component of te residual is simply u τ Iu τ X by Teorem 2.1. Hence tis term also as te role of bounding te lack of Galerkin ortogonality of Iu τ. We are now ready to state our main results in Teorems 5.1 and 5.2 below Global equivalence of norms. It is elpful to denote te time-localized dual norm of te residual by (5.5) R Y (Iu τ ) In X := sup (f, v) t Iu τ, v ( Iu τ, v) dt. v X, v X =1 Note tat R Y (Iu τ ) In X can always be bounded from above by te restriction of te Y -norm of te error u Iu τ to te time-step interval.

14 2824 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK Teorem 5.1 (equivalence of norms). Let te norm EY be defined by (5.1) and, for eac 1 n N, let te temporal data oscillation ηosc,τ n and te coarsening error ηc n be defined by ηc n := τ (5.6a) n(q n+1) (2q n+1) (2q {u n+3) τ (t n 1 ) P n [u τ (t n 1 )]}, (5.6b) [ηosc,τ n ] 2 := f(t) f τ (t) 2 H 1 (Ω) dt, were P n : H1 0 (Ω) V n denotes te elliptic ortogonal projection onto V n defined by ( P nw, v ) = ( w, v ) for all v V n. Ten, we ave (5.7) (u τ Iu τ ) 2 dt 8 R Y (Iu τ ) In 2 X + min { [ηc] n 2, 8[η osc,τ n ] 2}, were R Y (Iu τ ) In X is defined in (5.5). Furtermore, we ave (5.8) u Iu τ 2 Y u u τ 2 E Y 9 u Iu τ 2 Y + N min { [ηc] n 2, 8[ηosc,τ n ] 2}. We delay te proof of Teorem 5.1 until section 6 below. We empasize tat te coarsening term ηc n arises only in te equivalence of norms, and tat it does not need to be computed in practice since it does not appear in te a posteriori error estimators below; see, in particular, (5.10). Remark 5.2 (equivalence). Teorem 5.1 sows tat u u τ EY and u Iu τ Y are globally equivalent up to te minimum of temporal data oscillation and coarsening errors. A similar result due to (5.7) actually olds also on eac. In particular, one of our key contributions ere is to obtain polynomial-degree independent constants in (5.8). It is important to note tat altoug u u τ EY and u Iu τ Y are essentially globally equivalent, teir local distributions may differ. Remark 5.3 (relation to [45]). A similar result to (5.7) was previously obtained in te lowest-order case q n = 0 by Verfürt [45]; see, in particular, te bounds of [45, section 7] for wat is denoted tere τn 3 un un 1 2 1, wic is equivalent to (u τ Iu τ ) 2 dt wit q n = 0 in our notation. For iger polynomial degrees, we note tat Gaspoz et al. ave obtained independently an inequality of a similar kind as (5.7) in [21, Prop. 7], wit te difference tat (5.7) features a robust constant wit respect to te temporal polynomial degree and is sarper wit respect to te oscillation term Main a posteriori error estimate. We introduce te following a posteriori error estimators and data oscillation terms: n=1 (5.9a) (5.9b) (5.9c) (5.9d) (5.9e) ηf,k(t) n := σ τ (t) + Iu τ (t) K, ηj,k n := τ n(q n+1) (2q u n+1)(2q n+3) τ n 1 K, ηosc,,k(t) n := 2 K π 2 f τ (t) f τ (t) 2 K K T n, K K η osc,τ (t) := f(t) f τ (t) H 1 (Ω), η osc,init := u 0 Π u 0, 1 2,

15 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2825 were t, K T n, te equilibrated flux σ τ is defined in Definition 4.1, and were te data approximations f τ and f τ are, respectively, defined in section 4.2. Te two estimators ηf,k n and ηn J,K are our principal estimators, were ηn F,K measures, respectively, te lack of H(div)-conformity of te gradient of te reconstructed solution Iu τ, and were ηj,k n measures te lack of temporal conformity of te numerical solution u τ. Te term ηosc,,k n represents te data oscillation due to te spatial discretization, wereas η osc,τ represents te data oscillation due to te temporal discretization. We define te global a posteriori error estimators as N ηy 2 := { } (5.10a) + η osc,τ dt + [η osc,init ] 2, (5.10b) n=1 η 2 E Y := η 2 Y + N n=1 K T n [η n F,K + η n osc,,k] 2 K T n [η n J,K] 2. Notice tat in te absence of data oscillation, namely, if u 0 = Π u 0 and f = f τ = f τ (see Remark 4.1), ten η Y simplifies to ηy 2 = T 0 σ τ + Iu τ 2 dt, and η EY simplifies to ηe 2 Y = T 0 σ τ + Iu τ 2 + (u τ Iu τ ) 2 dt. Recall tat we write a b for two quantities a and b if a Cb wit a constant C depending only on te sape regularity of T n and T n, but oterwise independent of te mes size, time-step size, and polynomial degrees in space and time. Teorem 5.2 (E Y -norm a posteriori error estimate). Let u Y be te weak solution of (1.1), let u τ V τ denote te solution of te numerical sceme (3.4), and let Iu τ denote its temporal reconstruction, were te operator I is defined in (3.5). Let σ τ denote te equilibrated flux of Definition 4.1. Let EY be defined in (5.1), and let te a posteriori error estimators be defined in (5.9) wit η EY defined in (5.10). Ten, we ave te guaranteed upper bound (5.11) u u τ EY η EY. Moreover, for eac 1 n N and for eac K T n, te indicators satisfy te following local efficiency bound (5.12) [ηf,k] n 2 dt + [ηj,k] n 2 { u u τ 2 E a,n + [ηosc a,n ] 2}, Y a V K were E a,n is defined in (5.3), V Y K is te set of vertices of te element K, and te local data oscillation term ηosc a,n is defined by (5.13) [η a,n osc ] 2 := f Π a,n τ f 2 H 1 (ω a) dt. Furtermore, we ave te following global efficiency bound for u u τ EY : N [ ] N (5.14) [ηf,k] n 2 dt + [ηj,k] n 2 u u τ 2 E Y + osc ] 2. n=1 K T n [η a,n n=1 a V n Te proof of Teorem 5.2 is postponed to te following sections: te proof of te upper bound (5.11) is given in section 7, and te proof of te bounds (5.12) and (5.14) is te subject of section 8. Teorem 5.2 sows te local space-time efficiency of te estimators wit respect to u u τ EY.

16 2826 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK Remark 5.4 (temporal data oscillation). Te temporal data oscillation term η osc,τ is defined wit respect to a negative norm, as is usual in te literature [15, 45]. Similarly to [15, 45], tis temporal data oscillation term can be of te same order as te error in terms of te time-step size. Since tis term already appears in te upper bounds of te residual-based estimates of [45, (1.5)], it is seen tat tis issue is not related to te coice of equilibrated flux a posteriori error estimators, but is rater a part of te error estimation in te Y -norm. In practical computations, it is often advisable to determine a minimal temporal resolution for reducing tis term to witin a prescribed tolerance, in advance of solving te numerical sceme (3.4). Altoug te negative norm appearing in te definition of η osc,τ is noncomputable, tere are several possibilities for estimating it. First, we mention tat η osc,τ is bounded from above by C Ω f f τ wit C Ω te constant of te global Poincaré inequality, altoug tis can be pessimistic in practice. If f is a finite tensorial product of spatial and temporal functions, ten sarper bounds can be obtained by solving a set of independent coarse and low-order conforming approximations for elliptic problems, followed by equilibrated flux a posteriori error estimates to acieve guaranteed upper bounds. Finally, we also mention tat tis issue motivates a posteriori error estimators in oter norms; in particular, we sow in [18] tat X-norm a posteriori estimates benefit from data oscillation terms tat are of iger order by an additional factor of τ Extension to Y -norm estimates. As a consequence of te proof of Teorem 5.2, we can also sow guaranteed upper bounds and local-in-time and global-inspace efficiency of te estimators wit respect to u Iu τ Y, tereby generalizing te results to [45] to iger-order approximations. Corollary 5.3 (Y -norm a posteriori error estimate). Let te estimator η Y be defined by (5.10a). Ten, we ave (5.15) u Iu τ Y η Y. Furtermore, for eac 1 n N, we ave (5.16) [ ] [ηf,k] n 2 dt + [ηj,k] n 2 t (u Iu τ ) 2 H 1 (Ω) + (u Iu τ ) 2 dt K T n + min { [η n C] 2, 8[η n osc,τ ] 2} + osc ] 2. a V n [η a,n 6. Proof of equivalence between u u τ EY and u Iu τ Y. In tis section, we prove Teorem 5.1, along wit some corollary results, wic relate u u τ EY to u Iu τ Y. Our starting point involves te following two original bounds on te norms of te jumps, wic generalize one of te key results of Verfürt [45] for te lowest-order case q n = 0. In fact, our result sarpens and simplifies te proof of te result of [45] even in te lowest-order case. Lemma 6.1. For eac 1 n N, let P n : H1 0 (Ω) V n denote te elliptic ortogonal projection to V n defined by ( P nw, v ) = ( w, v ) for all v V n. Ten, for eac 1 n N, te jump u τ n 1 satisfies (6.1) τ n 8q n + 4 u τ n 1 2 R Y (Iu τ ) In 2 X + τ n 8q n + 4 {u τ (t n 1 ) P n [u τ (t n 1 )]} 2,

17 A POSTERIORI ERROR ANALYSIS FOR PARABOLIC PROBLEMS 2827 were R Y (Iu τ ) In X is defined in (5.5). Furtermore, we also ave te alternative bound τ n (6.2) 8q n + 12 u τ n ( R Y (Iu τ ) In 2 X + [ηn osc,τ ] 2). Proof. First, note tat (u τ Iu τ ) In = ( 1)qn 2 (L n q n+1 L n q n ) u τ n 1 belongs to te space Q qn+1( ; Ṽ n). We define te test function v τ := ( 1)qn 2 L n q n P n u τ n 1, wic belongs to Q qn ( ; V n ), and we use it in (3.8) for te numerical sceme, wic yields, by ortogonality of te Legendre polynomials and by te definition of te ortogonal projector P n, te identity v τ 2 dt = τ n 8q n + 4 P n u τ n 1 2 = ( (u τ Iu τ ), v τ ) dt I (6.3) n = (f t Iu τ, v τ ) ( Iu τ, v τ ) dt. Terefore, we ave v τ 2 dt R Y (Iu τ ) In 2 X. Tis bound yields te desired result (6.1) once it is combined wit (6.3) and te ortogonality relation P n u τ n 1 2 = u τ n 1 2 { u τ n 1 P n u τ n 1 } 2 = u τ n 1 2 {u τ (t n 1 ) P n [u τ (t n 1 )]} 2, were te last equality above follows from te facts tat u τ n 1 = u τ (t n 1 ) u τ (t + n 1 ) and tat u τ (t + n 1 ) V n. Tis completes te proof of te first bound (6.1). We now turn to te proof of (6.2); te main difference in te proofs of (6.1) and (6.2) is tat above we appealed to te numerical sceme using a discrete test function, wereas to establis (6.2), we sall now consider a iger-order polynomial function tat is not in te discrete test space. We define v on by v In := ( 1) qn 2 L n q n+1 u τ n 1, and ten we extend v by zero outside of, so tat v X. Ten, by ortogonality of te Legendre polynomial L n q n+1 to all polynomials of degree at most q n on, we ave te identities (f τ, v) dt = 0, ( t Iu τ, v) dt = 0, and ( u τ, v) dt = 0. Terefore, we obtain v 2 τ n dt = 8q n + 12 u τ n 1 2 = ( (u τ Iu τ ), v) dt I n = (f, v) ( t Iu τ, v) ( Iu τ, v) + (f τ f, v) dt. Te desired result (6.2) ten follows straigtforwardly from te above identity. Proof of Teorem 5.1. Te first inequality u Iu τ 2 Y u u τ 2 E Y is obvious from te definition of u u τ EY in (5.2). Recalling te definitions of ηj,k n in (5.9b) and ηc n in (5.6a), we deduce from (6.1) and (6.2) tat (6.4) [ηj,k] n 2 4(q n + 1) (2q n + 3) R Y (Iu τ ) In 2 X + [ηn C] 2 2 R Y (Iu τ ) In 2 X + [ηn C] 2, K T n and tat (6.5) [ηj,k] n 2 8(q n + 1) ( RY (Iu τ ) In 2 X (2q n + 1) + [ηn osc,τ ] 2) K T n 8 R Y (Iu τ ) In 2 X + 8[ηn osc,τ ] 2.

18 2828 ALEXANDRE ERN, IAIN SMEARS, AND MARTIN VOHRALÍK Terefore, we obtain (5.7) by taking te minimum of te rigt-and sides of te above bounds. Finally, we get (5.8) by summing te above inequality over all time steps and noting tat N n=1 R Y (Iu τ ) In 2 X = R Y (Iu τ ) 2 X u Iu τ 2 Y wic follows from (2.7). It is possible to obtain sligtly sarper variants of Teorem 5.1 under more specific assumptions. For instance, te following corollary sows tat u u τ EY is equivalent to u Iu τ Y witout any additional data oscillation, wenever te mes coarsening error is relatively small compared to te jumps. Corollary 6.2. Using te notation of Lemma 6.1, assume tat tere exists a constant θ [0, 1) suc tat [ u τ n 1 P n u τ n 1 ] 2 θ u τ n 1 2 for eac 1 n N. Ten, we ave (6.6) u Iu τ 2 Y u u τ 2 E Y 3 θ 1 θ u Iu τ 2 Y. Proof. Te result is a consequence of Lemma 6.1 and of te identity u τ n 1 P n u τ n 1 = u τ (t n 1 ) P n[u τ τ (t n 1 )], wic leads to n 8q u n+4 τ n θ R Y (Iu τ ) In 2 X. Adapting te proof of Teorem 5.1 ten yields (6.6). Note tat te case θ = 0 in Corollary 6.2 corresponds to te case of no coarsening. 7. Proof of te guaranteed upper bound. We prove ere (5.11) and (5.15). First, it is clear from (5.2) tat (5.15) immediately implies (5.11). Terefore, it remains to sow (5.15). Keeping in mind te equivalence identity (2.7) between norms of te errors and residuals, we turn our attention to bounds for te residual norm R Y (Iu τ ) X = sup v X\{0} B Y (u Iu τ, v)/ v X. To tis end, consider an arbitrary function v X suc tat v X = 1. Ten, we obtain R Y (Iu τ ), v = T 0 (f t Iu τ σ τ, v) (σ τ + Iu τ, v) dt, were we ave inserted te flux σ τ and used integration by parts over Ω. Next, we use (4.1), and we write f f τ = f f τ + f τ f τ. For any K T n, 1 n N, we deduce from (4.5) tat te function t (f τ (t) f τ (t), 1) K, wic is a real-valued polynomial of degree at most q n on, vanises identically on. Terefore, letting v K(t) denote te mean value of v(t) over te element K T n, wic is defined for a.e. t, we deduce from te Poincaré inequality tat (f τ (t) f τ (t), v(t)) K Kπ f τ (t) f τ (t) K v(t) K. Terefore, R Y (Iu τ ), v can be bounded as follows: R Y (Iu τ ), v = N (f f τ, v) (σ τ + Iu τ, v) dt N ηf,k v n K + π f τ (t) f τ (t) K v(t) K K T n n=1 n=1 K T n K + η osc,τ v dt N [ ] [ηf,k n + ηosc,,k] v n K + η osc,τ v dt n=1 K T n N { } η osc,τ v dt. n=1 K T n [η n F,K + η n osc,,k] 2