Technische Universität Graz

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1 echnische Universität Graz Coercive space time finite element methods for initial boundary value problems O. Steinbach, M. Zank Berichte aus dem Institut für Angewandte Mathematik Bericht 18/7

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3 echnische Universität Graz Coercive space time finite element methods for initial boundary value problems O. Steinbach, M. Zank Berichte aus dem Institut für Angewandte Mathematik Bericht 18/7

4 echnische Universität Graz Institut für Angewandte Mathematik Steyrergasse 3 A 81 Graz WWW: c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.

5 Coercive space time finite element methods for initial boundary value problems Olaf Steinbach, Marco Zank Institut für Angewandte Mathematik, U Graz, Steyrergasse 3, 81 Graz, Austria o.steinbach@tugraz.at, zank@math.tugraz.at Abstract We propose and analyse new space time Galerkin Bubnov type finite element formulations of parabolic and hyperbolic second order partial differential equations in finite time intervals. his approach is based, using Hilbert type transformations, on elliptic reformulations of first and second order time derivatives, for whiche Galerkin finite element discretisation results in positive definite and symmetric matrices. For the variational formulation of the heat and of the wave equation we prove related stability conditions in appropriate norms, and we discuss the stability of related finite element discretisations. Numerical results are given which confirm the theoretical results. 1 Introduction While for the analysis of parabolic and hyperbolic partial differential equations a variety of approaches such as Fourier methods, semigroups, or Galerkin methods is available, see, for example, [19, 3, 4, 7, 37, 39], standard approaches for the numerical solution are based on semi discretisations where the discretisation in space and time is splitted accordingly, see, e.g., [35] for parabolic partial differential equations, and [7, 8] for hyperbolic problems. More recently, there exist space time approaches as for example in [1, 5, 6, 9, 3, 36] for parabolic problems, and [3, 5, 1, 15, 4] for hyperbolic equations. In this work we introduce a new Fourier type method for the analysis of first and second order ordinary differential equations, and we transfer this approaco the corresponding parabolic and hyperbolic partial differential equations. he aim of this work is to provide space time Galerkin Bubnov type variational formulations where unique solvability follows from related coercivity estimates. his analysis may then serve not only as basis for the development and numerical analysis of adaptive space time finite element methods simultaneously in space and time, and for the construction of time parallel iterative solution strategies, but also for the analysis of related boundary integral equations methods for 1

6 the heat and wave equation, respectively, and the coupling of finite and boundary element methods. As a first model problem we consider the Dirichlet boundary value problem for the heat equation, α t u(x,t) x u(x,t) f(x,t) for (x,t) Q : Ω (,), u(x,t) for (x,t) Σ : Γ (,), (1.1) u(x,) for x Ω, where Ω R d, d 1,,3, is a bounded domain with, for d,3, Lipschitz boundary Γ Ω, α > is a given heat capacity constant, and f(x,t) is a given right hand side. Note that in the spatially one dimensional case d 1 we have Ω (a,b) and Γ {a,b}. A variational formulation of (1.1) is to find u L (,;H 1(Ω)) H1, (,;H 1 (Ω)) suchat α t u(x,t)v(x,t)dxdt+ x u(x,t) x v(x,t)dxdt (1.) Ω Ω f(x,t)v(x,t)dxdt is satisfied for all v L (,;H 1 (Ω)), where we assume f L (,;H 1 (Ω)). Note that we use the standard Bochner spaces, where u H 1, (,;H 1 (Ω)) satisfies u(x,) for x Ω. Related to the variational formulation (1.) we introduce the bilinear form a(u,v) : Ω [ ] α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) dxdt. (1.3) Since(1.) is a Galerkin Petrov variational formulation we need to establish an appropriate stability condition to ensure unique solvability, see also [13, 9, 3, 36]. A key ingredient in deriving the stability condition will be the definition of an appropriate norm as it is done in what follows. For u L (,;H 1(Ω)) H1, (,;H 1 (Ω)) we define w L (,;H 1 (Ω)) as the unique solution of the quasi static Dirichlet boundary value problem } x w(x,t) α t u(x,t) x u(x,t) for (x,t) Ω (,), (1.4) w(x,t) for (x,t) Γ (,), which is the unique solution of the variational formulation Ω x w(x,t) x v(x,t)dxdt Ω [ α t u(x,t)v(x,t)+ x u(x,t) x v(x,t) Ω Ω [ ] α t u(x,t) x u(x,t) v(x,t)dxdt ] dxdt a(u,v)

7 for all v L (,;H 1 (Ω)). When chosing v w this gives w L (,;H 1 (Ω)) and by duality we also obtain Ω Ω x w(x,t) dxdt [ ] α t u(x,t) x u(x,t) w(x,t)dxdt α t u x u L (,;H 1 (Ω)) w L (,;H 1 (Ω)), α t u x u L (,;H 1 (Ω)) α t u x u,v Q sup v L (,;H 1(Ω)) v L (,;H 1(Ω)) x w, x v L(Q) sup v L (,;H 1(Ω)) x v L (Q) x w L (Q) w L (,;H 1 (Ω)). In particular we have α t u x u L (,;H 1 (Ω)) w L (,;H 1 (Ω)) a(u,w) (1.5) as the energy norm for u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)). Indeed, α t u x u L (,;H 1 (Ω)) for u L (,;H 1 (Ω)) H 1,(,;H 1 (Ω)) implies u since the homogeneous heat equation with zero Dirichlet boundary conditions and zero initial conditions has only the trivial solution. An immediate consequence of (1.5) is the stability estimate α t u x u L (,;H 1 (Ω)) sup v L (,;H 1 (Ω)) a(u,v) v L (,;H 1 (Ω)) (1.6) for u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)). From (1.5) we further conclude w L (,;H 1(Ω)) a(u,w) [ ] α t u(x,t)w(x,t)+ x u(x,t) x w(x,t) dxdt Ω α t u L (,;H 1 (Ω)) w L (,;H 1(Ω)) + x u L (Q) x w L (Q) α t u L (,;H 1 (Ω)) + xu L (Q) w L (,;H 1(Ω)), i.e. for all u L (,;H 1 (Ω)) H 1 (,;H 1 (Ω)) we have α t u x u L (,;H 1 (Ω)) α t u L (,;H 1 (Ω)) + xu L (Q). (1.7) 3

8 For the reverse inequality and for u L (,;H 1 (Ω)) H 1,(,;H 1 (Ω)) we consider α t u x u L (,;H 1 (Ω)) a(u,w) a(u,u)+a(u,w u) [ ] α t uu+ x u x u dxdt+ x w x (w u)dxdt In particular, Ω α u() L (Ω) + xu L (Q) + x(w u) L (Q) + xu, x (w u) L (Q) x u L (Q) + x(w u) L (Q) xu L (Q) x (w u) L (Q) 1 [ ] x u L (Q) + x(w u) L (Q) 1 [ ] x u L (Q) + α tu L (,;H 1 (Ω)). (1.8) u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)) : Ω α t u L (,;H 1 (Ω)) + xu L (Q) (1.9) defines a norm in L (,;H 1 (Ω)) H1, (,;H 1 (Ω)) which is equivalent to (1.5), and from (1.6) we now conclude the stability condition 1 u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)) sup v L (,;H 1 (Ω)) a(u,v) v L (,;H 1 (Ω)) (1.1) for u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)). Note that in [3] the norm (1.9) was used within a finite element analysis of the variational formulation (1.). In particular, the bilinear form (1.3) is continuous satisfying a(u,v) u L (,;H 1 (Ω)) H1, (,;H 1 (Ω)) v L (,;H 1 (Ω)) (1.11) for u L (,;H 1 (Ω)) H1 (,;H 1 (Ω)) and v L (,;H 1 (Ω)). For v L (,;H 1 (Ω)) we finally define ũ(x,t) : t v(x,s)ds for x Ω, t [,]. By definition we have ũ L (,;H 1 (Ω)) H 1,(,;H 1 (Ω)). hen, a(ũ,v) Ω Ω [ ] α t ũ(x,t)v(x,t)+ x ũ(x,t) x v(x,t) dxdt [ ] α[ t ũ(x,t)] + x ũ(x) x t ũ(x,t) dxdt α t ũ L (Q) + 1 xũ() L (Ω) >. (1.1) 4

9 By using the stability condition (1.1), the continuity (1.11), and surjectivity (1.1) this implies unique solvability of the variational problem (1.), see, e.g., [6, 13]. he initial Dirichlet boundary value problem (1.1) therefore defines an isomorphism L : L (,;H 1 (Ω)) H 1,(,;H 1 (Ω)) [L (,;H 1 (Ω))]. (1.13) When considering the variational formulation (1.) and doing integration by parts in time, this leads to the adjoint variational formulation to find u L (,;H 1 (Ω)) suchat u(x,t)α t v(x,t)dxdt+ x u(x,t) x v(x,t)dxdt (1.14) Ω Ω f(x,t)v(x,t)dxdt is satisfied for all v L (,;H 1 (Ω)) H1, (,;H 1 (Ω)), where the test space now includes the final time condition v(x,) for x Ω, and where we now assume f [L (,;H 1 (Ω)) H1, (,;H 1 (Ω))]. As for the primal variational formulation (1.) we can establish unique solvability of the adjoint variational formulation (1.14), whichen implies an isomorphism L : L (,;H 1 (Ω)) [L (,;H 1 (Ω)) H 1,(,;H 1 (Ω))]. (1.15) Bothe primal variational formulation(1.) and the adjoint variational formulation(1.14) are Galerkin Petrov formulations where the test space is different from the ansatz space, in particular with respect to time. his motivates to consider variational formulations for the initial boundary value problem (1.1) where ansatz and test spaces are of the same order also in time. Using the isomorphisms (1.13) and (1.15) and some interpolation arguments one expects to consider test and ansatz spaces as subspaces of the anisotropic Sobolev space H 1,1/ (Q), e.g., [4, 19,, 3, 4]. In the case of an infinite time interval, i.e., such an approach was considered analytically in the PhD thesis of M. Fontes, [14], see also [1] for a related numerical analysis using wavelets. However, here we will consider only finite time intervals with <. In the case of time periodic boundary value problems, a related approach is considered in []. Althoughe numerical analysis of space time finite element methods for the variational formulation(1.) is well established, see, e.g., [1, 5, 6, 9, 3, 36], the analysis of boundary integral equations and related boundary element methods for the solution of the heat equation (1.1) relies on Galerkin Bubnov variational formulations in anisotropic Sobolev trace spaces of H 1,1/ (Q), see, e.g., [, 4]. In particular, instead of the stability condition (1.6) in the finite element analysis, an ellipticity estimate in the boundary element analysis is used. So we are interested in a unified approaco analyse both finite and boundary element methods within one framework, and allowing a numerical analysis also for the coupling of space time finite and boundary element methods. In addition to the initial boundary value problem (1.1) of the heat equation we also Ω 5

10 consider the related model problem for the wave equation, 1 c tt u(x,t) x u(x,t) f(x,t) for (x,t) Q : Ω (,), u(x,t) for (x,t) Σ : Γ (,), u(x,) t u(x,t) t for x Ω. (1.16) A standard approach for a space time finite element method to solve (1.16) is to consider an equivalent system with first order time derivatives, see, e.g., [3, 1]. Alternatively, one may consider variational formulations of the wave equation in (1.16) using integration by parts also in time, see, e.g., [5, 15, 4]. Here we will consider related variational formulations in suitable subspaces of H 1 (Q), and we will prove and discuss stability conditions in appropriate function spaces. he rest of this paper is organised as follows: In Sect. we consider simple first order ordinary differential equations to motivate the choice of a transformation operator to derive an elliptic and symmetric bilinear form for the first order time derivative. We discuss several properties of the Hilbert type transformation operator and we present some numerical results to illustrate the theoretical results. he results for the first order ordinary differential equations are extended in Sect. 3 to the heat equation in several space dimensions. We prove that the heat partial differential operator with zero Dirichlet boundary and initial conditions defines an isomorphism in certain anisotropic Sobolev spaces, implying a stability condition as required in the numerical analysis of the proposed Galerkin scheme. We comment on the stability of the numerical scheme and present some numerical results. Second order ordinary differential equations are considered in Sect. 4 where we introduce a different transformation operator which is not semi definite as in the case of first order equations. Hence we have to use different Sobolev norms to establish optimal stability estimates. As for the first order equations we provide a numerical analysis for the finite element discretisation, and we give some numerical results. Finally, in Sect. 5 we consider the space time variational formulation for the wave equation, we discuss the discretisation scheme, and we provide some numerical results for illustration. First order ordinary differential equations As a first model problem we consider for > the simple initial value problem t u(t) f(t) for t (,), u(), (.1) where we aim to derive and to analyse a coercive variational formulation which later will be used for the discretisation of time dependent partial differential equations which are of first order in time..1 Primal variational formulation If we define the Sobolev space } H, {v 1 (,) : H 1 (,): v(), 6

11 then the primal variational formulation of (.1) is to find u H 1,(,) suchat t u(t)v(t)dt f(t)v(t)dt for all v L (,). (.) Obviously it is sufficient to assume f L (,) in this case. Recall that u H, 1 (,) : tu L (,) [ t u(t)] dt defines a norm in H, 1(,). Lemma.1 he bilinear form a(, ) : H, 1(,) L (,) R, is bounded, i.e. a(u,v) : t u(t)v(t)dt, (.3) a(u,v) t u L (,) v L (,) for all u H 1,(,), v L (,), (.4) and satisfies the stability condition t u L (,) sup v L (,) a(u, v) v L (,) for all u H, 1 (,). (.5) Moreover, it holds v L (,) sup u H 1, (,) a(u,v) t u L (,) for all v L (,). (.6) Proof. he boundedness estimate (.4) is a direct consequence of the Cauchy Schwarz inequality. o prove (.5), for any u H, 1(,) we choose v tu L (,) to obtain i.e. a(u,v) t u(t)v(t)dt t u L (,) a(u,v) v L (,) Finally, for v L (,) we define u H 1,(,), u(t) t [ t u(t)] dt t u L (,) tu L (,) v L (,), sup v L (,) a(u, v) v L (,) v(s)ds, t u(t) v(t) for t [,]. 7.

12 hen, a(u,v) t u(t)v(t)dt implying the stability condition (.6), i.e. v L (,) a(u,v) t u L (,) [v(t)] dt v L (,) v L (,) t u L (,), sup u H 1, (,) a(u,v). t u L (,) As a consequence of Lemma.1, see, e.g., [6, Satz 3.6] or [13, Corollary A.45], we conclude unique solvability of the primal variational formulation (.), and the bilinear form (.3) implies, by using the Riesz representation theorem, a bounded and invertible operator satisfying B 1 : H 1,(,) L (,), u H 1, (,) B 1 u L (,). Dual variational formulation for all u H 1, (,). When using integration by parts, instead of the primal variational formulation (.) we may consider the dual variational formulation to find u L (,) suchat where u(t) t v(t)dt f(t)v(t)dt for all v H 1,(,), (.7) H, {v 1 (,) : H 1 (,): v() }, v H 1,(,) : [ t v(t)] dt. Now it is sufficient to assume f [H 1, (,)]. Lemma. he bilinear form a(, ) : L (,) H 1,(,) R, is bounded, i.e. a(u,v) : u(t) t v(t)dt, a(u,v) u L (,) t v L (,) for all u L (,), v H, 1 (,), (.8) and satisfies the stability condition Moreover, it holds u L (,) t v L (,) sup v H 1, (,) sup u L (,) a(u,v) t v L (,) a(u, v) u L (,) for all u L (,). (.9) for all v H, 1 (,). (.1) 8

13 Proof. he estimate (.8) is again a consequence of the Cauchy Schwarz inequality. o prove (.9), for any u L (,) we define v H 1, (,), hen, v(t) u(s)ds, t v(t) u(t) for t [,]. t a(u,v) u(t) t v(t)dt [u(t)] dt u L (,) u L (,) t v L (,), implying the stability condition (.9). o prove (.1), for any v H, 1 (,) we choose u t v L (,) to obtain a(u,v) u(t) t v(t)dt [ t v(t)] dt t v L (,) u L (,) t v L (,). As for the primal variational formulation we conclude unique solvability of the dual variational formulation (.7), whichen implies a bounded and invertible operator B : L (,) [H 1,(,)], satisfying u L (,) B u [H 1, (,)] for all u L (,)..3 Interpolation of operators Related to the initial value problem (.1) we consider the operator B 1 : H 1, (,) L (,) of the primal formulation (.), and the operator B : L (,) [H 1,(,)] of the dual formulation (.7). Hence, using interpolation arguments we may consider for s (,1) an operator B s : [H 1, (,),L (,)] s [L (,),[H 1, (,)] ] s, and we may ask for a representation of B s, in particular for s 1. Recall that the Sobolev space H 1/, (,) : [H,(,),L 1 (,)] 1/ is a dense subspace of H 1/ (,) with norm u H 1/, (,) [u(t)] dt+ [u(s) u(t)] dsdt+ s t [u(t)] dt. t For B 1 : H,(,) 1 L (,) we define the adjoint operator B 1: L (,) [H,(,)] 1 via u,b 1v (,) B 1 u,v L (,) for all u H,(,), 1 v L (,), 9

14 where, (,) denotes the duality pairing as extension of the inner product in L (,). hen we introduce A : B 1B 1 : H 1,(,) [H 1,(,)]. In particular for u H, 1 (,) we may consider the eigenvalue problem i.e. for all v H 1,(,) we have Au,v (,) B 1 u,b 1 v L (,) Au λu in [H 1,(,)], t u(t) t v(t)dt λ u(t)v(t) dt. Note that this is the variational formulation of an eigenvalue problem with mixed boundary conditions, tt u(t) λu(t) for t (,), u(), t u(). Hence we find ( (π ) t v k (t) sin +kπ ), λ k 1 ( π +kπ ), k,1,,3,... (.11) Recall that the eigenfunctions v k form an orthogonal basis in L (,) satisfying and in H 1, (,), t v k (t) t v l (t)dt λ k his motivates to consider for u H 1, (,) u(t) k u k sin +kπ, u k v k (t)v l (t)dt δ kl, (.1) v k (t)v l (t)dt 1 ( π ) δkl +kπ. u(t) sin +kπ dt, (.13) and by Parseval s identity we have ( (π ) ) ( t (π ) ) t u L (,) u k u l sin k l +kπ sin +lπ dt u k, (.14) as well as t u L (,) k k l u k u l t v k (t) t v l (t)dt 1 1 k ( π +kπ ) u k. (.15)

15 Hence, using interpolation, we define an equivalent norm in H 1/, (,), e.g., [3, 38], as well as an inner product, u H 1/, (,) 1 u,v H 1/, (,) 1 Analogously, we consider for w H 1/, (,), w(t) k k k w k cos +kπ, w k withe related norm and inner product, w H 1/, (,) 1 k ( π +kπ ) u k, (.16) ( π +kπ ) u k v k. ( π +kπ ) w k, w,z H 1/, (,) 1 Finally we introduce the dual space [H 1/, (,)] withe norm f [H 1/, (,)] sup w H 1/, (,) Lemma.3 For f [H 1/, (,)] we have f [H 1/, (,)] k w(t) cos +kπ dt, f,w (,) k w H 1/, (,). ( π +kπ ) w k z k. ( π +kπ ) 1f k (.17) with f k f,w k (,), w k (t) cos +kπ. 11

16 Proof. From the norm definition, using a series representation of w H 1/, (,), and with Hölder s inequality we first have i.e. f 1/ [H sup, (,)] w H 1/, (,) sup w H 1/, (,) ( 1 sup w H 1/, (,) k k ( k f,w (,) w H 1/, (,) w k f,w k (,) ) ( π ) 1/ +kπ w k w k f k k ( π +kπ ) w k f [H 1/, (,)] k ) 1/ ( k ( π +kπ ) 1f k. On the other hand, if the coefficients f k are given, we define and obtain ( k ) ( π ) 1/ 1f +kπ k k ( k l k f k w l w k ( π +kπ ) 1fk ( π +kπ ) 1f k ( π +kπ ) 1f k ( (π cos ( 1 k f,w (,) sup w 1/ H, (,) w H 1/, (,) ( π +kπ ) 1f k) 1/, f k w k k ) 1/ ( ( ) ) 1/ 1 π +kπ [w k ] k ) ) ( t (π ) ) t +kπ cos +lπ dt ( ) ) 1/ π +kπ [w k ] f,w (,) w H 1/, (,) f [H 1/, (,)]. 1

17 his concludes the proof. he variational formulation of the initial value problem (.1) is to find u H 1/, (,) such that t u,w (,) f,w (,) for all w H 1/, (,), (.18) where f [H 1/, (,)] is given. Note that (.18) is a Galerkin Petrov variational formulation with different trial and test spaces. Hence we have to establish an appropriate stability condition which is equivalent to an ellipticity estimate for the bilinear form t u,h v (,) with some transformation operator H : H 1/, (,) H 1/, (,) to be specified..4 ransformation operator o motivate the particular definition of the operator H : H 1/, (,) H 1/, (,) we write, by using (.13), t u(t) 1 ( ( π (π ) ) t u k )cos +kπ +kπ k as distributional derivative, i.e. for w H 1/, (,) we have t u,w (,) 1 ( ( π (π ) t u k )cos +kπ +kπ If we define k w(t) (H u)(t) : we conclude the ellipticity estimate t u,h u (,) 1 1 k l k l ( π ) u k u l +kπ ) w(t) dt. u l cos +lπ, (.19) ( (π ) ) ( t (π ) ) t cos +kπ cos +lπ ( π +kπ ) u k u H 1/, (,). (.) Remark.1 H u H 1/, (,) as given in (.19) is the unique solution of the variational problem H u,z 1/ H tu,z, (,) (,) for all z H 1/, (,). he definition of the transformation operator H therefore coincides withe definition of the optimal test space as used, e.g. in discontinuous Galerkin Petrov methods [9]. Indeed, for u(t) k ( π ) t u k sin +kπ 13 dt

18 we use the ansatz w(t) (H u)(t) k w k cos +kπ, and the test function to obtain w,z H 1/, (,) 1 z(t) k k z k cos +kπ ( π +kπ ) w k z k t u,z 1 for all z k, from which we conclude w k u k for k,1,,... k ( π +kπ ) u k z k By construction we have w H u H 1/, (,), and H : H 1/, (,) H 1/, (,) is norm preserving, i.e. H u H 1/, (,) u H 1/, (,) for all u H 1/, (,). Vice versa, if w H 1/, (,) is given, w(t) k w k cos +kπ, w k the inverse transformation operator reads u(t) (H 1 w)(t) k w(t) cos +kπ dt, w k sin +kπ. Next, we are going to prove some properties of the transformation operator H. First, we consider a commutation property withe time derivative operator t. Lemma.4 For u H 1/, (,) we have t H u,v (,) H 1 tu,v (,) for all v H 1/, (,). Proof. For an arbitrary ϕ C ([,]) with ϕ() we first compute (H ϕ)(t) k ϕ k cos +kπ, ϕ k ϕ(t) sin +kπ dt, 14

19 and therefore t (H ϕ)(t) 1 follows. On the other hand, implies t ϕ(t) 1 k (H 1 tϕ)(t) 1 k ( ( π (π ) ) t ϕ k )sin +kπ +kπ ( ( π (π ) ) t ϕ k )cos +kπ +kπ k ( ( π (π ) ) t ϕ k )sin +kπ +kπ, i.e. t H ϕ H 1 tϕ for all ϕ C ([,]) with ϕ(). Now the assertion follows by completion. Next we prove that H is unitary. Lemma.5 For u H 1/, (,) and w H 1/, (,) there holds H u,w L (,) u,h 1 w L (,). Proof. For u H 1/, (,) and w H 1/, (,) we have u(t) u k sin +kπ, w(t) and (H u)(t) k k Hence we compute H u,w L (,) u k cos +kπ, (H 1 k l (H u)(t)w(t)dt u k w l u k w k k k l u k w l l w)(t) w l cos +lπ, l cos +kπ sin +kπ u(t)(h 1 w)(t)dt u,h 1 w L (,). w l sin +lπ. cos +lπ dt sin +lπ dt Using Lemma.4 and Lemma.5 we conclude the following symmetry relation. 15

20 Corollary.6 For u,v H 1/, (,) there holds t u,h v (,) H u, t v (,) u,v H 1/, (,). Proof. For ϕ,ψ C ([,]) with ϕ() ψ() we first have H ϕ() H ψ() and therefore t ϕ,h ψ L (,) H 1 tϕ,ψ L (,) t H ϕ,ψ L (,) (H ϕ)(t)ψ(t) + H ϕ, t ψ L (,) H ϕ, t ψ L (,) holds. Now the assertion follows by completion. he next property of H is required when considering, instead of (.1), more general differential equations. Lemma.7 here holds Proof. By using we have v(t) k v,h v L (,) v,h v L (,) for all v H 1/, (,). (.1) v k sin +kπ, (H v)(t) 1 1 π k l v k v l k l k l v k v l ( (π sin [ sin l ) ) t +kπ ( (k +l+1)π t v k v l [ (k +l+1)π cos k l v l cos +lπ, ( (π cos 1 [ v k v ] l 1 ( 1) k+l+1, k +l+1 ) ) t +lπ dt ) ( +sin (k l)π t )] dt ( (k +l+1)π t )] 16

21 where the second integral is ignored due to symmetry. When splitting k and l into odd and even indices, i.e. k i,i+1, l j,j +1, this gives v,h v L (,) [ v i v j π i+j +1 + v ] i+1v j+1 i+j +3 i j M M 1 1 ] [v π lim i v j x i+j dx+v i+1 v j+1 x i+j+ dx M i j ( 1 M ) π lim ( 1 M ) v i x i dx+ v i+1 x i+1 dx. M i Remark. he matrix H as used in the previous proof, i.e. i H[j,i] 1 i+j +1 for i,j,1,...,n, is a Hilbert matrix [17] which is positive definite, but ill conditioned. But for our purpose it is sufficient to use that (.1) is non negative. Next we will have a closer look on the definition of the transformation operator H to see its relation withe well known Hilbert transform, see, e.g., [18]. Lemma.8 he operator H as defined in (.19) allows the integral representation (H u)(t) K(s, t)u(s) ds as a Cauchy principal value integral where the kernel function is given as [ ] K(s,t) 1 1 sin ( 1 ) + π s t sin ( ). (.) π s+t Proof. In fact, (H u)(t) k k ( (π ) t u k cos +kπ ) (( π ) s u(s) sin ) +kπ dscos u(s)k(s, t) ds ( (π +kπ ) t ) 17

22 with K(s,t) 1 k k (( π ) ( s (π sin ) +kπ cos [ ( (π ) ) s t sin +kπ +sin ) t +kπ ( (π ) ) s+t +kπ )]. By using k (( π ) ) sin +kπ x we further conclude the representation (.). 1 1 sin ( for x,,4,... π x) Remark.3 For fixed s,t (,),s t, we may consider to conclude lim K(s,t) 1 s π(s t)(s+t) (H u)(t) 1 π u(s) s s ts+t ds, where the kernel function shows for s t the same behaviour as for the Hilbert transform (Hu)(t) 1 π u(s) s t ds for which all the previous properties are well known, see, e.g., [18]..5 Variational formulations For the solution of the initial value problem (.1) we consider the variational formulation (.18) to find u H 1/, (,) suchat t u,h v (,) f,h v (,) for all v H 1/, (,), (.3) where f [H 1/, (,)] is given. Since the bilinear form t u,h v (,) is bounded, i.e. for u,v H 1/, (,) there holds t u,h v (,) t u 1/ [H H v u v,, }{{ (,)] 1/ H, } (,) H 1/, (,) H 1/, (,) B 1/ u 1/ [H, (,)] and elliptic, see (.), we conclude unique solvability of the variational formulation (.3). 18

23 Remark.4 From the ellipticity estimate (.) we also conclude the stability condition u tu,h u (,) 1/ H sup, (,) H u 1/ H, (,) w H 1/, (,) t u,w (,) for all u H 1/, (,), w 1/ H, (,) and from which we conclude unique solvability of the Galerkin Petrov formulation to find u H 1/, (,) suchat t u,w (,) f,w (,) for all w H 1/, (,). (.4) Next we consider a conforming finite element discretisation for the variational formulation (.3). For a time interval (,) and a discretisation parameter N N we consider nodes t < t 1 < t < < t N 1 < t N, finite elements τ l (t l 1,t l ) of local mesh size h l t l t l 1, l 1,...,N, and a related finiteelementspacesh 1(,)ofpiecewiselinearcontinuousbasisfunctionsϕ k, k,...,n, with global mesh size h maxh l. hen the finite element discretisation of the variational formulation (.3) is to find u h V h : Sh 1(,) H1/ (,) span{ϕ k } N k1 suchat t u h,h v h L (,) f,h v h (,) for all v h V h. (.5) Using standard arguments, e.g. [31], we conclude unique solvability of (.5) as well as the a priori error estimates u u h H σ, (,) ch s σ u H s (,), (.6) when assuming u H s (,) for some s [1,], and for σ, 1,1. Note that for σ 1 (.6) is a consequence of Céa s lemma and the approximation property of Sh 1 (,), while for σ we use the Aubin Nitsche trick, and for σ 1 we have to use an inverse inequality, i.e. we have to assume a globally quasi uniform mesh in this case. he Galerkin Bubnov finite element formulation (.5) is equivalent to the linear system of algebraic equations K h u f with a symmetric and positive definite stiffness matrix K h defined by K h [j,k] t ϕ k,h ϕ j L (,) for k,j 1,...,N. As numerical example we consider the solution u(t) sin ( 9π t) for t (,) (,) 4 where the right-hand side is f(t) 9π cos( 9π t). For the discretisation we consider a 4 4 sequence of finite element spaces Sh 1(,) of uniform mesh size h /N, and N j+1, j,...,7. Since the solution u is smooth, we use s within the error estimate (.6) to conclude second order convergence in L (,), and linear convergence in H 1 (,), respectively. his behaviour is confirmed by the numerical results as given in able 1. In addition, wepresent theminimal andmaximal eigenvalues ofthestiffnessmatrixk h aswell as the resulting spectral condition number of K h which behave as expected for a first order differential operator. Note that these results correspond to the Galerkin discretisation of a 19

24 N u u h L eoc t (u u h ) L eoc λ min (K h ) λ max (K h ) κ (K h ) able 1: Numerical results for the Galerkin Bubnov formulation (.5). hypersingular boundary integral operator in boundary element methods for second order elliptic partial differential equations, see, e.g., [31]. he evaluationof thetransformedbasisfunctions H ϕ k canbedoneby using thedefinition (.19). Althoughthepiecewise linearbasisfunctionsϕ k havelocalsupport, thetransformed basis functions H ϕ k are global, see Fig. 1, and therefore the stiffness matrix K h is dense. As in the case of the hypersingular boundary integral operator one may use different techniques such as adaptive cross approximation [8] to accelerate the computations, but this is far behind the scope of this contribution. Figure 1: ransformed basis functions H ϕ k, k 1,...,N, N 4.

25 Instead of the initial value problem (.1) we now consider for µ > the first order linear equation t u(t)+µu(t) f(t) for t (,), u(), (.7) and the related variational formulation to find u H 1/, (,) suchat t u,h v (,) +µ u,h v L (,) f,h v (,) for all v H 1/, (,), (.8) where f [H 1/, (,)] is given. When combining (.) and (.1) this gives t v,h v (,) +µ v,h v L (,) t v,h v (,) v H 1/, (,) for all v H 1/, (,), i.e. the bilinear form of the variational problem (.8) is bounded and elliptic, implying unique solvability of (.8). For the solution u H 1/, (,) of the variational problem (.8) we have u H 1/, (,) t u,h u (,) t u,h u (,) +µ u,h u L (,) f,h u (,) f [H 1/, (,)] H u H 1/, (,), implying u f 1/ H. (.9), (,) [H 1/, (,)] For the analysis of the heat equation we also need to have appropriate estimates for the solution u in L (,). Lemma.9 Let u H 1/, (,) be the unique solution of the variational problem (.8), where f [H 1/, (,)] is given. hen, where u L (,) k f k µ + 1 ( π f k : f,w k (,), w k (t) : cos +kπ. +kπ), (.3) Proof. Let (f n ) n N L (,) be a sequence with lim f f n 1/ n [H. We write, (,)] f n L (,) as f n (t) f n,k cos +kπ, f n,k f n (t)cos +kπ dt. (.31) k Let u n H 1/, (,)bethe weak solution ofthe differential equation (.7)with right hand side f n. It follows analogously to (.9) that u u n H 1/, (,) f f n [H 1/, (,)] 1

26 and therefore u n u in H 1/, (,) and u n u in L (,) as n. Because of f n L (,) and using (.31) we have the representation t t u n (t) e µ(s t) f n (s)ds f n,k e µs cos(a k s)dse µt k f n,k µ +a k k and we obtain, when computing all integrals, u n L (,) k f n,k µ +a k Now the assertion follows as n. [a k sin(a k t)+µcos(a k t) µe µt ], a k 1 1 µ [ 1+e µ ] ( k f n,k µ +a k ) Remark.5 From (.3) we immediately conclude the estimate ( u L (,) 3 π ) f +kπ k f [H., 1 (,)] k ( π +kπ ), f n,k. µ +a k Moreover, when we assume f L (,), (.3) gives u L (,) f µ k 1 µ f L (,), i.e. µ u L (,) f L (,). k he Galerkin Bubnov discretisation of (.8) is to find u h V h suchat t u h,h v h L (,) +µ u h,h v h L (,) f,h v h (,) for all v h V h. (.3) As for the initial value problem (.1) we have unique solvability of (.3), but related a priori error estimates depend on µ in general, requiring a sufficient small mesh size o ensure convergence for large µ. Remark.6 Instead of the Galerkin Bubnov variational formulation (.8) we may also consider the Galerkin Petrov formulation to find u H 1/, (,) suchat t u,w (,) +µ u,w L (,) f,w (,) for all w H 1/, (,), (.33) where the ellipticity of t v,h v (,) +µ v,h v L (,) implies a related stability estimate from which unique solvability of (.33) follows. For the finite element discretisation of the Galerkin Petrov variational formulations (.4) and (.33) we have to define a suitable test space W h H 1/, (,). A first choice is to use W h : Sh 1(,) H1/, (, ). Althoughe discrete systems are always uniquely solvable, since the stiffness matrices are regular lower triangular, the resulting scheme is never stable when considering (.7). he construction of a more suitable test space is, in particular when considering partial differential equations such as the heat equation, more challenging. k

27 3 Heat equation As model problem for a parabolic partial differential equation we consider the Dirichlet problem for the heat equation, t u(x,t) x u(x,t) f(x,t) for (x,t) Q : Ω (,), u(x,t) for (x,t) Σ : Γ (,), (3.1) u(x,) for x Ω, where Ω R d, d 1,,3, is a bounded domain with, for d,3 Lipschitz boundary Γ Ω. o write down a variational formulation we need to have suitable Sobolev spaces. In addition to the eigenfunctions v k (t) and eigenvalues λ k as given in (.11) we consider the eigenfunctions φ i H 1 (Ω) and associated eigenvalues µ i, i N, of the spatial Dirichlet eigenvalue problem x φ µφ in Ω, φ on Γ, φ L (Ω) 1. Recall that the eigenfunctions φ i form an orthonormal basis in L (Ω), and an orthogonal basis in H 1 (Ω). In addition, we have < µ 1 µ µ 3... and µ i as i. For a function u L (Q) we therefore find the representation u(x,t) u i,k v k (t)φ i (x) i1 k U i (t)φ i (x), U i (t) i1 u i,k v k (t) (3.) k withe coefficients Note that we have and u i,k u(x,t)v k (t)φ i (x)dxdt Ω ( (π ) ) t sin +kπ u L (Q) U i L (,) u H 1 (Q) i1 i1 u(x,t)φ i (x)dxdt. Ω i1 k u i,k [ ] t U i L (,) +µ i U i L (,) i1 k [ 1 ( π ) ] +kπ +µi u i,k. 3

28 his motivates to define the norm, for u H 1 (Q) with u(,) u Σ, u H 1,1/ ;, (Q) : i1 [ ] U i +µ H 1/, (,) i U i L (,) i1 k and to introduce the anisotropic Sobolev space H 1,1/ ;, (Q) : [ 1 ( π +kπ )+µ i ] u i,k, { u L (Q): u H 1,1/ ;, (Q) < }. Note that H 1,1/ ;, (Q) H 1/, (,;L (Ω)) L (,;H 1 (Ω)). Analogously, we introduce H 1,1/ ;, (Q) H1/, (,;L (Ω)) L (,;H 1 (Ω)), which is equipped withe norm w H 1,1/ ;, (Q) i1 k [ 1 ( π +kπ )+µ i ] w i,k, and w i,k ( (π ) ) t cos +kπ w(x,t)φ i (x)dxdt. Ω Lemma 3.1 For the dual norm of f [H 1,1/ ;, (Q)] we have f [H 1,1/ ;, (Q)] i1 k [ 1 ( π ) ] 1 +kπ +µ i f i,k (3.3) with f i,k f,w kφ i Q. 4

29 Proof. From the norm definition, using a series representation of w H 1,1/ ;, (Q), and with Hölder s inequality we first have i.e. f,w Q f 1,1/ [H sup ;, (Q)] w w H 1,1/ ;, (Q) 1,1/ H ;, (Q) w i,k f,w k φ i Q sup w H 1,1/ ;, (Q) ( sup w H 1,1/ ;, (Q) ( i1 k [ 1 f [H 1,1/ ;, (Q)] i1 ( i1 i1 k k [ 1 k [ 1 ( π +kπ )+µ i ] w i,k i1 w i,k f i,k k ) 1/ ( π +kπ )+µ i ] w i,k ( π ) ] 1 1/ +kπ +µ i fi,k), i1 k [ 1 ( π ) ] 1 +kπ +µ i f i,k. he lower estimate follows as in the proof of Lemma.3, we skip the details. According to the previous sections we consider the variational formulation of (3.1) to find u H 1,1/ ;, (Q) suchat ) 1/ t u,v Q + x u, x v L (Q) f,v Q (3.4) is satisfied for all v H 1,1/ ;, (Q), where f [H 1,1/ ;, (Q)] is given, and, Q denotes the duality pairing as extension of the inner product in L (Q). heorem 3. he variational formulation (3.4) implies an isomorphism L: H 1,1/ ;, (Q) [H 1,1/ ;, (Q)], satisfying u H 1,1/ ;, (Q) Lu [H 1,1/ ;, (Q)] for all u H 1,1/ ;, (Q). (3.5) Proof. For the solution u of the variational problem (3.4) we use the ansatz (3.) where U i H 1/, (,)areunknownfunctionstobedetermined. Whenchoosing, forafixedj N, 5

30 v(x,t) : V(t)φ j (x) with V H 1/, (, ) as test function, the variational formulation (3.4) leads to find U j H 1/, (,) suchat is satisfied for all V H 1/, (,). It holds t U j,v (,) +µ j U j,v L (,) f,vφ j Q (3.6) f,vφ j Q f [H 1,1/ ;, (Q)] Vφ j H 1,1/ ;, (Q) 1+µ j f [H 1,1/ ;, (Q)] V H 1/, (,) for all V H 1/, (,) and so, F j,v (,) : f,vφ j Q fulfils F j [H 1/, (,)]. he unique solvability of (3.6) follows analogously as for (.7). So, we have for every j N a unique solution U j H 1/, (, ) of the variational formulation (3.6), satisfying U j H 1/, (,) t U j,h U j (,) For M N we define and we conclude u M H 1/, (,;L (Ω)) t U j,h U j (,) +µ j U j,h U j L (,) f,φ j H U j Q. u M (x,t) M U j (t)φ j (x), j1 M U j M f,φ H 1/, (,) j H U j Q j1 f,h u M Q j1 Hence, using (.3) for f [H 1,1/ ;, (Q)] H u M H 1,1/ ;, (Q) f [H 1,1/ ;, (Q)] u M H 1,1/ ;, (Q). f i,k F i,w k (,) f,φ iw k Q we obtain u M L (,;H 1 (Ω)) M µ i U i L (,) i1 M i1 M i1 µ i µ i + 1 f i,k ( π +kπ) 1 1 f i,k f µ i + ( π +kπ) [H 1,1/ k k ;, (Q)], 6

31 where we have used Withis we have a a +b a+b 1 (a+b) a+b for < a,b R. u M H 1,1/ ;, (Q) u M H 1/, (,;L (Q)) + u M L (,;H 1 (Ω)) f [H 1,1/ ;, (Q)] u M H 1,1/ ;, (Q) + f [H 1,1/ ;, (Q)], and therefore u M H 1,1/ ;, (Q) f [H 1,1/ ;, (Q)] follows for all M N. he last inequality yields the bound u H 1,1/ ;, (Q) lim M M i1 [ ] U i +µ H 1/, (,) i U i L (,) lim M u M H 1,1/ ;, (Q) 4 f [H 1,1/ ;, (Q)] <, and thus, u H 1,1/ ;, (Q) with lim M u M u in H 1,1/ ;, (Q). he existence of a solution of the variational formulation(3.4) is proven by inserting the constructed function u into the variational formulation (3.4) and using the approximating sequence (u M ) M N. he uniqueness of a solution of the variational formulation (3.4) is a consequence of the uniqueness of the coefficient functions U j. Corollary 3.3 As a direct consequence of (3.5) we immediately conclude the stability estimate 1 u sup t u,w Q + x u, x w L (Q) (3.7) H 1,1/ ;, (Q) w w H 1,1/ ;, (Q) 1,1/ H ;, (Q) for all u H 1,1/ ;, (Q). he variational formulation (3.4) is equivalent to find u H 1,1/ ;, (Q) suchat t u,h v Q + x u, x H v L (Q) f,h v Q (3.8) is satisfied for all v H 1,1/ ;, (Q), where the operator H acts only on the time variable t. he stability estimate (3.7) implies the stability estimate 1 u H 1,1/ ;, (Q) sup v H 1,1/ ;, (Q) t u,h v Q + x u, x H v L (Q) v H 1,1/ ;, (Q) (3.9) 7

32 for all u H 1,1/ ;, (Q), and therefore unique solvability of the variational formulation (3.8) follows. When using some conforming space time finite element space V h H 1,1/ ;, (Q), the Galerkin variational formulation of (3.8) is to find u h V h suchat t u h,h v h L (Q) + x u h, x H v h L (Q) f,h v h Q (3.1) is satisfied for all v h V h. Since the related finite element stiffness matrix is positive definite, unique solvability of (3.1) follows for any conforming choice of V h. However, to perform the temporal transformation H easily, and to be able to present an a priori error analysis, here we will consider a space time tensor product finite element space only. Let W hx span{ψ i } M i1 H1 (Ω) be some spatial finite element space, e.g., of piecewise linear or bilinear continuous basis functions ψ i which are defined with respect to some admissible and globally quasi uniform finite element mesh with mesh size h x. As before, V ht Sh 1 t (,) H 1/, (,) span{ϕ k } N k1 is the space of piecewise linear functions which are defined with respect to some globally quasi uniform finite element mesh with mesh size. Hence we can introduce the tensor product space time finite element space V h : W hx V ht. For a given v H 1/, (,;L (Ω)) we define the H 1/, projection Q 1/ v L (Ω) V ht as the unique solution of the variational problem t Q 1/ v,h v ht L (Q) t v,h v ht Q for all v ht L (Ω) V ht. Moreover, for v L (,;H 1(Ω)) we define the H1 projection v W hx L (,) as the unique solution of the variational problem x v(x,t) x v hx (x,t)dxdt x v(x,t) x v hx (x,t)dxdt Ω for all v hx W hx L (,). It turns out that Q 1/ v V h is well defined when assuming t v L (,;H 1 (Ω)) and xv H 1/, (,;L (Ω)), respectively, and that the projection operators Q 1/, and partial derivatives t, x commute in space and time [38]. heorem 3.4 Let u H 1,1/ ;, (Q) and u h V h be the unique solutions of the variational problems (3.8) and (3.1), respectively. If u is sufficiently regular, and the spatial domain Ω is assumed to be either convex or has a smooth boundary Γ, then there hold the error estimates and u u h H 1/, (,;L (Ω)) c 1 h 3/ t u H (,;L (Ω)) +c h 3/ x u H 1/, (,;H 3/ (Ω)) Ω +c 3 h 1/ t h x t x u L (Q) +c 4 h 3/ t t x u L (Q) (3.11) u u h L (Q) c 1 h t u H (,;L (Ω)) +c h x u L (,;H (Ω)) +c 3 h x t x u L (Q) +c 4 h x t u L (,;H (Ω)) +c 5 h t x u H (,;L (Ω)). (3.1) 8

33 Proof. Withe norm representation in H 1/, (,;L (Ω)), the positivity (.1), and the Galerkin orthogonality of the variational formulations (3.8) and (3.1), we have for v h Q 1/ u V h, using the definitions of the projections Q 1/ and, and integration by parts spatially, i.e. u h Q 1/ u H 1/, (,;L (Ω)) t (u h Q 1/ u),h (u h Q 1/ u) Q t (u h Q 1/ u),h (u h Q 1/ u) Q + x (u h Q 1/ u), x H (u h Q 1/ u) L (Q) t (u Q 1/ u),h (u h Q 1/ u) Q + x (u Q 1/ u), x H (u h Q 1/ u) L (Q) t (u u),h (u h Q 1/ u) Q + x (u Q 1/ u), x H (u h Q 1/ u) L (Q) t (u u),h (u h Q 1/ u) Q (3.13) x (u Q 1/ u),h (u h Q 1/ u) Q u u H 1/, (,;L (Ω)) u h Q 1/ u H 1/, (,;L (Ω)) + x (u Q 1/ u) [H 1/, (,;L (Ω))] u h Q 1/ u H 1/, (,;L (Ω)), Hence we have u h Q 1/ u H 1/, (,;L (Ω)) u u h H 1/, (,;L (Ω)) u u H 1/, (,;L (Ω)) + x(u Q 1/ u) [H 1/, (,;L (Ω))]. u Q 1/ u H 1/, (,;L (Ω)) + u h Q 1/ u H 1/, (,;L (Ω)) u Q 1/ u H 1/, (,;L (Ω)) + u u H 1/, (,;L (Ω)) + x(u Q 1/ u) [H 1/, (,;L (Ω))] u Q 1/ u 1/ H, (,;L (Ω)) + u Q1 h x u 1/ H ( ) + I Q 1/ )(u u H 1/, (,;L (Ω)), (,;L (Ω)) + u u H 1/, (,;L (Ω)) + x(u Q 1/ u) [H 1/, (,;L (Ω))], and the energy error estimate (3.11) follows from standard error estimates for the involved projection operators. 9

34 With a Poincaré Friedrichs type inequality and relation (3.13) we also have 1 c u h Q 1/ u L (Q) u h Q 1/ u H 1/, (,;L (Ω)) which implies and therefore t (u u),h (u h Q 1/ u) Q x (u Q 1/ u),h (u h Q 1/ u) L (Q) t (u u) L (Q) u h Q 1/ u L (Q) + x (u Q 1/ u) L (Q) u h Q 1/ u L (Q), u h Q 1/ u L (Q) c t (u u) L (Q) +c x (u Q 1/ u) L (Q), u u h L (Q) u Q 1/ u L (Q) + u h Q 1/ u L (Q) u Q 1/ u L (Q) +c t (u u) L (Q) +c x (u Q 1/ u Q 1/ u L (Q) + u u L (Q) + ( ) I Q 1/ )(u u +c t (u u) L (Q) +c x (u Q 1/ u) L (Q). u L (Q) L (Q) Finally, (3.1) follows again from standard error estimates for the projection operators. As numerical example we consider the solution u(x,t) sin ( 5π t) sin(πx) for (x,t) Q 4 with Q : (, 1) (, ). For a uniform discretisation of the Galerkin variational formulation (3.1) withe tensor product space time finite element space V h W hx V ht we use the mesh sizes h x 1 and h M t with M N N j, j 1,...,6. Since the solution u is smooth, we expect a second order convergence in L (Q), see (3.1), and a linear order convergence in H 1 (Q). Note that the latter follows by standard arguments when using the H 1 (Q) projection and an inverse inequality. he predicted convergence orders are confirmed by the numerical results given in able. M,N dof h x u u h L eoc u u h H 1 eoc able : Convergence rates of the Galerkin Bubnov formulation (3.1). 3

35 Remark 3.1 Numerical results [38] indicate that the stability constant c S of the discrete inf sup condition c S u h sup t u h,h v h L(Q) + x u h, x H v h L(Q) 1,1/ H ;, (Q) v h V h v h 1,1/ H ;, (Q) for all u h V h is mesh dependent, i.e. c S O( ). However, it seems to be possible to derive almost optimal energy error estimates also in this case. Since this is far behind the scope of this paper, this will be discussed elsewhere. 4 Second order ordinary differential equations As in (.1) we consider the initial value problem tt u(t) f(t) for t (,), u() t u(). (4.1) When multiplying the differential equation with a test function w satisfying w(), integrating over(, ), and applying integration by parts once, this results in the variational formulation to find u H, 1 (,) suchat t u(t) t w(t)dt f,w (,) (4.) is satisfied for all w H 1, (,), where f [H1, (,)] is given. he bilinear form a(u,w) : t u(t) t w(t)dt for u H, 1 (,), w H1,(,) (4.3) is obviously bounded and therefore it remains to establish some stability or ellipticity estimate to ensure unique solvability of the variational formulation (4.). For this we use the concept of an optimal test function, see Remark.1. Lemma 4.1 Let u H, 1(,) be given. he unique solution w H u H, 1 (,) of the variational problem is given as w(t) k t w(t) t v(t)dt t u(t) t v(t)dt for all v H,(,), 1 (4.4) w k cos +kπ, w k π +kπ t u(t)sin +kπ dt. (4.5) 31

36 Proof. When using the ansatz and test functions ( (π ) ) ( t (π ) ) t w(t) w k cos +kπ, v(t) cos +lπ k within the variational formulation (4.4) and the orthogonality (.1), this gives w l π +lπ t u(t)sin +lπ dt. As for the transformation operator H : H 1/, (,) H 1/, (,) we state some properties for H : H, 1(,) H1,(,) as defined in (4.5). Lemma 4. he operator H as defined in (4.5) is norm preserving satisfying H u H 1, (,) u H 1, (,) for all u H 1, (,). Proof. Let w H u as defined in (4.5), and when using (.15) this gives w H 1, (,) tw L (,) 1 k ( π +kπ ) w k. On the other hand, for z t u L (,) we consider the series representation z(t) k and with (.14) we have z k sin +kπ, z k z(t) sin +kπ dt, Now the assertion follows from w k u H 1, (,) tu L (,) z L (,) π +kπ π +kπ ( (π t u(t)sin ) ) t +kπ dt ) dt ( (π ) t z(t) sin +kπ zk. k π +kπ z k. Lemma 4.3 For u,v H 1,(,) there holds the symmetry relation a(u,h v) a(v,h u). 3

37 Proof. When using the definition (4.4) this gives a(u,h v) t u(t) t (H v)(t)dt t (H u)(t) t (H v)(t)dt t (H v)(t) t (H u)(t)dt t v(t) t (H u)(t)dt a(v,h u). Now we are in the position to prove some ellipticity estimate for the bilinear form (4.3). heorem 4.4 he bilinear form as given in (4.3) is elliptic, i.e. a(u,h u) t u L (,) for all u H 1,(,). Proof. Due to the construction of w H u we have a(u,h u) t u(t) t w(t)dt and the assertion follows by using Lemma 4.. t w(t) t w(t)dt t w L (,), Lemma 4.5 For u H, 1 (,) there holds the representation (H u)(t) u() u(t). (4.6) Proof. For the coefficients in (4.5) we obtain, by using integration by parts, w k π +kπ t u(t)sin +kπ dt [ ( (π ) ) t π +kπ u(t) sin +kπ 1 ( π ) ( (π ) ) ] t +kπ u(t) cos +kπ dt ( π ) π +kπu()sin +kπ u(t) cos +kπ dt u()e k u k, where u k u(t) cos +kπ dt 33

38 and e k cos +kπ dt π ( π ) sin +kπ +kπ are the Fourier coeffecients of the given function u(t) and of the constant function e(t) 1, respectively. Hence we have (H u)(t) w k cos +kπ k u() e k cos +kπ u k cos +kπ k u() u(t). Remark 4.1 From the representation (4.6) we easily conclude all the properties we have shown before. However, the form u,h u L (,) k u(t)[u() u(t)]dt is indefinite, i.e. a result as (.1) for the transformation H does not hold for H. For a finite element discretisation of the variational formulation (4.) we use the same notations as in Section. In particular, we have to find u h V h : S 1 h (,) H1,(,) suchat t u h, t H v h L (,) f,h v h (,) for all v h V h. (4.7) As before we have unique solvability of (4.7), the a priori error estimate (.6) remains valid, where for σ 1 this corresponds to the energy error estimate, while for σ we have to apply a Nitsche type argument. For the numerical example we consider the solution u(t) sin ( 5 πt) for t (,) 4 with. he numerical results are given in able 3, where we observe optimal order of convergence as predicted. he stiffness matrix of the Galerkin Bubnov finite element formulation (4.7) is symmetric, see Lemma 4.3, and positive definite, see heorem 4.4, and its spectral behaviour is as known for finite element discretisations of second order partial differential equations. Moreover, due to (4.6), we have for the piecewise linear basis functions ϕ k H, 1(,), k 1,...,N, (H ϕ k )(t) ϕ k (t) for k 1,...,N 1, and 1 for t [,t N 1 ], (H ϕ N )(t) t for t (t N 1,], t N 1 34

39 N u u h L eoc t (u u h ) L eoc λ min (K h ) λ max (K h ) κ (K h ) able 3: Numerical results for the Galerkin Bubnov formulation (4.7). and hence H V h span{ϕ k } N 1 k. (4.8) Instead of (4.1) we now consider the second order linear equation, for µ ν >, tt u(t)+µu(t) f(t) for t (,), u() t u(), (4.9) and the variational formulation to find u H 1,(,) suchat a(u,h v) : t u(t) t (H v)(t)dt+µ u(t)(h v)(t)dt f,h v (,) (4.1) is satisfied for all v H 1, (,), where f [H1, (,)] is given. heorem 4.6 For given f [H 1, (,)] the variational formulation (4.1) admits a unique solution u H 1,(,) satisfying u H 1, (,) c f [H 1, (,)]. Proof. By using the Riesz representation theorem we can rewrite the variational problem (4.1) as operator equation Au+µCu f, where A: H 1, (,) [H1, (,)] defined via Au,v t u, t H v L (,) for u,v H 1,(,) is elliptic, and hence, invertible, and C: H 1, (,) [H1, (,)] defined via Cu,v u,h v L (,) for u,v H 1,(,) is compact. Hence we can apply the Fredholm alternative and it remains to ensure the injectivity of A + µc. Let u H, 1 (,) be a solution of the homogeneous equation (A+µC)u, i.e. t u, t w L (,) µ u,w L (,) for all w H,(,). 1 35

40 his is the weak formulation of the eigenvalue problem tt u(t) µu(t) for t (,), u() t u(), which only admits the trivial solution u. While the result of heorem 4.6 ensures unique solvability of the variational formulation (4.1), it does not include an explicit dependence on the parameter µ. Hence we will provide a stability estimate from which we can conclude such a result. Lemma 4.7 For u H, 1 (,) there holds the stability estimate +ν tu L (,) sup v H 1, (,) a(u,v). (4.11) t v L (,) Proof. For given u H 1,(,) and suitable chosen w H,(,) we consider the test function v : H u+w H 1,(,). hen, if a(u,v) t u(t) t [ u(t)+w(t)]dt+µ [ t u(t)] dt [ t u(t)] dt u(t) t w(t) t u(t) t w(t)dt+µ + u(t)[u() u(t)+w(t)]dt u(t) tt w(t)dt +µ u(t)[u() u(t)+w(t)]dt u(t)[u() u(t)+w(t)]dt [ t u(t)] dt u() t w() [ ( )] + u(t) tt w(t)+µ u() u(t)+w(t) dt [ t u(t)] dt, tt w(t)+µw(t) µ[u(t) u()] for t (,), w() t w() is satisfied. Using µ ν we obtain w(t) ν t ( ) sin ν(s t) [u(s) u()]ds, 36

41 and therefore follows. With t w(t) ν t ( ) cos ν(s t) [u(s) u()]ds ( ) νsin ν(s t) [u(t) u()] ν we further conclude i.e. Withis we finally have and therefore t ( ) sin ν(s t) s u(s)ds [ t w(t)] ν [ t ν t ν [ t w(t)] dt ν t t +ν ( ) ] sin ν(s t) s u(s)ds ) sin (ν(s t) ds ) sin (ν(s t) ds t t t ) sin (ν(s t) dsdt ν 1 cos (ν) 1+ν 4 ν 1 4 ν [ t u(t)] dt, t v L (,) t w t u L (,) t w L (,) 1 ν tu L (,). t w L (,) + t u L (,) ( ) sin ν(s t) s u(s)ds [ s u(s)] ds [ t u(t)] dt [ t u(t)] dt [ t u(t)] dt (1+ 1 ν ) t u L (,), a(u,v) t u L (,) +ν tu L (,) t v L (,) follows, which implies the stability condition as stated. While heorem 4.6 implies unique solvability of the variational formulation (4.1) we can use the stability condition (4.11) to conclude a bound for the solution u which explicitely depends on ν. 37

42 Corollary 4.8 For the unique solution u H,(,) 1 of the variational formulation (4.1) there holds t u L (,) (1+ 1 ) ν f [H 1, (,)]. (4.1) Remark 4. We consider the initial value problem (4.9) for f(t) sin(νt) withe solution u(t) 1 [ ] sin(νt) νt cos(νt), ν t u(t) 1 tsin(νt). For this we compute t u L (,) 1 1 [ ν ν 6ν cos(ν)sin(ν) 48ν 3 ] 6ν cos (ν)+3cos(ν)sin(ν) as ν. On the other hand we determine w H, 1 (,) as unique solution of the boundary value problem tt w(t) f(t) for t (,), t w() w(), i.e. t w(t) 1 ν [ ] cos(νt) 1. Hence we compute f [H, 1 (,)] t w L (,) 1 1 [ ] 3ν +cos(ν)sin(ν) 4sin(ν) 3 ν 3 as ν. In particular we have t u L (,) f [H 1, (,)] 1 6 ν as ν, which shows that the estimate (4.1) is sharp with respect to the order of ν and, respectively. While for f [H 1, (,)] the bound (4.1) shows an explicit dependence on ν µ, we can prove an estimate independent of µ when assuming f L (,). Lemma 4.9 For given f L (,) the unique solution u H 1,(,) satisfies ν u H 1, (,) +µ u L (,) 1 f L (,). (4.13) 38

43 Proof. For the solution u and its first order derivatitve we find the representations and Hence we compute [ t u(t)] +ν [u(t)] t and therefore we obtain u(t) 1 ν t u(t) [ t t t ) cos (ν(t s) ds t t ( ) sin ν(t s) f(s)ds ( ) cos ν(t s) f(s)ds. ( ) [ t cos ν(t s) f(s)ds] + t [f(s)] ds t u H 1, (,) +µ u L (,) [f(s)] ds+ t [f(s)] ds, ( ) ] sin ν(t s) f(s)ds ) sin (ν(t s) ds {[ t u(t)] +µ[u(t)] } dt tdt t [f(s)] ds 1 f L (,). [f(s)] ds Remark 4.3 As in Remark 4. we consider problem (4.9) for f(t) sin(νt) withe solution u(t) and its derivative t u(t) 1 tsin(νt), i.e. t u L (,) 1 4 3, f L (,) 1 1 [ ] ν cos(ν)sin(ν) 1 ν. Hence we conclude t u L (,) f L (,) 1 1, i.e. the estimate (4.13) is sharp with respect to the order of. he Galerkin Bubnov finite element formulation of the equivalent variational formulation (4.1) is to find u h V h : Sh 1(,) H1, (,) suchat a(u h,h v h ) t u h, t H v h L (,) +µ u h,h v h L (,) f,h v h (,) (4.14) is satisfied for all v h V h. Unique solvability and related error estimates follow as for the numerical solution of elliptic operator equations with compact perturbations, which is based on a discrete stability condition. 39