A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems

Transcription

1 A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems EMMANUIL H. GEORGOULIS AND OMAR LAKKIS Abstract. We derive energy-norm a posteriori error bounds for an Euler timestepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler timestepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which prompts the derivation of new a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.. Introduction Adaptive methods for partial differential equations (PDE s) of evolution type have become a staple in improving the efficiency in large scale computations. Since the 98 s many adaptive methods have been increasingly based on a posteriori error estimates, which provide a sound mathematical case for adaptive mesh refinement based on error estimators which can be decomposed in spatially and temporally local error indicators. In the context of parabolic equations, a posteriori error estimates have been derived for various norms since early 99 s [EJ9, Pic98]. Inspired by the milestones set recently for the mathematical theory of convergence for adaptive methods in elliptic problems [MNS2, BDD4, CKNS7], there has been a recent push for similar results for parabolic problems calling to a closer understanding of a posteriori error estimates [CJ4, Ver3, BBM5, LM6, e.g.]. Most results in this area cover simple time-stepping schemes and a conforming space discretization. The extant literature on a posteriori error control for nonconforming spatial methods can be grouped in a handful of works [SW5, EP5, YC6, NS5, CY7]. In [SW5] a posteriori L 2 (H )-norm error bounds for a spatially semidiscrete method via interior penalty discontinuous Galerkin (IPDG) methods are derived and used heuristically in the implementation of the fully discrete scheme. In [EP5, YC6] L 2 (L 2 )-norm error bounds for IPDG are obtained using duality techniques, while in [NS5], a posteriori error bounds are presented for a fully discrete method consisting of a backward Euler time-stepping and linear Crouzeix Raviart elements in space. Note that none of the papers in the literature, to our knowledge, cover the case of a posteriori energy-norm error bounds for fully discrete schemes with DG methods, which is the chief objective of our paper. We both acknowledge and thank the Nuffield Foundation s Young Researcher Grant scheme and the EPSRC Multiscale Network, for bringing us together to start this work, and the Computational Mathematics Junior Programme at Hausdorff Insitute for Mathematics in Bonn, which hosted us during the final write-up phase.

2 2 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS Discontinuous Galerkin (DG) methods are an important family of nonconforming finite element methods for elliptic, parabolic and hyperbolic problems dating back to 97 s and early 98 s [Nit7, RH73, Bak77, Whe78, Arn82]. DG methods have undergone substantial development in the recent years [RWG99, RW, HSS2, and references, e.g.]. Various survey papers on DG methods can be found in the literature [CKS, ABCM2, e.g.]. The practical interest in DG methods owes to their flexibility in mesh design and adaptivity, in that they cover meshes with hanging nodes or locally varying polynomial degrees. DG methods are thus ideally suited for hp-adaptivity and provide good local conservation properties of the state variable. Moreover, in DG methods the local elemental bases can be chosen freely for the absence of interelement continuity requirements, yielding very sparse in many cases even diagonal mass matrices even with high precision quadrature. Note also that DG methods popular for their very good stability properties in transport- or convection-dominated problems [CKS]; the a posteriori error analysis of convection-dominated problems is beyond the scope of our study and we concentrate on diffusion-only parabolic equations. Our main results are a posteriori error bounds in the energy norm for a family of fully discrete approximations of the following PDE problem in 2 we gather the functional analysis notation and background... Problem (linear parabolic boundary-initial value problem). Given an open (possibly curvilinear) polygonal domain Ω R d, d = 2, 3, a real number T >, two (generalized) functions f L (, T ; L 2 (Ω)) and a L (Ω (, T )) d d, (.) such that a(x, t) is symmetric positive definite for almost all (x, t) Ω [, T ], find a function u L 2 (, T ; H (Ω)), and such that t u L (, T ; H (Ω)) (.2) t u (a u) = f on Ω (, T ], u() = u on Ω, and u Ω (t) =, for t (, T ]. (.3) In 2 we propose a class of numerical methods for solving this problem. These methods consist in a backward Euler time-stepping scheme in combination with various choices of spatial DG methods. Our emphasis is on the widely applied IPDG method [Arn82, RWG99, HSS2]. We consider the notation of Problem. to be valid throughout the paper and u denotes the solution of problem (.3). Although the assumption f L (, T ; L 2 (Ω)) may be weakened provided a posteriori error estimates for the corresponding spatial finite element method can be obtained for such weak data we refrain from doing it for simplicity s sake. In fact, we consider f to be piecewise continuous in time with a finite number of time-discontinuities and with the implied constraints on the time partition, to be discussed in 2.4. The matrix-valued function a(, t), for each t (, T ) is allowed to have jump discontinuities; the set of spatial discontinuities of a will be considered to be aligned with the finite element meshes. For simplicity, we shall assume that a is continuous in time, but a finite number of discontinuities can be accounted for easily, as long as these occur at the points of the time partition in the fully discrete scheme. The PDE (.3) is assumed to be uniformly elliptic in the sense that the supremum and the infimum of the set { } (a(x, t)ζ) ζ/ ζ 2 : ζ R d, (x, t) Ω (, T ] (.4) are both positive real numbers. As for the boundary values, we remark that our approach can be appropriately modified in order to extend homogeneous to general

3 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 3 time-dependent Dirichlet boundary values. Under the assumptions made so far, we have that the solution to (.3) exists and satisfies u C(, T ; H (Ω)) and t u L 2 (, T ; L 2 (Ω)) [LSU67]. In line with a unified approach to a posteriori error analysis for elliptic-problem DG methods [Ain5, Ain7, CGJ8] our discussion will be presented first in an abstract setting. Our results are then shown to be applicable to a wide class of DG methods provided that a posteriori error bounds for the corresponding steady-state problem are available. We stress that, although we focus a posteriori error bounds for spatial DG methods, our abstract results can be applied to a wider class of nonconforming methods, other than DG provided they satisfy certain requirements which we briefly describe next. Consider given a particular nonconforming finite element space S. () For each Z S it is possible to decompose it as Z = Z c + Z d such that Z c H (Ω) S, (.5) where Z c and Z d are called Z s conforming part and the nonconforming part, respectively. This decomposition is an analytic device and is not needed for computational purposes. The only requirement on this decomposition is the ability to quantify certain norms of Z d in terms of Z, as found in the literature [BHL3, KP3, HSW7, e.g.], as well as our Lemma 4.3. (2) Given a function z H (Ω), and let Z S be the corresponding Ritzprojection via the finite element method, it is possible to bound the norm of the error Z z, using a posteriori error estimators for the steady state problem. A key tool in our a posteriori error analysis is the elliptic reconstruction technique [MN3]. Roughly speaking, the elliptic reconstruction technique, as far as energy estimates are concerned, allows to neatly separate the time discretization analysis form the spatial one. This technique, which has been adapted to tackle fully-discrete schemes via energy methods for conforming methods [LM6], is extended in this work to the nonconforming setting, to all methods that meet the two requirements above. Briefly said, the idea of elliptic reconstruction denoting by u the solution of (.3) and U that of the discrete problem consists in building an auxiliary function w, called the elliptic reconstruction of U, which satisfies two key properties: (a) a PDE-like relation binds the parabolic error u w with data quantities only involving w U and the problem s data, f, a, and u, (b) the function U is the Ritz projection of w onto S. Note that w is an analysis-only device, despite its name it is not an object that is computable and this is not needed. We believe that it is possible to obtain similar a posteriori error estimates, for each single method at hand, by working directly, i.e., without using an elliptic reconstruction technique, but this will inevitably lead to further complications which may render the analysis quite involved, especially for the fully discrete scheme. This prejudice of ours is testified by the somewhat surprising lack of previous rigorous results in the literature. Finally, we point out that it is possible to follow a similar approach to ours in order to derive a posteriori error estimates in lower order functional spaces such as L (, T ; L 2 (Ω)). We remark that new estimators arise in the derivation of fully discrete a posteriori error bounds, due to the time-dependent diffusion tensor considered in this work, compared to [LM6] where only time-independent diffusion coefficients are addressed.

4 4 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS The following is an outline of this article. After introducing the notation and the method in 2, the elliptic reconstruction is used to develop an abstract framework for spatially semidiscrete schemes in 3 and their fully discrete counterpart in 5. The actual error estimators for each particular method are then consequences of our abstract framework and specific elliptic error estimators such as the ones presented in [BHL3, KP3, BGC5, HSW7, HSW6]. Moreover, in 4 we prove a posteriori bounds for the corresponding steady state problem of (.) for IPDG, thus extending existing results [BHL3, KP3, HSW7, HSW6] to the case of general (non-diagonal) diffusion tensor, with minimal regularity assumptions on the exact solution. These a posteriori bounds are then combined with the general framework presented in in 3 and 5 to deduce fully computable bounds for the DG-approximation error of the parabolic problem. 2. Preliminaries 2.. Functional analysis tools. Given an open subset ω R d, we denote by L p (ω), p, the Lebesgue spaces of functions with summable p-powers on ω. The corresponding norms L p (ω); the norm of L 2 (ω) will be denoted by ω for brevity; by, ω we write the standard L 2 -inner product on ω. When ω = Ω we omit the subindex. We denote by H s (ω), the standard Hilbertian Sobolev space of index s R; H (ω) signifies the subspace of H (ω) of functions with vanishing trace on the boundary ω. The Poincaré Friedrichs inequality v Ω C PF v Ω for v H (Ω) (2.) turns the H (Ω) seminorm into a norm on H (Ω). We consider thus Ω to be the norm on H (Ω). We will use also H (Ω), the dual space of H (Ω), equipped with the duality brackets. Namely, if f H (Ω) then for each φ H (Ω) its value on φ is denote by g φ which coincide with g, φ if g L 2 (ω). Thus the norm of g is given by g H (Ω) := sup g, φ φ H (Ω) {} φ. (2.2) The duality pairing ( ) H, H allows us to define, for each fixed t (, T ] the elliptic operator A (t) : H (Ω) H (Ω) where ( ) A (t)v φ := a(t) v, φ = (a(x, t) v(x)) φ(x) dx (2.3) for φ H (Ω), v H (Ω). Here, and throughout the paper, we use the shorthand f(t) = f(, t), for a function f : [, T ] Ω R. Note that thanks to the uniform parabolic assumption given by (2.3) and the Lax Milgram Theorem, the definition of the operator A (t) is well defined and yields an isomorphism between H (Ω) and H (Ω) [Eva98]. In other words, the operator A (t) induces a bounded coercive bilinear form Ω (v, w) H (Ω) H (Ω) A (t)v w R, (2.4) which we will extend later to a larger nonconforming space. We stress from the outset that although the bilinear form in (2.4) will be extended later to larger spaces, the operator A (t) will not and it will only act on H (Ω) functions throughout the discussion. For p +, we also define the spaces L p (, T, X), with X being a real Banach space with norm X, consisting of all measurable functions v : [, T ] X,

5 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 5 for which ( T ) /p v L p (,T ;X) := v(t) p X dt < +, for p < +, v L (,T ;X) := ess sup t T v(t) X < +, for p = +. (2.5) Finally, we denote by C(, T ; X) the space of continuous functions v : [, T ] X with norm v C(,T ;X) := max t T v(t) X < Finite element spaces. Let T be a subdivision of Ω into disjoint open sets, which we (abusively) call elements. We assume the T to be parametrized by mappings F κ, for each κ T, where F κ : ˆκ κ is a diffeomorphism with nonsingular Jacobian, and ˆκ is the reference element or reference square. The above mappings are such that Ω = κ T κ. We often use the word mesh for subdivision, and we say that a mesh is regular if it has no hanging nodes; otherwise the mesh is irregular. Unless otherwise stated, we allow the mesh to be -irregular, i.e., for d = 2, there is at most one hanging node per edge, typically its center; for d = 3 a corresponding concept is available. For an integer p, we denote by P p (ˆκ), the set of all polynomials on ˆκ of degree p, if ˆκ is the reference simplex, or of degree p in each coordinate direction, if ˆκ is the reference cube. We consider the discontinuous Galerkin finite element space S = S p (T ) := {v L 2 (Ω) : v κ F κ P p (ˆκ), κ T }. (2.6) By Γ we denote the union of all sides of the elements of the subdivision T (including the boundary sides). We think of Γ as the union of two disjoint subsets Γ = Γ Γ int, where Γ is the union of all boundary sides. Let two elements κ +, κ T have a common a side e := κ + κ Γ int. Define the outward normal unit vectors n + and n on e corresponding to κ + and κ, respectively. For functions q : Ω R and φ : Ω R d uniformly continuous on each of κ ±, but possibly discontinuous across e, we define the following quantities. For q + := q κ +, q := q κ and φ + := φ κ +, φ := φ κ, we set {q} e := 2 (q+ + q ), {φ} e := 2 (φ+ + φ ), q e := q + n + + q n, φ e := φ + n + + φ n ; (2.7) if e κ + Γ, these definitions are modified to {q} e := q +, {φ} e := φ +, q e := q + n +, φ e := φ + n +. (2.8) We introduce the meshsize as the function h : Ω R, by h(x) = diam κ, if x κ and h(x) = {h}, if x Γ. The shape-regularity of the subdivision T is defined as µ(t ) := sup κ T h κ r κ, (2.9) where r κ is the radius of the largest ball that fits entirely in κ. We shall use the gradient s regular part operator, T, of an elementwise differentiable function v defined by ( T v) κ := (v κ ) for κ T. (2.) Note that the full distributional gradient, v, consists of an extra term taking into account the jumps of v across the edges: v = T v + v Γint. (2.)

6 6 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS Finally, we consider some shorthand notation for quantities involving the diffusion tensor a. In particular, we define the elementwise constant functions a, a : Ω [, T ] R by a (, t) κ := a(, t) 2 2 L (κ) and a (, t) κ := ( a(, t)) 2 2 L (κ), (2.2) for κ T, and a = {a }, a = ({/a }), on Γ, where 2 denotes the Euclideaninduced matrix norm. Finally, let α (t) := max x Ω a (x, t) and α (t) = min x Ω a (x, t). (2.3) 2.3. Spatial DG discretization. Introduce the DG space S := S + H (Ω), and a corresponding DG bilinear form B : S S R, which we assume to be an extension of the bilinear form defined by (2.4), viz., B(t; v, z) = A (t)v z for all v, z H (Ω), t (, T ]. (2.4) Though B is time-dependent, we do not write it explicitly in the semidiscrete case and omit the t. The space S is equipped with a DG norm, denoted and depending on the method at hand, which extends the energy norm, i.e., v = a(, t) v for all v H (Ω), (2.5) for t [, T ]. Also here, the norm is time-dependent, but this dependence is not explicitly written. A norm equivalence between the energy norm and a(, t) in H (Ω), uniformly with respect to t, suffices for all the bounds presented below to hold, modulo a multiplicative constant; but, we eschew this much generality for clarity s sake. The semidiscrete DG method in space for problem (.3), reads as follows: Find U C, (, T ; S) such that t U, V + B(U, V ) = f, V for V S, t [, T ]. (2.6) We stress that these assumptions are satisfied by many DG methods for second order elliptic problems available in the literature, possibly by using inconsistent formulations [ABCM2]. Moreover, (2.4) and (2.5) are satisfied by IPDG (along with the corresponding energy norm) considered below as a paradigm. Assumption (2.4) implies the consistency of the bilinear form B on H (Ω), i.e., t u, v + B(u, v) = f, v, for all v H (Ω), (2.7) where u is the exact (weak) solution to the initial-boundary value problem (.3). Note that this is the same as writing t u + A u = f. (2.8) 2.4. Fully discrete solution. To further discretize in time, consider an increasing time partition {t n } n=,...,n, and the corresponding timesteps τ n = t n t n, for n =,..., N. We assume that the points of time discontinuities of a fall in {t n }, that t = and t N = T. For each n =,..., N, we consider that S n is a DG finite element space of fixed degree p built on a partition T n, which may be different from T n when n. In 5.3, we will say more about the sequence of meshes and the compatibility relations among them. Let f n (x) := f(x, t n ) and let U be the projection (or an interpolation) of u onto the finite element space S. We say that {U n } n=,...,n is a fully discrete solution of (.3) if, for each n =,..., N we have that U n S n satisfies ( U n U n ) /τ n, V + B n (U n, V ) = f n, V for all V S n, (2.9)

7 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 7 Since the elliptic operator A (and the bilinear form B) depend on time, in the fully discrete setting, we denote their value at time t by A (t) and B(t), respectively and when t = t n we take A n := A (t n ) and B n := B(t n ). Noting that the term U n can be replaced by Π n U n, where Π n : L 2 (Ω) S n is the orthogonal projection, we consider a slightly more general situation where Π n U n in (2.9) is replaced by I n U n. Here I n : S n S n is a general data transfer operator, depending on the particular implementation. The operator I n may coincide with Π n, but it may be an interpolation operator for example. The general fully discrete Euler scheme then reads ( U n I n U n ) /τ n, V + B n (U n, V ) = f n, V for all V S n. (2.2) We have taken f n = f(t n ), but we could take a more general approximation than f(t n ), for example, a good choice is also given by f n := t n t n f(s) ds, for which a suitable modification of our arguments leads to similar results. 3. Abstract a posteriori bounds for the semidiscrete problem We derive next an abstract a posteriori error bound for the quantity ( T u U L 2 (,T ;S ) := u(t, ) U(t, ) 2) /2, (3.) where denotes the appropriate (space) energy norm. In the a posteriori error analysis below, we shall make use of the idea of elliptic reconstruction operators introduced in [MN3] for the semidiscrete problem and extended to fully discrete problems in [LM6]. 3.. Definition (elliptic reconstruction and discrete operator). Let U be the (semidiscrete) DG solution to the problem (2.6). We define the elliptic reconstruction w H (Ω) of U to be the solution of the elliptic problem B(w, v) = AU Πf + f, v for all v H (Ω), (3.2) where Π : L 2 (Ω) S denotes the orthogonal L 2 -projection on the finite element space S, and A : S S denotes the discrete DG operator defined by AZ, V = B(Z, V ) for all V S, (3.3) for each Z S. Note that this is valid on [, T ] Remark (the role of the elliptic reconstruction). The elliptic reconstruction is well defined. Indeed, AU S is the unique L 2 -Riesz representation of a linear functional on the finite dimensional space S and the existence and uniqueness of (weak) solution of (3.2), with data AU Πf + f L 2 (Ω), follows from the Lax Milgram Theorem in view of (2.4). To avoid clutter it will be handy to write (3.2) The key property of w is that the DG solution U of the semidiscrete timedependent problem (2.6) is also the DG solution of the steady-state boundaryvalue problem (3.2). Indeed, let W S be the DG-approximation to w, defined by the finite dimensional linear system B(W, V ) = AU Πf + f, V, (3.4) for all V S, which implies B(W, V ) = AU, V = B(U, V ) for all V S, i.e., W = U Definition (error, elliptic and parabolic parts). We shall decompose the error as follows: e := U u = ρ ɛ, where ɛ := w U, and ρ := w u, (3.5)

8 8 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS where w = w(t) denotes the elliptic reconstruction of U = U(t) at time t [, T ]. We call ɛ the elliptic error and ρ the parabolic error Lemma (semidiscrete error relation). Let u be the solution of Problem., U denote the solution of the DG scheme (2.6). Then, we have t e, v + B(ρ, v) = for all v H (Ω). (3.6) Proof For each v H (Ω) we have t e, v + B(ρ, v) = t U, v + B(w, v) f, v = t U, v + AU Πf + f, v f, v = t U + AU, Πv f, Πv, Πv =, (3.7) where in the first equality we used (2.7); in the second and fourth equalities, we made use of Definition 3.; in the third equality the properties of the orthogonal L 2 -projection onto S are used; finally, the last equality follows from (2.6) Definition (conforming-nonconforming decomposition). In the theory developed below, we shall consider the decomposition of the DG solution U S into conforming (continuous) and nonconforming (discontinuous) parts as follows U = U c + U d, (3.8) where U c S c := H (Ω) S and U d := U U c S. Note that at this point we do not specify any particular decomposition, thus keeping the choice of such a decomposition at our disposal. Let e c := e U d = U c u H (Ω), and ɛ c := ɛ + U d = w U c H (Ω). (3.9) 3.6. Theorem (long-time a posteriori error bound for DG). Let u and U be the exact weak solution of (.3) and the DG solution of the problem (2.6), respectively. Let w be the elliptic reconstruction of U as in Definition 3.. Assuming (2.4) and (2.5), and that a decomposition of the form (3.8) is available, then the following error bound holds U u L 2 (,T ;S ) 3 w U L 2 (,T ;S ) + 2 U d L 2 (,T ;S ) + 2 t U d / (3.) α L 2 (,T ;H (Ω)) + u U c (). Proof Set v = e c in (3.6); then, in view of (3.9), we have t e c, e c + B(ρ, ρ) = B(ρ, ɛ c ) t U d, e c. (3.) Recalling (2.4), (2.5) and that ρ, e c H (Ω) for every t [, T ], using the Cauchy Schwarz inequality, and the duality pairing ( H, H ), we arrive to 2 d t e c 2 + ρ 2 ρ ɛ c + t U d H (Ω) e c. (3.2) Also, (3.9) implies e c H (Ω) ( ɛ c + ρ )/ α. (3.3) Setting I := ɛ c, I 2 := t U d / α H (Ω) in (3.2) and rearranging lead to 2 d t e c 2 + ρ 2 ρ (I + I 2 ) + I I 2, (3.4) which implies d t e c 2 + ρ 2 4(I 2 + I2 2 ). (3.5) Integration on [, T ] and taking square roots yields ρ L 2 (,T ;S ) e c () + 2 ɛ c L 2 (,T ;S ) + 2 t U d / α L 2 (,T ;H (Ω)). (3.6) The assertion follows using triangle inequality on (3.5) and (3.9).

9 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN Theorem (short-time a posteriori error bound for DG). Let the assumptions of Theorem 3.6 hold. Then the following error bound holds: u U L 2 (,T ;S ) ( 3/2 + ) w U L 2 (,T ;S ) + 3/2 U d L 2 (,T ;S ) Proof Let T [, T ] be such that Then, (3.) implies + 2 t U d L (,T ;L 2 (Ω)) + 2 u U c (). (3.7) e c (T ) = max t T e c(t) =: E c. (3.8) 2 d t e c 2 + ρ 2 ρ ɛ c + E c t U d, (3.9) which, after integration on [, T ], yields 2 E2 c + ρ 2 L 2 (,T ;S ) 2 e c() 2 + ρ L 2 (,T ;S ) ɛ c L 2 (,T ;S ) + E c t U d L (,T ;L 2 (Ω)), (3.2) or 4 E2 c 2 e c() ɛ c 2 L 2 (,T ;S ) + tu d 2 L (,T ;L 2 (Ω)). (3.2) Going back to (3.9), upon integration with respect to t between [, T ], we obtain 2 ρ 2 L 2 (,T ;S ) 2 e c() ɛ c 2 L 2 (,T ;S ) + 4 E2 c + t U d 2 L (,T ;L 2 (Ω)), (3.22) which, in conjunction with (3.2) gives ρ 2 L 2 (,T ;S ) e c() ɛ c 2 L 2 (,T ;S ) + 4 tu d 2 L (,T ;L 2 (Ω)) ; (3.23) the final bound now follows using the triangle inequality on (3.5) and (3.9) Remark (long- versus short-time bounds). We note that the crucial difference between bounds (3.) and (3.7) is that in the latter the L -accumulation term t U d L (,T ;S ) is present; this implies t U d L (,T ;L 2 (Ω)) T t U d L 2 (,T ;L 2 (Ω)), (3.24) which may be preferable if T <, but can be inefficient for long-time integration. On the other hand, the corresponding term in (3.) is t U d / α L 2 (,T ;H (Ω)), which can be a bit less inefficient when the diffusion tensor a varies substantially on Ω Remark (elliptic a posteriori error estimates). The bounds (3.) and (3.7) are not (yet) explicitly a posteriori bounds: w U L 2 (,T ;S ) still needs to be bounded by a computable quantity. To this end, given g L 2, consider the elliptic problem: find z H (Ω) such that (a z) = g in Ω, z = on Ω, whose solution can be approximated by the following DG method: find Z S such that B(Z, V ) = g, V for all V S. If assume that an a posteriori estimator functional E exists, i.e., (3.25) (3.26) z Z E (Z, a, g, T ), (3.27)

10 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS then we can computably bound w U L 2 (,T ;S ) in (3.) and (3.7) through ( T w U L 2 (,T ;S ) E (U, a, AU Πf + f, T ) 2) /2. (3.28) A posteriori bounds for various DG methods have been studied, under different assumptions on data and admissible finite element spaces, by many authors [BHL3, KP3, HSW7, Ain7, HSW6, CGJ8]. Thus Theorems 3.6 and 3.7 can be applied to any DG and more generally to any non-conforming method satisfying (2.4) and (2.5), and for which (3.27) is available. The object of 4 is to address this for IPDG. 4. A posteriori error bounds for the interior penalty DG method Here we extend the energy-norm a posteriori bounds for the family IPDG methods [BHL3, KP3, HSW7, cf.] for the Poisson problem, to the case of the general diffusion problem (3.25). We stress that our results can be generalized as to allow for nonhomogeneous or mixed boundary conditions following [HSW7, KP3, resp.]. 4.. Definition (IPDG method). For z, v S, the bilinear form B : S S R for the IPDG method for the problem (3.25) can be written as ( B(z, v) := (a T z) T v + θ{aπ v} z {aπ z} v +σ z v ), (4.) Ω Γ for θ {,, }, where Π : [L 2 (Ω)] d S d denotes also the orthogonal L 2 - projection operator onto S d, and the penalty function σ : Γ R is defined by σ := C a,µ(t )a, (4.2) h where the constant C a,µ(t ) > depends on the shape-regularity of the mesh T and on the smallest possible C a > such that C a a κ + a κ C a, (4.3) for every pair of elements κ + and κ sharing a common side. We also define the corresponding energy norm for the IPDG method by ( v := a T v 2 + /2, σ v Γ) 2 (4.4) for v S. Note that for v S, we have Π v = v and, therefore, B can be reduced to the more familiar form ( B(z, v) := (a T z) T v + θ{a v} z {a z} v + σ z v ), (4.5) Ω Γ for z, v S [Arn82, ABCM2, RWG99, HSS2, cf.]. Observe that both (2.4) and (2.5) hold for the particular B and defined above Remark (conforming part of a nonconforming finite element function). The space-discontinuous finite element space S contains the conforming (continuous) finite element space S c = S H (Ω) as a subspace. The approximation of functions in S by functions in S c will play an important role in our derivation of the a posteriori bounds. This can be quantified in the following result, which is an extension of [KP7, Thm. 2.]. For other similar results we refer to [SW5, BHL3, HSW7, BE7].

11 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 4.3. Lemma (bounding the nonconforming part via jumps). Suppose T is regular (no hanging nodes) and a is elementwise (weakly) differentiable. Then, for any function Z S there exists a function Z c S c such that and Z Z c C h Z Γ, (4.6) a T (Z Z c ) C 2 σ Z Γ, (4.7) where C, C 2 > constants depending on the shape-regularity, on the maximum polynomial degree of the local basis and on C a. The proof, omitted here, follows closely that of [KP3, Thm. 2.2]. Lemma 4.3 can be proved for irregular (hanging-nodes) meshes [KP3, Thm. 2.3], in which case C and C 2 depend on the maximum refinement and coarsening levels L max Lemma (a posteriori bounds for IPDG method for elliptic problem). Let T be a regular and a is elementwise (weakly) differentiable. Let z and Z be given by (3.25) and (3.26). Then where E IP (Z, a, g, T ) := CK T a z Z E IP (Z, a, g, T ), (4.8) ( h/ a (g + T (a T Z)) + h/a a T Z Γint + σ Z Γ ), and K T a := max Ω a /a, where C > depends only on µ(t ) and C a. (4.9) Proof Our proof is inspired by [KP3, HSW7]. Denoting by Z c S c the conforming part of W as in Lemma 4.3, we have e := z Z = e c + e d, where e c := z Z c, and e d := Z c Z, (4.) yielding e c H (Ω). Thus, we have B(z, e c ) = g, e c. Let Π : L 2 (Ω) R denote the orthogonal L 2 -projection onto the elementwise constant functions; then Π e c S and we define η := e c Π e c. We also have B(e, e c ) = B(z, e c ) B(Z, e c ) = g, e c B(Z, η) B(Z, Π e c ) = g, η B(Z, η), (4.) which implies a e c 2 = B(e c, e c ) = g, η B(Z, η) B(e d, e c ). (4.2) For the last term on the right-hand side of (4.2), we have B(e d, e c ) a T e d a e c + a κ h(π ec ) κ e e d / h s, 2 s Γ κ=κ +,κ (4.3) where κ + and κ are the (generic) elements having e as common side. Using the inverse estimate of the form hv e C V κ for V = Π e c, and the stability of the L 2 -projection, we arrive to B(e d, e c ) a T e d a e c + CK T a a e c σ e d Γ. (4.4) Finally, noting that e d = Z, and making use of (4.7) we conclude that B(e d, e c ) CK T a a e c σ W Γ. (4.5)

12 2 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS To bound the first two terms on the right-hand side of (4.2), we begin by an elementwise integration by parts yielding ( g, η B(Z, η) = g + T (a T Z) ) η {η} a Z ds Ω Γ int (4.6) + θ{aπ η} Z ds σ Z η ds. Γ The first term on the right-hand side of (4.6) can be bounded as follows: ( ) g + (a Z) η h/ a (g + T (a T Z)) a h η ; (4.7) Ω upon observing that h η κ C e c κ, this becomes ( ) g + (a Z) η CK T a h/ a (g + T (a T Z)) a e c. (4.8) Ω For the second term on the right-hand side of (4.6), we work analogously to (4.3) to deduce {η} a Z ds CKa T a e c h/a a T Z Γint. (4.9) Γ int For the third term on the right-hand side of (4.6), we observe that η = e c and, working analogously to (4.3), we obtain θ{aπ η} Z CKa T θ a e c σ Z Γ, (4.2) Γ and finally, for the last term on the right-hand side of (4.6), we get σ η Z CKa T a e c σ Z Γ. (4.2) Γ The result follows combining the above relations Theorem (a posteriori bounds for IPDG method for parabolic problem). Let u, U be the exact weak solution of (.3), and the IPDG solution of the problem (2.6), respectively, and let a be elementwise (weakly) differentiable. Then, the following error bound holds: T u U 2 L 2 (,T ;S ) C ( E 2 IP (U, a, AU Πf + f, T ) + α Γ h t U 2 ) Γ + C ( u U() + h U() ) 2. (4.22) If we assume also that u, U C(, T ; H (Ω)) H (, T ; L 2 (Ω)), then the following bound also holds: T ( T u U 2 L 2 (,T ;S ) C EIP(U, 2 a, AU Πf + f, T ) + C ) 2 h t U Γ + C ( u U() + h U() ) 2. (4.23) Proof The results follow immediately from combining Theorems 3.6 and 3.7 with Lemma 4.4, in conjunction with (4.6). Finally, we give a result on useful properties of the IPDG bilinear form and of the norm, which will be useful in 5.

13 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN Lemma (continuity of B and stability of the L 2 -projection). Consider the notation of 4 and let B and denote the IPDG bilinear form (4.5) and the DG-norm (4.4). Then for Z, V S we have B(Z, V ) CK T a Z V. (4.24) Moreover, for v H (Ω), the L 2 -projection is DG-norm-stable, i.e., Πv CK T a v. (4.25) Proof We omit the proof of (4.24) which mimicks that of (4.5). For stability, note Πv 2 = a T (Πv Π v) 2 + σ v Πv 2 Γ C ( a 2 h (Πv Π v) 2 + a 2 h (v Πv) 2 + a 2 T (v Πv) 2) C ( a 2 h (v Π v) 2 + a 2 T (v Πv) 2) which implies (4.25). C (a /a ) 2 T v 2, (4.26) 5. A posteriori error bound for the fully discrete scheme In this section we discuss the abstract error analysis for the fully discrete scheme defined in (2.4). 5.. The elliptic reconstruction and the basic error relation. Extend the sequence {U n } into a continuous piecewise linear function of time: U() = U and U(t) = l n (t)u n + l n (t)u n, (5.) for t [t n, t n ], and n =,..., N, where the functions l n and l n are the Lagrange basis functions l n (t) := t t n [tn,t τ n] + t n+ t [tn,t n τ n+]. (5.2) n+ Using these time extensions and the (time and mesh dependent) discrete elliptic operator A n of Definition 3. with respect to S n, defined by A n Z S n such that A n Z, V = B n (Z, V ) for all V S n, (5.3) we can write the scheme (2.2) in the following pointwise form: t U(t) + A n U n = I n U n U n + Π n f n, (5.4) for all t (t n, t n ), n =,..., N. We like to warn at this point that we use the same symbol U(t) to indicate the fully discrete solution time-extension in this section, and the semidiscrete solution in 3. This should cause no confusion as long as the two cases are kept in separate sections. For each fixed t [, T ], and the corresponding n =,..., N such that t (t n, t n ], we define the time-dependent elliptic reconstruction to be the function w(t) H (Ω) satisfying w(t) = l n (t)w n + l n (t)w n +, (5.5) where w n H (Ω) is the elliptic reconstruction of U n defined implicitly as the (weak) solution of the elliptic problem with data A n U n Π n f n + f n, A n w n = A n U n Π n f n + f n, (5.6)

14 4 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS and w+ n is the forward elliptic reconstruction of U n, defined as the solution of the problem A n w+ n = A n + I n U n, (5.7) where the operator A n + : S n S n is defined by A n + Z S n such that A n + Z, V = B n + (Z, V ) for all V S n, (5.8) B+ n being the nonconforming bilinear form corresponding to A n, but with respect to the space S n (in contrast to B n which is defined with respect to S n ). For instance, for IPDG, we have B n + (Z, V ) := κ T n κ (a(t n ) Z) V ( + θ{a(tn ) V } Z {a(t n ) Z} V + σ n Z V ). (5.9) Γ n Using this definition of w on [t n, t n ], the equation (5.4), implies t U(t) + A (t)w(t) = I n U n U n + f n + A (t)w(t) A n w n. (5.) Subtracting the exact equation from this identity we obtain t [U u]+a [w u] = ( I n U n U n ) /τ n +(f n f)+(a w A n w n ) (5.) for all t (t n, t n ) and n =,..., N, this leads to the following technical basis of this section Lemma (fully discrete error relation). With the notation introduced in this section, let e = U u (full error), ρ := w u (parabolic error) and ɛ := w U (elliptic error). Then we have t e + A ρ = ( I n U n U n ) /τ n + f n f + A w A n w n, (5.2) on [t n, t n ], for all n =,..., N. Proof Replace the new notation for the errors in (5.) Mesh interaction, DG spaces and decomposition. The domain Ω s subdivisions {T n } n=,...,n are assumed to be compatible in the sense that given two consecutive meshes, say T n and T n, either T n is a local refinement of T n or viceversa. For each n =,..., N, we denote by T ˇ n the coarsest common refinement of T n and T n. The finite element space corresponding to T n being S n, we shall be using the space Šn which is the finite element space with respect to T ˇ n. Furthermore we denote by S n := S n + H (Ω) and by S := N n= S n, the minimal space that contains all these spaces. These spaces are equipped with the same type of norms give as in Section 2.3. The conforming-nonconforming decomposition of U, that we shall be using is performed as follows: (a) For each given t n, with n =,..., N we assume that the following two decompositions exist for U n, U n = U n d + U n c with respect to the mesh ˇ T n, and U n = U n d+ + U n c+ with respect to the mesh ˇ T n+. (5.3) Note that if the mesh changes, in general, Ud n and U n need not be equal d+ functions.

15 (b) A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 5 For each t (t n, t n ], n =,..., N, we define U d (t) := l n (t)u n d+ + l nu n d and U c (t) := l n (t)u n c+ + l n (t). (5.4) 5.4. Definition (a posteriori error indicators). We set here some notation that is useful to state the main results concisely. We make some assumptions in the process. For each time interval [t n, t n ], with n =,..., N, we introduce a posteriori error indicators as follows. (a) We assume that there exist C els, C dgc > such that and Π n v C els v, for all v H (Ω), (5.5) B n (Z, V ) C dgc Z V, for all Z, V S n. (5.6) The time-stepping indicator is given by θ n := C elsc dgc 3 I n U n U n. (5.7) (b) The time data-approximation indicator is β n := ( tn t n 2 /2 f(t n ) f(s) H (Ω) ds). (5.8) τ n α (s) (c) (d) The mesh-change (or coarsening) indicator is defined as I n U n U n ( ) /2 H γ n := (Ω) tn. (5.9) τ n τ n t n α The parabolic nonconforming part indicator is given by Ud n U n ( ) /2 d+ H δ n := (Ω) tn, (5.2) τ n τ n t n α and the elliptic nonconforming part indicator defined as ( δ n := Ud n 2 + U n 2) /2 d+. (5.2) (e) The space (or elliptic) error indicator is given by ε n := E (U n, a(t n ), A n U n Π n f n + f n, T n ), (5.22) where E is a particular choice of an energy-norm elliptic error estimator for the given spatial method. Furthermore the forward elliptic error indicator, due to mesh change, is given by (f) ε + n := E (I n U n, a(t n ), A n + I n U n, T n ). (5.23) Consider first the auxiliary function of time λ a,n (s) := a(s)a(t n ) a(s) a(t n ), s [t n, t n ] (5.24) 2 L (Ω) where the inner matrix norm is the Euclidean-induced one. This definition is possible thanks to a s being symmetric positive definite. The function λ a,n is identically zero if the operator is time-independent, otherwise it acts like the

16 6 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS numerator of a s normalized Hölder-continuity ratio.) Then we may define the following operator approximation indicators ( ζ n := An U n ) /2 H (Ω) tn α n l τ nλ 2 2 a,n, (5.25) n t n A n ζ n + I n U n ( ) /2 H := (Ω) tn α n l τ n λ 2 2 a,n, (5.26) n t n ζn := [ ( A n + A n] I n U n ) /2 tn ln 2 H, (5.27) (Ω) τ n α t n ζ n := ζ n + ζ n + ζ n. (5.28) 5.5. Remark (computing the H (Ω) norm). The norm H (Ω) appearing in the indicators is usually not computable, so for practical purposes this has to be replaced by the L 2 (Ω) norm times the Poincaré Friedrichs constant C PF defined in (2.) which implies the dual inequality v H (Ω) C PF v, for all v L 2 (Ω). (5.29) Note that this will not deteriorate most of the indicators. The only indicators that may be affected by this change are δ n and δ n, and it may be possible to provide a sharp bound for negative Sobolev norms of U d, but this seems to remain an open question at the time of writing Remark (computing A n U n and similar terms). The operators A n and A n + appearing in Definition 5.4, can be realized in two ways in practice: (a) To save time, one can use the fully discrete scheme in pointwise form (5.4) to evaluate some of these terms. For example A n U n = Π n f n ( U n I n U n ) /τ n. (5.3) (b) The corresponding stiffness matrix could be computed and applied to the argument. This seems to be necessary for A n Remark (mesh-change prediction). The mesh-change indicator γ n can be precomputed in a given computation. Indeed, this term does not use explicitly any quantity deriving from the solution of the n-th Euler time-step (2.2). This term is usually computable when a precise operator I n is available and it involves only local matrix-vector operations on each group of elements to be coarsened Remark (an alternative time-stepping indicator). An equally valid definition for the time-stepping estimator θ n can be given by θ n := 3 A n + I n U n A n U n H (Ω). (5.3) This alternative definition has the advantage of having no constants, but it is more complicated to compute and it must be reduced to the L 2 (Ω) norm by using the Poincaré Friedrichs inequality. A good side effect of this alternative choice is that in this case the indicator ζ n vanishes; all other estimators remain unchanged Theorem (abstract a posteriori energy-error bound for Euler DG). Let {U n } n be the solution of (2.2) and U its time-extension as defined by (5.) and w the elliptic reconstruction as defined by (5.5). Then, with reference to Definition 5.4,

17 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 7 for each m =,..., N, we have u U L 2 (,t m;s ) u() U c () + 2η p,m + ( ) /2 m 2η e,m + δ nτ n, (5.32) where the parabolic-error estimator is defined as ( m ) /2 η p,m := (θ n + ζ n + β n + γ n + δ n ) 2 τ n and the elliptic estimator is defined by (5.33) ( m /2 ( η e,m := ε 2 n + ε + 2)) n. (5.34) We spread the proof in paragraphs The energy identity. As in the proof of Theorem 3.6 to get an energy identity out of (5.2), we will test with the error s conforming part Start with combining (5.2) and definition (5.5) to get e c := e U d = ρ + ɛ c. (5.35) t e c + A ρ = t U d + ( I n U n U n ) /τ n + f n f + A w A n w n (5.36) Testing the above relation with e c we obtain the following energy identity 2 d t e c 2 + ρ 2 = t e c, e c + B (ρ, ρ) =B (ρ, ɛ c ) + t U d, e c + A w A n w n e c + ( I n U n U n ) /τ n + f n f, e c. (5.37) Integrating (5.37) from to t m (, T ], for an integer m, m N fixed, we may write the integral form of the energy identity 2 e c(t) 2 + tm tm + tm + + tm ρ 2 = 2 e c() 2 + tm B (ρ, ɛ c ) N A w A n w n e c [tn,t n] N ( I n U n U n ) /τ n + f n f, e c [tn,t n] t U d, e c = : I + I (t m ) + I 2 (t m ) + I 3 (t m ) + I 4 (t m ). (5.38) To obtain the a posteriori error bound for scheme (2.2), we now bound each of I i (t m ), i =,..., 4 (I needs no bounding) appearing in relation (5.38), in terms of either a-posteriori-computable or left-hand-side quantities. A term that substantially distinguishes the fully discrete case from the semidiscrete one discussed in 3 is the time-discretization term I 2 (t m ), so we start by bounding this term.

18 8 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS 5.. Time discretization estimate. To bound I 2 (t m ) we start by working out the first factor of the integrand as follows A (s)w(s) A n w n = A (s) [ l n (s)w n + l n (s)w+ n ] A n w n =l n (s) [A (s) A n ] w n + l n (s) [ A (s) A n ] w+ n + l n (s) ( (5.39) A n w+ n A n w n) Since w n and e c are both in H (Ω), we may bound the first term with [A (s) A n ]w n e c (s) = ((a(s) a(t n )) w n ) e c (s) Ω ( a(s) ( ) ) a(tn = a(s)a(t n ) a(s) a(t n ) ) w n e c (s) Ω a(s)a(t n ) a(s) a(t n ) a(s) e c (s) a(t n ) w n 2 L (Ω) =λ a,n (s) w n e c (s). The second factor above can be bounded as follows w n 2 = A n w n w n = A n U n, w n = A n U n H (Ω) wn H (Ω) An U n H (Ω) α n w n, where the last step owes to the fact that (5.4) (5.4) Thus w n A n U n H (Ω) /αn α n wn 2 B n (w n, w n ) = A n w n w n. (5.42) and we obtain tn t n l n (s) [A (s) A n ] w n e c (s) ds An U n H (Ω) α n ( tn = ζ n τn e c L 2 ([t n,t n];s ). t n l n (s) 2 λ a,n (s) 2 ds ) /2 ( tn t n e c 2 ) /2 (5.43) Similarly, we obtain tn l n (s) [ A (s) A n ] w+ n e c (s) ds t n A n + I n U n ( ) /2 ( tn tn H (Ω) α n l n (s) 2 λ a,n (s) 2 ds e c 2 t n t n = ζ n τn e c L 2 ([t n,t n];s ). (5.44) ) /2 To estimate the resultant of the integrand s third term in (5.39), we recall the elliptic reconstruction s definition and note that in view of (5.4) we may write, for n 2, that A n w+ n A n w n e c = A n + I n U n A n U n, e c = [ A n + A n] I n U n, e c + A n [ I n U n U n], Π n e c (5.45)

19 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN 9 given that A n I n U n, A n U n S n. The first term on the right-hand side of (5.45) is simply bounded by [ A n + A n] I n U n [, e c A n + A n] I n U n e H (Ω) c, (5.46) and thus, recalling definition (5.27), we have tn t n l n [ A n + A n] I n U n, e c ζ n τn e c L 2 ([t n,t n];s ). (5.47) The second term on the right-hand side of (5.45) can be given a simpler expression as follows: A n [ I n U n U n], Π n e c =B n ( I n U n U n, Π n e c ) C dgc I n U n U n Π n e c C dgc C els I n U n U n e c (5.48) thanks to the stability of Π n with respect to the energy norm assumed in (5.5). Therefore, recalling definition (5.7), we obtain tn t n l n A n [ I n U n U n], Π n e c θn τn e c L 2 ([t n,t n];s ) (5.49) The time error estimate follows: m I 2 (t m ) (ζ n + θ n ) ( ρ L 2 (t n,t n;s ) + ɛ c L 2 (t n,t n;s ) ). (5.5) 5.2. The other error estimates. To bound the spatial error term, I (t) in (5.38), we simply consider I (t m ) = tm B (ρ, ɛ c ) tn ρ ɛ c, (5.5) with the aim of absorbing the first factor in the left-hand side of (5.38) and using an elliptic error estimator to bound the second term. The term I 3 (t m ) in (5.38) which takes into account data approximation: m tn I n U n U n I 3 (t m ) = + f n f, e c. (5.52) t n The first term can be bounded by using the ( H (Ω), H (Ω) ) pairing, as we did with the time-estimator above: m tn τ n t n ( I n U n U n ) /τ n + f n f, e c m (γ n + β n ) τ n e c L 2 (t n,t n;s ), (5.53) where we have used definitions (5.8) and (5.9). Hence we obtain the bound m I 3 (t m ) (β n + γ n ) ) τ n ( ρ L 2 (t n,t n;s ) + ɛ c L 2 (t n,t n;s ). (5.54) We estimate the last term on the right-hand side of (5.38). This term can be bounded in two different ways. For conciseness, we expose only the estimate that

20 2 EMMANUIL H. GEORGOULIS AND OMAR LAKKIS yields smaller accumulation over long integration-times: tm m tn I 4 (t m ) = t U d, e c t U d H (Ω) e c t n m ) δ n τn ( ρ L 2 (t n,t n;s ) + ɛ c L 2 (t n,t n;s ), by recalling (5.2). (5.55) 5.3. Concluding the proof of Theorem 5.9. Combining the energy relation (5.38) with the bounds (5.5), (5.5), (5.54) and (5.55), we obtain ρ 2 L 2 (t n,t n;s ) 2 e c() m (θ n + ζ n + β n + γ n + δ n ) τ n ɛ c L 2 (t n,t n;s ) m ( (θ n + ζ n + β n + γ n + δ n ) ) τ n + ɛ c L 2 (t n,t n;s ) ρ L 2 (t n,t n;s ), which is an inequality of the form (5.56) a 2 c 2 + d b + (d + b) a, (5.57) where a, b, d R m+ and c R are appropriately chosen. It follows that a max { c, d } + d + b, (5.58) which, using the notation introduced in the statement of the theorem, implies ρ L 2 (,t m;s ) 2 e c () + 2η p,m + ɛ c L 2 (,t m;s ). (5.59) To close the estimate, the last term on the right-hand side of (5.59) is bounded by The first term yields ɛ c L 2 (,t m;s ) ɛ L 2 (,t m;s ) + U d L 2 (,t m;s ). (5.6) ɛ 2 L 2 (,t m;s ) = m 2 Simliarly the second term yields tn tn m t n l n (w n + U n )l n ɛ n c 2 l n ε + n 2 + l n ε 2 n t n m ( ε + 2 n + ε 2 ) n τn = 2 η2 e,m. U d 2 L 2 (,t m;s ) 2 (5.6) m δ nτ 2 n. (5.62) Merging these inqeualities with (5.59) and using the triangle inequality we obtain we obtain (5.32). e L 2 (,t m;s ) ɛ L 2 (,t m;s ) + ρ L 2 (,t m;s ) (5.63)

21 A POSTERIORI CONTROL IN PARABOLIC DISCONTINUOUS GALERKIN Remark (short-time integration). In the spirit of Theorem 3.7, it is possible to modify Theorem 5.9 and the appropriate indicators as to accommodate a short time-integration version of this result where L -accumulation in time replaces the L 2 -accumulation for certain estimators. Over shorter time-intervals this provides a tighter bound Theorem (a posteriori energy-error bound for Euler IPDG). Under the same assumptions of Theorem 5.9, assuming we employ the IPDG method in space as described in 4, the error bound (5.32) holds with the estimators η p,m, η e,m and m δ 2 nτ n explicitly computable as follows: (i) for the elliptic indicators ε n and ε + n, replace E by E IP, as defined in (4.9), into (5.22) and (5.23), respectively; (ii) for the nonconforming part indicators δ n and δ n, respectively, (cf. 5.3 and Lemma 4.3), we replace Ud n U n, U n d+ d and U n, (5.64) d+ respectively, by H (Ω) (iii) C PF C ȟn U n U n, C2 ˇσn U n, (5.65) where ȟn is the meshsize function of Tn ˇ and ˇσ n is related to it via (4.2); replace all H (Ω) norms by C PF times the L 2 (Ω) norm. References [ABCM2] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2/2), no. 5, (electronic). MR [Ain5] Mark Ainsworth, A synthesis of a posteriori error estimation techniques for conforming, non-conforming and discontinuous Galerkin finite element methods, Recent advances in adaptive computation, Contemp. Math., vol. 383, Amer. Math. Soc., Providence, RI, 25, pp. 4. MR MR (26m:65234) [Ain7], A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal. 45 (27), no. 4, (electronic). MR MR [Arn82] Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 9 (982), no. 4, MR 83f:6573 [Bak77] Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 3 (977), no. 37, MR MR43742 (55 #4737) [BBM5] A. Bergam, C. Bernardi, and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp. 74 (25), no. 25, 7 38 (electronic). MR MR (27c:6572) [BDD4] Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (24), no. 2, MR MR2577 (25d:65222) [BE7] Erik Burman and Alexandre Ern, Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Math. Comp. 76 (27), no. 259, 9 4 (electronic). MR MR [BGC5] Rommel Bustinza, Gabriel N. Gatica, and Bernardo Cockburn, An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems, J. Sci. Comput. 22/23 (25), MR MR24293 (26e:6522) [BHL3] Roland Becker, Peter Hansbo, and Mats G. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg. 92 (23), no. 5-6, MR MR (23k:6535) [CGJ8] Carsten Carstensen, Thirupathi Gudi, and Max Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Preprint, Humboldt Universität, Berlin, 28.