A Posteriori Error Estimation for hp-version Time-Stepping Methods for Parabolic Partial Differential Equations

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1 Numerische Mathematik manuscript No. (will be inserted by the editor) A Posteriori Error Estimation for hp-version Time-Stepping Methods for Parabolic Partial Differential Equations Dominik Schötzau 1, Thomas P. Wihler 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z, Canada, schoetzau@math.ubc.ca Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 301 Bern, Switzerland, wihler@math.unibe.ch The date of receipt and acceptance will be inserted by the editor Numerische Mathematik, Vol. 115, pp , 010 Summary The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cg) and the discontinuous Galerkin (dg) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments. Key words Parabolic problems, a posteriori error estimation, hpversion, continuous Galerkin time-stepping, discontinuous Galerkin time-stepping. Mathematics Subject Classification (1991): 65M60, 65J10 1 Introduction Galerkin time-stepping methods for initial-value problems were introduced, for example, in [13, 14, 17,]. These methods are based Supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Correspondence to: Thomas P. Wihler

2 D. Schötzau and T. P. Wihler on variational formulations of the initial-value problems and provide piecewise polynomial approximations in time. The approximation can be chosen to be either continuous or discontinuous at the time discretization points, thereby giving rise to the so-called continuous Galerkin (cg) and discontinuous Galerkin (dg) time-stepping methods, respectively. For both approaches, the discrete Galerkin formulations decouple into local problems on each time-step, and the discretizations can therefore be understood as implicit one-step schemes. The cg and dg time-stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3,5,10,11,16]. The application of cg and dg approaches to the time discretization of parabolic partial differential equations (PDEs) has been studied in [6 9,15,8] and the references therein. The variational character of the cg and dg methods allows for arbitrary variation in the size of the time-steps and the local approximation orders. Therefore, they can be extended naturally to hp-version Galerkin schemes; see, e.g., [3,31] for hp-version cg and dg discretizations of initial-value problems for ODEs. The main feature of these hp-methods is their ability to approximate smooth solutions with possible local singularities at high algebraic or even exponential rates of converge. In particular, exponential convergence can be achieved in the numerical approximation of problems with start-up singularities; see [4,5, 30] for linear parabolic PDEs and [4] for Volterra integro-differential equations with weakly singular kernels. Furthermore, it has been proved in [0,1] that the combination of hp-version time-stepping with suitable wavelet spatial discretizations results in log-linear complexity solution algorithms for nonlocal evolution processes involving pseudo-differential operators. In addition, the discretization of high-dimensional parabolic problems, using sparse grids in space, has been analyzed in [9]. In order to obtain a posteriori error estimates for the cg and dg methods several techniques have been proposed in the literature. For example, a posteriori error estimation based on duality techniques can be found in, e.g., [3, 10, 16] and the references therein. An alternative approach, applied to dg methods, has been recently presented in [19]. Here, the analysis is based on an appropriate reconstruction of the dg solution which allows the dg variational formulation to be rewritten in strong form, and subsequently, enables the application of natural energy arguments. In this paper, we shall use the h-version approach presented in [19], and extend it to the hp-version cg and dg methods. Specifically, we present a posteriori error estimators for both the hpversion cg and dg schemes which are fully explicit with respect to

3 A Posteriori Error Estimation for hp-version Time-stepping Methods 3 the size of the time-steps and polynomial degrees. The main novelties of the present analysis are: 1) a complete hp-characterization of the reconstruction operator from [19] and its difference (measured in a suitable norm) to the dg solution, and ) the application of a similar approach to the hp-version cg scheme. The resulting error estimators are tested within an hp-adaptive algorithm based on local Sobolev regularity estimation, as proposed in [1] for elliptic problems. Our numerical experiments indicate that exponential rates of convergence can be achieved for smooth problems with start-up singularities (as induced, for example, by incompatible initial data). Let us discuss some notation that will be used throughout the paper: The inner product and associated norm of a Hilbert space V are denoted by (, ) V and V, respectively. Furthermore, V signifies the dual space of V. Additionally, the duality pairing in V V is denoted by, V V and the dual norm by V. For an interval I = (a, b), the space C 0 (I; V ) consists of all functions u : I V that are continuous on I with values in V. It is endowed with the standard maximum norm u C 0 (I;V ) = max u(t) V. t I Moreover, L (I; V ) signifies the space of (classes of) measurable functions u : I V so that u(t) V is square-integrable over I (with respect to the Lebesgue measure on I). We notice that L (I; V ) is a Hilbert space with the inner product (u, v) L (I;V ) = (u(t), v(t)) V dt. The norm induced by (, ) L (I;V ) is ( u L (I;V ) = I I ) 1 u(t) V dt. Moreover, we recall that L (I; V ) = L (I; V ). For u L (I; V ) and v L (I; V ), we then have the duality pairing u, v L (I;V ) L (I;V ) = u(t), v(t) V V dt. Let us now introduce the linear parabolic problems considered in this work. This shall be done within an abstract Hilbert space setting. More precisely, let X H be two Hilbert spaces with continuous embedding. We further suppose that X is dense in H. Upon identification of H with its dual space H, the following Gelfand triple I

4 4 D. Schötzau and T. P. Wihler of Hilbert spaces is obtained: X H = H X. For T > 0 and given data u 0 H, g L ((0, T ); X ), we consider the parabolic problem u (t) + Au(t) = g(t), t (0, T ), u(0) = u 0. (1) Here, A : X X is a linear elliptic operator (in space) defined by Au, v X X = a(u, v) u, v X, for a bilinear form a : X X R that is assumed to be continuous and coercive. More precisely, there are two constants α, β > 0 such that a(u, v) α u X v X u, v X, () a(u, u) β u X u X. (3) The standard weak formulation of the parabolic problem (1) is to find u(t) such that u(0) = u 0 and u (t), v X X + a(u(t), v) = g(t), v X X for all v X and t (0, T ). Under the above assumptions, this variational problem has a unique weak solution u satisfying u L ((0, T ); X) C 0 ([0, T ]; H), u L ((0, T ); X ). Additionally, there holds the stability estimate u C 0 ([0,T ];H) + u L ((0,T );X) + u L ((0,T );X ) C ( g L ((0,T );X ) + u 0 H ), for a constant C > 0 that only depends on α and β in () and (3); see [18, Theorem 4.1]. Finally, we denote by û 0 X a generic approximation of u 0 H. Remark 1 We remark that, if a satisfies a Gårding inequality of the form a(u, u) β u X γ u H u X, then the corresponding parabolic problem can be cast into our setting by applying the substitution u = e γt ũ.

5 A Posteriori Error Estimation for hp-version Time-stepping Methods 5 The outline of the rest of the paper is as follows: In Section, we introduce the hp-version time-stepping methods for the parabolic model problem (1). In Section 3, we shall present our hp-version a posteriori error estimates which constitute the main results of this work. Furthermore, Section 4 contains the proof of an hp-version approximation property that is crucial in the analysis of dg methods. Additionally, we display a series of numerical results in Section 5. Finally, we present some concluding remarks in Section 6. Time-Stepping Methods In this section, we shall recall and discuss the hp-version formulations of the continuous and discontinuous Galerkin time-stepping methods from [3, 4] and [31], respectively, for the time (semi-) discretization of the parabolic problem (1). We consider a partition M = { } M m=1 of the time interval (0, T ) into M open subintervals = (t m 1, t m ), m = 1,,..., M, obtained from a set of nodes 0 = t 0 < t 1 < t <... < t M 1 < t M = T. We set k m = t m t m 1 and refer to as the m th time-step. Furthermore, to each time-step we assign a polynomial degree r m 0 and store these numbers in the vector r = {r m } M m=1. In the sequel, for an integer l, we write r ± l to denote the degree vector {r m ± l} M m=1. Additionally, we use the notation P r (I; X) to denote the space of all polynomials of degree at most r N 0 on I with coefficients in X, i.e., P r (I; X) = { p C 0 (I; X) : p(t) = r x i t i, x i X Then, for a partition M and a degree vector r, we introduce the discontinuous Galerkin space VdG r (M; X) = { U L ((0, T ); X) : U Im P rm ( ; X), 1 m M }, and the continuous Galerkin space V r cg(m; X) = V r dg (M; X) C0 ([0, T ]; X) = { U C 0 ([0, T ]; X) : U Im P rm ( ; X), 1 m M }. }.

6 6 D. Schötzau and T. P. Wihler Furthermore, for a piecewise continuous function U, we define the one-sided limits of U in X or H at the node t m by U + m = lim s 0 U(t m + s), U m = lim s 0 U(t m s). The jump of U at t m, 1 m M 1, is defined by [[U ] m = U + m U m. Note that, if U is continuous at t m, we have [U ] m = 0..1 Continuous Galerkin Time-Stepping For given partition M and a degree vector r, the hp-version continuous Galerkin (cg) method for the approximation of (1) is to find U cg V r+1 cg (M; X) such that U cg(0) = û 0 X and B cg (U cg, V ) = F cg (V ) V VdG r (M; X). (4) Here, for U V r+1 cg (M; X), V V r dg (M; X), the continuous Galerkin forms are given by B cg (U, V ) = F cg (V ) = M { (U, V ) H + a(u, V ) } dt, M g(t), V X X dt. m=1 m=1 It is well-known that the cg method in (4) is consistent and uniquely solvable; see, e.g., [13, 14,31]. Furthermore, since the test functions are discontinuous, the variational problem (4) decouples into local problems on each time-step, giving rise to an implicit one-step time marching scheme. Indeed, suppose that the approximate solution U cg is given on the time-steps I 1,..., 1, then U cg Im P rm+1 ( ; X) on is found by solving { (U cg, V ) H + a(u cg, V ) } dt = g(t), V X X dt for all V P rm ( ; X), with the additional condition that U + cg,m 1 = U cg,m 1. Here, we use the convention that U cg,0 = û 0 X.

7 A Posteriori Error Estimation for hp-version Time-stepping Methods 7. Discontinuous Galerkin Time-Stepping Given a partition M and a degree vector r, the hp-discontinuous Galerkin (dg) method for the approximation of (1) reads: Find U dg (M; X) such that V r dg B dg (U dg, V ) = F dg (V ) V VdG r (M; X). (5) Here, for U, V VdG r (M; X), the discontinuous Galerkin forms are given by B dg (U, V ) = F dg (V ) = M m=1 + { (U, V ) H + a(u, V ) } dt M ([U ] m 1, V m 1 + ) H + (U 0 +, V 0 + ) H, m= M m=1 g(t), V X X dt + (û 0, V + 0 ) H. The dg method in (5) is consistent and has a unique solution; see [8, Chapter 1] or [3,4]. Similarly to the cg method, it can also be interpreted as an implicit one-step time-stepping scheme. Suppose again that the approximate solution U dg is given on the time-steps I 1,..., 1. Then U dg Im P rm ( ; X) on is found by solving { (U dg, V ) H + a(u dg, V ) } dt + ([U dg ] m 1, V m 1 + ) H = g(t), V X X dt for all V P rm ( ; X), where we let [U dg ] 0 = û 0 X. 3 A Posteriori Error Estimation In this section, we shall develop hp-version a posteriori error estimates for the time-stepping schemes in (4) and (5). In order to prepare the tools for the ensuing analysis, we shall first recall two families of specialized polynomial bases and prove some auxiliary results related to L -projections. Then, in Sections 3.3 and 3.4, the cg and dg methods shall be analyzed, respectively.

8 8 D. Schötzau and T. P. Wihler 3.1 Polynomial Bases We shall express functions in the space VdG r (M; X) in terms of stepwise Legendre series. To this end, let K i (t) denote the standard Legendre polynomial of degree i 0 on the unit interval Î = ( 1, 1), normalized such that K i (1) = 1; cf., e.g., [1]. These polynomials satisfy the orthogonality relation K i (t) K j (t) dt = γ i δ i,j i, j 0, (6) bi with γ i = 1 i+1 and δ i,j denoting the Kronecker symbol. Furthermore, there holds K i ( 1) = ( 1) i. The reference interval Î can be mapped onto = (t m 1, t m ), 1 m M, by the use of the affine mapping F m : Î, ˆt t = F m (ˆt) = k m ˆt + t m 1 + t m. (7) On, the mapped Legendre polynomials are then defined by K m i (t) = K i (F 1 m (t)), t. A simple change of variables shows that K m i (t)k m j (t) dt = k m γ i δ i,j i, j 0. (8) Let now U be a piecewise polynomial in VdG r (M; X). On each timestep, it is a polynomial of degree r m. Hence, it can be expanded in the form U Im (t) = r m u m i K m i (t), t, with coefficients u m i X. From the orthogonality property (8) of the Legendre polynomials, it follows that U L (;X) = k m r m γ i u i m X. We shall also consider the integrated Legendre polynomials. On the unit interval Î = ( 1, 1) we define ( Q i (t) = γ i 1 Ki (t) K ) i (t), i 0, (9) where we set K 1 = K = 0 and γ 1 = 1. The function Q i (t) is a polynomial of degree i and the set { Q i (t)} r forms a basis of

9 A Posteriori Error Estimation for hp-version Time-stepping Methods 9 the polynomial space P r (Î; R). Note that Q 0 (t) = 1, Q1 (t) = t, and Q i (±1) = 0 for i. Moreover, well-known properties of the Legendre polynomials, see, e.g., [7, Section A.4], and the orthogonality conditions in (6) readily imply the following result. Lemma 1 For any i, there holds Q i (t) = t 1 K i 1 (τ) dτ. Furthermore, for i, j 0, the following relations hold: γi 1 γ i if i = j = 0, 1, γ Q i (t) Q i 1 (γ i + γ i ) if i = j, j (t) dt = γ i 1 γ i 3 γ i if i = j +, bi γ i 1 γ i+1 γ i if j = i +, 0 otherwise. 3. L -Projection Next, we define the L -projection for functions in L ( ; X ) and L ((0, T ); X ). We begin by showing the following auxiliary result. Lemma If p P rm ( ; X ) satisfies p, q L (;X ) L (;X) = 0 then p is the zero polynomial. q P rm ( ; X), Proof We expand p in the Legendre basis and have p(t) = r m j=0 p jk m j (t), with coefficients p j X. For 0 i r m and v X, consider the polynomial q(t) = vki m (t) P rm ( ; X). Then the orthogonality (8) of the Legendre polynomials yields r m 0 = p(t), q(t) X X dt = p j, v X X Kj m (t)ki m (t) dt = k m γ i p i, v X X. j=0 Since 0 i r m and v X is arbitrary, we conclude that This completes the proof. p i = 0 0 i r m.

10 10 D. Schötzau and T. P. Wihler For a function u L ( ; X ), we define its L -projection Πm,rm u to the space P rm ( ; X ) by requiring that Π,rm m u, V L (;X ) L (;X) = u, V L (;X ) L (;X) (10) for all V P rm ( ; X). In view of Lemma, the L -projection is uniquely defined. In fact, the following result holds true. Lemma 3 For u L ( ; X ), we have with u i X given by (Π,rm m u)(t) = r m u i K m i (t), u i, v X X = 1 k m γ i u, vk m i L (;X ) L (;X), v X. Proof We first remark that u i, v X X 1 k m γ i u L (;X ) vk m i L (;X). Therefore, since vki m L (I = m;x) v X Ki m (t) dt = k m γ i v X, the functional u i belongs to X. Let now U(t) = r m and let V P rm ( ; X) be given by V (t) = r m j=0 u i K m i (t), v j K m j (t), with coefficients v j X. From the orthogonality relation (8) and the definition of u i, we then obtain r m U, V L (;X ) L (;X) = u i, v j X X Ki m (t)kj m (t) dt = = i,j=0 r m r m k m γ i u i, v i X X u, v i K m i L (;X ) L (;X) = u, V L (;X ) L (;X).

11 A Posteriori Error Estimation for hp-version Time-stepping Methods 11 Hence, U is the L -projection. This completes the proof. Next, let us consider the L -projection of functions in L ( ; X) that are of the form R u(t) = u j Kj m (t), (11) j=0 with R N, i.e., only finitely many coefficients u j X in the above sum differ from zero. Lemma 4 Let u L ( ; X) be given by the series (11). Then we have Πm,rm (Au) = A(Πm rm u), where Πm rm signifies the L -projection from L ( ; H) to P rm (, H) with respect to the inner product (, ) L (;H). Proof We first note that Hence, (Π rm m u) (t) = r m A(Π rm m u) = On the other hand, we have Au = Applying Lemma 3, we obtain u i K m i (t) L ( ; X). (1) r m Au i K m i (t). (13) R Au j Kj m (t). j=0 Π,rm m (Au) = where, for any v X, there holds r m y i K m i (t), (14) yi, v X X = 1 Au, vki m k m γ L (;X ) L (;X) i = 1 Au, vki m X k m γ X dt i = 1 R Au j, v X k m γ X Kj m (t)ki m (t) dt, i j=0

12 1 D. Schötzau and T. P. Wihler i = 0, 1,..., r m. Thus, using the orthogonality property (8), yields y i, v X X = Au i, v X X v X. Therefore, by Lemma, we have that yi = Au i, i = 0, 1,..., r m. Comparing (13) and (14) completes the proof. as Finally, we define the following global L -projections elementwise Π,r Im = Π,rm m, Π r Im = Π rm m, 1 m M. (15) 3.3 Error Estimation for the cg Time-Stepping Method Let u be the solution of (1) and U cg V r+1 cg (M; X) the cg approximation from (4). On each time-step, the function U cg is a polynomial of degree r m + 1. Hence, it can be expanded in the form U cg (t) = r m+1 u m i K m i (t), t, (16) with coefficients u m i X. We now define the error measure { } E cg = max E 1,cG, E,cG, E 3,cG, (17) where E 1,cG = u U cg C 0 ([0,T ];H), α E,cG = u U cg L ((0,T );X), α E 3,cG = u Πr U cg L ((0,T );X). Next, we introduce the elemental error indicator η cg,m = k m r m + 3 um r m+1 X, m = 1,,..., M, (18) where u m r m+1 is the Legendre coefficient of order r m + 1 in the expansion (16). We further set η cg = M ηcg,m. (19) m=1 The following estimate is our main result for the continuous Galerkin time-stepping method. It shows that the error indicator η cg gives rise to a reliable and efficient a posteriori error estimate, up to data approximation terms.

13 A Posteriori Error Estimation for hp-version Time-stepping Methods 13 Theorem 1 Let η cg be defined in (19). Then the error measure (17) satisfies the upper bound E cg β α η cg + α g Π,r g L ((0,T );X ) + u 0 û 0 H, and the lower bound α 8 η cg E cg. Remark The estimates in Theorem 1 are fully explicit with respect to both the step sizes and the local polynomial degrees, and the constants occurring in the error bounds are explicitly given in dependence on the continuity and coercivity constants α, β of the bilinear form a(, ); cf. () and (3). The proof of Theorem 1 will follow along the lines of [19, Theorem 3.3 and Theorem 3.8]. Before carrying it out in detail, we adapt [19, Lemma 3.] to our setting. This leads to the following auxiliary result. Lemma 5 There holds a(u 1 u, u U ) α max { u U 1 X, u U } β X + α U 1 U X for any u, U 1, U X. Proof Using the coercivity and continuity of the elliptic form a(, ) in () and (3), we obtain a(u 1 u, u U ) = a(u 1 u, u U 1 ) + a(u 1 u, U 1 U ) α u U 1 X + β u U 1 X U 1 U X. The inequality ab ε a + 1 ε b, with ε = α /β, then gives a(u 1 u, u U ) α u U 1 X + β α U 1 U X. (0) Similarly, there holds a(u 1 u, u U ) = a(u u, u U ) + a(u 1 U, u U ) α u U X + β α U 1 U X. The inequalities (0) and (1) now imply the desired bound. (1) We are now ready to show Theorem 1.

14 14 D. Schötzau and T. P. Wihler Proof (Theorem 1) The cg solution U cg V r+1 cg (M; X) satisfies T 0 { T (U cg, V ) H + AU cg, V X X} dt = g, V X X dt for all V VdG r (M; X). We note that U cg V dg r (M; X). Furthermore, we have (U cg, V ) H = U cg, V X X for all V VdG r (M; X). Using the definition of the L -projection Π,r, see (10) and (15), and Lemma, there holds: 0 U cg + Π,r AU cg = Π,r g in V r dg (M; X ), U cg (0) = û 0. Furthermore, applying Lemma 4 leads to We conclude that U cg + AΠ r U cg = Π,r g, U cg (0) = û 0. (u U cg ) + A(u Π r U cg ) = g Π,r g in L ((0, T ); X ), (u U cg )(0) = u 0 û 0. Testing the above equation with (u U cg ) gives 1 d dt u U cg H a(π r U cg u, u U cg ) Here, we have used the fact that 1 = g Π,r g, u U cg X X. d dt u U cg H = (u U cg ), u U cg X X. Moreover, from Lemma 5, it follows that where a(π r U cg u, u U cg ) α M(t) β α U cg Π r U cg X, Therefore, M(t) = max { (u U cg )(t) X, (u Π r U cg )(t) X}. 1 d dt u U cg H + α M(t) β α U cg Π r U cg X + g Π,r g, u U cg X X. ()

15 A Posteriori Error Estimation for hp-version Time-stepping Methods 15 The inequality ab ε a + 1 ε b, with ε = /α, then shows that g Π,r g, u U cg X X g Π,r g X M(t) 1 α g Π,r g X + α 4 M(t). (3) We now combine the inequalities in () and (3), subtract the term α 4 M(t) on both sides, and multiply the resulting inequality by. We obtain d dt u U cg H + α M(t) β α U cg Π r U cg X + α g Π,r g X. We recall that u U cg C 0 ([0, T ], H). Hence, integrating the above estimate over (0, t) for 0 t T and observing the initial condition lead to (u U cg )(t) H + α M L ((0,t);X) β α U cg Π r U cg L ((0,T );X) Since this holds for any 0 t T, we obtain + α g Π,r g L ((0,T );X ) + u 0 û 0 H. E cg β α U cg Π r U cg L ((0,T );X) + α g Π,r g L ((0,T );X ) + u 0 û 0 H. (4) Furthermore, using the triangle inequality, we immediately obtain the lower bound U cg Π r U cg L ((0,T );X) 4 α ( α u U cg L ((0,T );X) + α u Πr U cg L ((0,T );X) ) (5) 8 α E cg. Then, from the solution representation in (16) and from (1), it follows that Hence, we have Π r U cg (t) = r m u m i K m i (t), t, U cg (t) Π r U cg (t) = u m r m+1k m r m+1(t), t,

16 16 D. Schötzau and T. P. Wihler and U cg Π r U cg L (;X) = k mγ rm+1 u m r m+1 X. (6) The proof of Theorem 1 now follows from (4) (6). 3.4 Error Estimation for the dg Time-Stepping Method We shall now derive an hp-version a posteriori error estimate for the dg method (5). For this purpose, consider a dg function U VdG r (M; X). Then, following [19, Section.1], we define a reconstruction Û = R(U) VcG (M; X) of U by requiring that r+1 (Û, V ) H dt = (U, V ) H dt + ([U ] m 1, V ) H, (7) for all V P rm ( ; X), and Û m 1 + = U m 1. On the first element, we take Û 0 + = û 0. From [19, Lemma.1], it follows that Û is well-defined and continuous on [0, T ]. We note that the operator R is only required for the purpose of the analysis and does not need to be computed in practice. A second main result of this paper is the complete hp-version characterization of the difference U Û appearing in the a posteriori error analysis of the dg method (5). More precisely, we will prove the following identities. Theorem Let U VdG r (M; X). Then, for 1 m M, we have U Û L (I = k r m + 1 m;x) m (r m + 1)(r m + 3) [U ] m 1 X. Consequently, there holds U Û L ((0,T );X) = M m=1 k m r m + 1 (r m + 1)(r m + 3) [U ] m 1 X. The proof of this result will be given in Section 4. Let now u be the solution of the parabolic equation (1) and U dg VdG r (M; X) the discontinuous Galerkin approximation from (5). We define the error measure { } E dg = max E 1,dG, E,dG, E 3,dG, (8)

17 A Posteriori Error Estimation for hp-version Time-stepping Methods 17 where E 1,dG = α u U dg L ((0,T );X), E,dG = u ÛdG C 0 ([0,T ];H), α E 3,dG = u ÛdG L ((0,T );X). Applying the ideas in [19], and proceeding similarly to the analysis of the cg method, the following hp-version a posteriori error estimates are obtained. Proposition 1 The error measure (8) satisfies the upper bound E dg β α U dg ÛdG L ((0,T );X) + α g Π,r g L ((0,T );X ) + u 0 û 0 H, and the lower bound α 8 U dg ÛdG L ((0,T );X) E dg. Proof Recalling the definition of the dg scheme and of the reconstruction ÛdG = R(U dg ) of U dg, there holds T 0 { } (Û dg, V ) H + AU dg, V X X dt = T 0 g, V X X dt for all V VdG r (M; X). Then, since Û dg, AU dg VdG r (M; X ), it follows from the definitions (10) and (15) of the L -projection that (u ÛdG) + A(u U dg ) = g Π,r g in L ((0, T ); X ), (u ÛdG)(0) = u 0 û 0. Testing this equation with u ÛdG gives 1 d dt u ÛdG H a(u dg u, u ÛdG) = g Π,r g, u ÛdG X X. Upon setting M(t) = max the bound in Lemma 5 ensures that { } (u U dg )(t) X, (u ÛdG)(t) X, a(u dg u, u ÛdG) α M(t) β α U dg ÛdG X.

18 18 D. Schötzau and T. P. Wihler Furthermore, similarly to (3), it holds that g Π,r g, u ÛdG X X 1 α g Π,r g X + α 4 M(t). Hence, as in the proof of the a posteriori error estimate for the cg method, see () and (3), it follows that d dt u ÛdG H + α M(t) β α U dg ÛdG X + α g Π,r g X. Therefore, integrating from 0 to t leads to (u ÛdG)(t) H + α M L ((0,t);X) β α U dg ÛdG L ((0,T );X) + α g Π,r g L ((0,T );X ) + u 0 û 0 H. This readily implies the lower bound. The upper bound follows from the triangle inequality U dg ÛdG L ((0,T );X) 4 α ( α u U dg L ((0,T );X) + α u ÛdG L ((0,T );X) 8 α E dg, which completes the proof. For m = 1,,... M, we now introduce the elemental error indicator ηdg,m = k r m + 1 m (r m + 1)(r m + 3) [U dg ] m 1 X, (9) and set M ηdg = ηdg,m. (30) m=1 The combination of Theorem and Proposition 1 yields the following hp-version a posteriori error estimate for the dg time-stepping scheme (5). Theorem 3 Let η dg be defined in (30). Then the error measure (8) satisfies the upper bound E dg β α η dg + α g Π,r g L ((0,T );X ) + u 0 û 0 H, and the lower bound α 8 η dg E dg. )

19 A Posteriori Error Estimation for hp-version Time-stepping Methods 19 We note that, as for the cg method, the above a posteriori error estimates are fully explicit with respect to the time-steps, the local polynomial degrees, and the stability constants α and β corresponding to the bilinear form a(, ); cf. () and (3). 4 Proof of Theorem In this section, we will present the proof of Theorem. In Section 4.1, we will derive a representation formula for the difference U Û in terms of a lifting operator. Section 4. focuses on the properties of (more general) polynomial lifting operators. Finally, in Section 4.3 we will complete the proof of Theorem. 4.1 A Representation Formula We shall first derive a representation formula for the reconstruction error U Û occurring in Theorem. Let U VdG r (M; X) be a dg function. As in the unifying framework proposed in [] for the analysis of discontinuous Galerkin methods for elliptic problems, we rewrite the jumps of U in terms of lifting operators. Specifically, for 1 m M, we define the lifting by requiring that L m : V r dg (M; X) Prm ( ; X) (L m (U), V ) H dt = ([[U ] m 1, V + m 1 ) H V P rm ( ; X), (31) where we use [U ] 0 = U 0 + û 0 on the first element. The following identity holds. It will be derived from a more general result and will be proved at the end of Section 4.. Proposition The lifting L m (U) from (31) is well-defined and we have L m (U) L (I = (r m + 1) m;x) [U ] m 1 X for any U VdG r (M; X). In order to relate the reconstruction Û = R(U) of a dg function U defined in (7) to the lifting L m (U), we notice that, for U, V k m

20 0 D. Schötzau and T. P. Wihler V r dg (M; X), the discontinuous Galerkin form B dg can be expressed as B dg (U, V ) = M m=1 { (U + L m (U), V ) H + a(u, V ) } dt. More importantly, the defining properties of the reconstruction Û in (7) can be written as (Û, V ) H dt = (U + L m (U), V ) H dt V P rm ( ; X), and with U 0 Û + m 1 = U m 1, (3) = û 0. Hence, since Û P rm ( ; X), we have Û = U + L m (U) on. Integrating this equation over (t m 1, t) for t m 1 t t m, we obtain Û(t) Û + m 1 = U(t) U + m 1 + t t m 1 L m (U) dτ. Then, applying (3), we obtain the representation formula t U(t) Û(t) = [U ] m 1 L m (U) dτ, t m 1 t m 1 t t m, (33) for 1 m M, where [U ] 0 = U + 0 û 0 on the first element. 4. Lifting operators In this section, we introduce and analyze generic lifting operators. In addition, we will prove the stability result in Proposition. We shall first consider lifting operators on the unit interval Î = ( 1, 1). Let r 0 and z X be fixed. We denote by L r z the polynomial in Pr (Î; X) that satisfies ( L r z(t), V (t)) H dt = (z, V r ( 1)) H V P (Î; X). (34) bi

21 A Posteriori Error Estimation for hp-version Time-stepping Methods 1 Lemma 6 The lifting L r z is well-defined and can be represented in the form L r z(t) = z r ( 1) i (i + 1) K i (t). Furthermore, the identity holds. L r z L ( I;X) b = z X (r + 1) Proof By expanding L r z(t) in Legendre polynomials, we have r L r z(t) = a i Ki (t), with coefficients a i X to be determined. In view of the orthogonality property (6) of the Legendre polynomials, we see that a i = 1 L r γ z(t) K i (t) dt. i Therefore, (a i, v) H = 1 γ i bi ( L r z(t), v K i (t)) H dt v X. bi Testing (34) with V (t) = v K i (t), v X, and noting that K i ( 1) = ( 1) i, yields that (a i, v) H = 1 ( L r γ z(t), v K i (t)) H dt = 1 (z, v i bi γ K i ( 1)) H i = ( 1)i γ i (z, v) H. Since this holds for any v X and X is dense in H, we obtain a i = z 1 ( 1) i = z γ i (i + 1)( 1)i. This shows the first part of the lemma. Moreover, using this representation of L r z(t) and the orthogonality properties of the Legendre polynomials, we conclude that L r z L ( b I;X) = z X 4 r = z X (r + 1). This proves the second claim. γ i (i + 1) = z X r (i + 1)

22 D. Schötzau and T. P. Wihler Let us now consider the interval = (t m 1, t m ). For r m 0 and z X, we define the lifting L rm m,z P rm ( ; X) by (L rm m,z(t), V (t)) H dt = (z, V (t m 1 )) H Lemma 7 We have and L rm m,z(f m (ˆt)) = k m Lr m z (ˆt), ˆt Î, L rm m,z L (;X) = k m L rm z L ( b I;X), where F m is the element mapping in (7). V P rm ( ; X). Proof Let V P rm (Î; X). Define V Prm ( ; X) by V (F m (ˆt)) = V (ˆt). Then, the definition of the lifting operators and a change of variables lead to ( L rm z (ˆt), V (ˆt)) H dˆt = (z, V ( 1)) H = (z, V (t m 1 )) H bi = (L rm m,z(t), V (t)) H dt = k m (L rm m,z(f m (ˆt)), V (ˆt)) H dˆt. bi Since this holds for any V rm P (Î; X), the first claim follows. Moreover, this result and the change of variables t = F m (ˆt) show that L rm m,z(t) X dt = k m = k m bi bi L rm m,z(f m (ˆt)) X dˆt L rm z (ˆt) X dˆt. This completes the proof. We are now ready to prove Proposition. Proof (Proposition ) Consider a function U VdG r (M; X), a timestep = (t m 1, t m ) and a polynomial degree r m 0. Taking z = [U ] m 1, it follows directly that L m (U) = L rm m,z = L rm m,[[u]] m 1.

23 A Posteriori Error Estimation for hp-version Time-stepping Methods 3 The claim in Proposition now follows by combining Lemma 6 and Lemma 7. More precisely, we have L m (U) L (I = m;x) Lrm m,z L (I = m;x) L rm z k m which is the identity in Proposition. = (r m + 1) [U ] m 1 k m X, L ( b I;X) 4.3 Reconstruction Error In this subsection, we present an hp-version analysis for the difference U Û in (33) and prove Theorem. Again, we first consider the unit interval Î = ( 1, 1). As before, let r 0 and z X be fixed. Then, motivated by (33), we define Ê r z(t) = with L r z defined in (34). t 1 Lemma 8 For r 0, we have Êr z(t) = z b i = L r z(τ) dτ + z, t Î, (35) ( r+1 ) b i Qi (t) with { 1 if i = 0, ( 1) i (i 1) if i 1. Here, Q i is the i th integrated Legendre polynomial from Lemma 1. Proof Integrating the representation of L r z in Lemma 6 yields ( t r L r z(τ) dτ = z ) t ( 1) i (i + 1) K i (τ) dτ 1 1 ( = z r+1 ) t ( 1) i 1 (i 1) K i 1 (τ) dτ. i=1 Using the first identity in Lemma 1 and the fact that K 0 (t) = 1, we conclude that ( ) t L r z(τ) dτ = z r t + ( 1) i 1 (i 1) Q i (t). 1 i= 1

24 4 D. Schötzau and T. P. Wihler Then, ( ) Êz(t) r = z r+1 1 t + ( 1) i (i 1) Q i (t). i= Since Q 0 (t) = 1 and Q 1 (t) = t, the claim follows. Let us consider the lowest-order cases where r = 0 and r = 1, respectively. It can be readily seen that We then obtain Ê0 z L ( b I;X) = z X 4 Ê1 z L ( b I;X) = z X 16 Êz 0 (t) = z (1 t), Ê 1 z (t) = z 4 ( 1 t + 3t ). (1 t) dt = bi 3 z X, (1 t + 3t ) dt = 4 bi 15 z X. For general r 0, the following result holds. (36) (37) Lemma 9 For r 0, we have Êr z L ( b I;X) = r + (r + 1)(r + 3) z X. Proof We start from the representation in Lemma 8, and note that Êr z L ( I;X) b = z r+1 X b i b j Q i (t) 4 Q j (t) dt, r 0. i,j=0 bi To simplify notation, let us define T r = r+1 i,j=0 b i b j M i,j, M i,j = bi Q i (t) Q j (t) dt. Hence, we need to show that T r = 8r + 8 (r + 1)(r + 3). (38) We prove (38) by induction. From (37) we see that T 0 = 8 3, T 1 = 16 15,

25 A Posteriori Error Estimation for hp-version Time-stepping Methods 5 and hence, relation (38) holds true for r = 0 and r = 1. Suppose now that it holds for r 1. We need to prove that T r+1 = To that end, we note that T r+1 = r+ i,j=0 b i b j M i,j = Since M i,j = M j,i, we obtain 8r + 16 (r + 3)(r + 5). (39) r+1 i,j=0 r+1 b i b j M i,j + b i b r+ M i,r+ r+1 + b r+ b j M r+,j + b r+m r+,r+. j=0 r+1 T r+1 = T r + b r+ b i M i,r+ + b r+m r+,r+. From Lemma 1 we conclude that and b r+ r+1 b i M i,r+ = 4b r+ b r γ r 1 γ r+1 γ r, Therefore, it follows that b r+m r+,r+ = b r+γ r+1(γ r+ + γ r ). T r+1 = T r 4b r+ b r γ r 1 γ r+1 γ r + b r+γ r+1(γ r+ + γ r ). Using the induction assumption that (38) holds, and the definition of γ i, we obtain T r+1 = 8r + 8 (r + 1)(r + 3) 4b r+ b r (r 1)(r + 1)(r + 3) + 4b r+ (r + 1)(r + 3)(r + 5). Note that, from Lemma 8, we have b r = ( 1) r (r 1). Using this identity and elementary manipulations yields (39).

26 6 D. Schötzau and T. P. Wihler Next, we scale the results above to an interval = (t m 1, t m ) of length k m and a polynomial degree r m 0. Similarly to (35), we define the function Em,z rm on by t Em,z(t) rm = L rm m,z(τ) dτ + z, t, t m 1 with L rm m,z defined above. Recalling that F m is the elemental mapping in (7), we have the following identity. Lemma 10 There holds Moreover, E rm m,z(f m (ˆt)) = Êrm z (ˆt), ˆt Î. E rm m,z L (;X) = k m Êrm z L ( b I;X). Proof Changing variables and the relation in Lemma 7 yield Fm(ˆt) Em,z(F rm m (ˆt)) = = k m = F m( 1) ˆt ˆt 1 1 L rm z L rm m,z(τ) dτ + z L rm m,z(f m (ˆτ)) dˆτ + z (ˆτ) dˆτ + z = Êrm z (ˆt). This shows the first claim. Furthermore, using this identity, we have that Em,z(t) rm X dt = k m E rm m,z(f m (ˆt)) X dˆt This completes the proof. = k m bi bi Êrm z (ˆt)) X dˆt. Let us now turn to the proof of Theorem. Proof (Theorem ) Let U VdG r (M; X). Consider the interval = (t m 1, t m ) and the polynomial degree r m 0. Setting z = [U ] m 1, we conclude from equations (31), (33) and the definition of Em,z rm that U(t) Û(t) = Erm m,z(t) = E rm m,[[u]] m 1 (t), t.

27 A Posteriori Error Estimation for hp-version Time-stepping Methods 7 Hence, by using Lemma 9 and Lemma 10, we obtain U Û L (I = m;x) Erm m,z L (I = k m m;x) Êrm z = k m L ( b I;X) r m + (r m + 1)(r m + 3) [U ] m 1 X. This shows Theorem. 5 Numerical Experiments In this section we shall illustrate the performance of the error estimators η cg and η dg from Theorem 1 and Theorem 3 for the cg and dg schemes, respectively, within an hp-adaptive refinement algorithm. Specifically, given the numerical solution on the entire interval (0, T ), we use the local error estimators η cg,m and η dg,m in (18) and (9), respectively, to identify those time-steps on which large errors occur. More precisely, an element is marked for refinement if η cg,m > θ max i η cg,i, respectively η dg,m > θ max i η dg,i. Here, θ is a parameter which we set to be 0.5 in all of our numerical experiments. Subsequently, for each of the marked elements a decision is made whether h- or p-refinement is applied. To that end, we use a local smoothness estimation technique as presented in, e.g., [1]. Starting from a coarse initial time partition and a low-order polynomial degree vector, this procedure is repeated until a given tolerance τ is met. Our goal is to show that, for the examples considered, the error estimators for the cg and dg approximations over the whole interval (0, T ) decay at the same (asymptotic) rate as the actual error measures in (17) and (8), respectively, and that exponential convergence is achieved even for solutions with start-up singularities. For detailed implementational techniques for hp-version dg time-stepping methods we refer to [4, 9]. We discretize our model problems using a high-order finite element method in space so that the spatial errors are dominated by the errors resulting from the cg and dg time discretizations. For our numerical experiments, we shall consider the homogeneous heat equation in one space dimension, u t (x, t) u xx (x, t) = 0, (x, t) (0, 1) (0, 1), (40) subject to Dirichlet boundary conditions u(x, t) = 0 (x, t) {0, 1} (0, 1),

28 8 D. Schötzau and T. P. Wihler and the initial condition u(x, 0) = u 0 (x) x (0, 1). Here, the Hilbert spaces from Section 1 are the standard Sobolev spaces of order zero and one: H = L (0, 1) and X = H0 1 (0, 1). Furthermore, the elliptic operator is A = d (in the weak sense). dx Hence, α = β = 1 in () and (3), respectively. We will investigate the above problem numerically for different choices of u 0. Specifically, we shall look at Example 1 : u (1) 0 (x) = sin(πx), Example : u () 0 (x) = x(1 x), Example 3 : u (3) 0 (x) = 1. We denote by u (i) the solution corresponding to the initial data u (i) 0, i = 1,, 3. The solution u (1) is given by u (1) (x, t) = e πt sin(πx); it is arbitrarily smooth in x and t. The solutions u () and u (3) can be readily expressed in terms of Fourier series; they both have start-up singularities at t = 0. More precisely, it can be seen (see [4, p. 868]) that u () H 5 4 ε ( (0, 1); H 1 0 (0, 1) ), u (3) H 1 4 ε ( (0, 1); H 1 0 (0, 1) ), for any ε > 0. Here, H s ((0, 1); H0 1 (0, 1)) denotes the Sobolev space of order s of functions on (0, 1) with values in H0 1 (0, 1). In Figures 1 3 we plot the time meshes and local polynomial degrees resulting from the hp-adaptive dg time discretizations of Examples 1 3. The error tolerance is chosen to be τ = 10 6 for Example 1, τ = 10 5 for Example, and τ = 10 1 for Example 3. Furthermore, in Examples 1 and, we use 4 elements in space and a uniform spatial polynomial degree of 10. For Example 3, geometric mesh refinement in space was applied (with a theoretically optimal factor of 0.17) to appropriately resolve the incompatibility of the initial datum at x = 0 and x = 1; cf. [6]. The horizontal axis represents the time partition, and on the vertical axis, the local polynomial degrees are displayed. As expected, the smooth solution in Example 1 is approximated on a uniform time mesh, and higher polynomial degrees are used. For Examples and 3, due to the singular behavior of the solution at the start, the time partition is geometrically refined at t = 0. (Note that we have used a logarithmic scale on the horizontal axis.) This is in correspondence with the a priori error analysis in [4]. Furthermore, we notice that the polynomial degrees tend to decrease away

29 A Posteriori Error Estimation for hp-version Time-stepping Methods local polynomial degree time mesh Fig. 1. Example 1: Adaptively refined time mesh and polynomial degrees for the dg method local polynomial degree time mesh Fig.. Example : Adaptively refined time mesh and polynomial degrees for the dg method. from t = 0, which is due to the fact that the right-hand side of (40) is zero, and hence, the solution is flattened out quickly. Similar results (particularly for Examples 1 and ) are obtained for the cg method.

30 30 D. Schötzau and T. P. Wihler local polynomial degree time mesh Fig. 3. Example 3: Adaptively refined time mesh and polynomial degrees for the dg method. Figures 4 6 display the behavior of the errors (i.e., the error measures in (17) and (8)) and the error estimators η cg and η dg of the cg and dg time discretizations for Examples 1 3. Here, the numerical solutions are compared to the known exact solutions, and integrals are computed using Gaussian quadrature of sufficiently high order. In addition, the efficiency indices, i.e., the ratio between the error estimators and the actual errors, are shown. All graphs are presented in a semi-logarithmic coordinate system. The horizontal coordinate axes correspond to the number of degrees of freedom N in the time discretization (more precisely, N for Example 1, and N 1 for Examples and 3; cf. [4]). We see that the errors decay exponentially. Moreover, the efficiency indices are consistently between 1 and, thereby indicating the sharpness and asymptotic correctness of the estimators for the given examples. The slow initial decay of the errors for the cg method in Example 3 may be explained by the fact that the cg time discretization is less dissipative than the dg scheme, and therefore, large errors at {x = 0, 1} {t = 0} caused by the incompatibility of the initial conditions might be smoothed out at a comparatively low rate during the initial refinements.

31 A Posteriori Error Estimation for hp-version Time-stepping Methods error measure error estimator efficiency index N error measure error estimator efficiency index N Fig. 4. Example 1: Performance of cg (top) and dg (bottom).

32 3 D. Schötzau and T. P. Wihler error measure error estimator efficiency index N 1/ error measure error estimator efficiency index N 1/ Fig. 5. Example : Performance of cg (top) and dg (bottom).

33 A Posteriori Error Estimation for hp-version Time-stepping Methods error measure error estimator efficiency index N 1/ 10 1 error measure error estimator efficiency index N 1/ Fig. 6. Example 3: Performance of cg (top) and dg (bottom).

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