Technische Universität Graz

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1 echnische Universität Graz A Stabilized Space ime Finite Element Method for the Wave Equation Olaf Steinbach and Marco Zank Berichte aus dem Institut für Angewandte Mathematik Bericht 18/5

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3 echnische Universität Graz A Stabilized Space ime Finite Element Method for the Wave Equation Olaf Steinbach and Marco Zank Berichte aus dem Institut für Angewandte Mathematik Bericht 18/5

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 A Stabilized Space ime Finite Element Method for the Wave Equation Olaf Steinbach and Marco Zank Abstract We consider a space time variational formulation of the wave equation by including integration by parts also in the time variable. A standard finite element discretization by using lowest order piecewise linear continuous functions then requires a CFL condition to ensure stability. o overcome this restriction, and following the work of A. A. Zlotnik (1994), we consider, in the case of space time tensor product discretizations, a stabilized variational problem which is unconditionally stable. We provide a stability and error analysis, and some numerical results which confirm the theoretical findings. 1 Introduction While for the analysis of parabolic and hyperbolic partial differential equations a variety of approaches such as Fourier and Laplace methods, semigroup theory, or Galerkin methods, is available, see, for example, [9, 1, 11, 14, 1, ], standard approaches for the numerical solution are in most cases based on semi discretizations where the discretization in space and time is split accordingly, see, e.g., [19] for parabolic problems, and [5, 6, 15] for hyperbolic equations. More recently, there exist space time approaches as for example in [1,, 1, 13, 16, 17, ] for parabolic problems, and [3, 4, 7, 8, 3] for hyperbolic equations, see also [18] where the space time discretization of the wave equation requires some CFL condition. In this work we introduce a stabilized finite element method for a second order ordinary differential equation and we transfer this approach to the corresponding hyperbolic partial differential equation. As model problem we consider the Dirichlet problem for the wave equation, Olaf Steinbach, Marco Zank Institut für Angewandte Mathematik, U Graz, Steyrergasse 3, 81 Graz, Austria. o.steinbach@tugraz.at, zank@math.tugraz.at 1

6 Olaf Steinbach and Marco Zank tt u(x,t) x u(x,t) = f (x,t) for (x,t) Q := (, ), u(x,t) = for (x,t) Σ := Γ (, ), (1) u(x,) = t u(x,) = for x, where R d, d = 1,,3, is a bounded domain with Lipschitz boundary Γ =, > is a finite time and f is a given right hand side. he variational formulation of (1) is to find u H 1,1 ;, (Q) := L (, ;H 1()) H1, (, ;L ()) H 1 (Q) such that t u, t w L (Q) + xu, x w L (Q) = f,w Q () is satisfied for all w H 1,1 ;, (Q) := L (, ;H 1()) H1, (, ;L ()) H 1 (Q), where f L (Q) is given. Here, we use the standard Sobolev and Bochner spaces with the subspaces { } H,(, 1 ;L ()) := v H 1 (, ;L ()): v(,) = and { } H,(, 1 ;L ()) := v H 1 (, ;L ()): v(, ) =. Furthermore,, Q denotes the duality pairing as extension of the inner product in L (Q). Note that the initial condition u(,) = is considered in the strong sense, whereas the initial condition t u(,) = is incorporated in a weak sense. It is well known that for f L (Q) there exists a unique solution u H 1,1 ;, (Q) of the variational formulation (), see [9, heorem 3. in Chapter IV], and [18]. Nevertheless, a conforming tensor product space time discretization of () by piecewise multilinear continuous functions requires the CFL condition [18] h t 1 d h x, (3) where h t and h x are the uniform mesh sizes in time and space, see also Remark 1 in Section 3. o gain a deeper understanding of the CFL condition (3) a corresponding scalar ordinary differential equation tt u(t) + µ u(t) = f (t) for t (, ), u() = t u() =, (4) where µ > and f are given, is analyzed and an unconditionally stable finite element method for (4) is introduced. Note that µ > is related to the eigenvalues of the Laplace operator for homogeneous Dirichlet conditions. Instead of the variational formulation () we consider a stabilized formulation which generalizes the approach of [3]. his stabilization is first discussed for the scalar differential equation (4), and then transferred to the wave equation (1). he rest of this paper is organized as follows: In Section we consider the second order ordinary differential equation (4), where we show unique solvability. In addition, we prove a discrete inf sup condition to get an unconditionally stable numerical scheme and error estimates. We present some numerical examples to illustrate the theoretical

7 A Stabilized Space ime Finite Element Method for the Wave Equation 3 results. In Section 3 we extend the ideas of Section to the scalar wave equation, where we get an unconditionally stable finite element method for the wave equation. We present a numerical analysis of the discretization scheme, an error estimate with respect to L (Q) and we provide some numerical results for illustration. Second order ordinary differential equations As a model problem we consider the second order linear equation for µ >, tt u(t) + µ u(t) = f (t) for t (, ), u() = t u() =, (5) and the variational formulation to find u H, 1 (, ) such that a(u,w) = f,w (, ) (6) is satisfied for all w H 1, (, ), where > and f [H1, (, )] are given, and where the bilinear form is a(u,w) := t u, t w L (, ) + µ u,w L (, ). Note that, (, ) denotes the duality pairing as extension of the inner product in L (, ), and the Sobolev spaces { } H,(, 1 ) := v H 1 (, ): v() =, { } H,(, 1 ) := v H 1 (, ): v( ) = are endowed with the inner products and with the induced norm u,v H 1, (, ) := u,v H, 1 (, ) := t u(t) t v(t)dt, u H 1 (, ) := tu L (, ) = [ t u(t)] dt. he dual space [H, 1 (, )] is characterized as completion of L (, ) with respect to the norm f [H 1, (, )] := sup f,w (, ). w H, 1 (, ) w H 1 (, ) For v H, 1 (, ) we define w(t) = (H v)(t) := v( ) v(t), i.e. w H, 1 (, ). hen the variational formulation (6) is equivalent to the variational formulation to find u H, 1 (, ) such that

8 4 Olaf Steinbach and Marco Zank t u, t H v L (, ) + µ u,h v L (, ) = f,h v (, ) (7) is satisfied for all v H, 1 (, ). Since the bilinear form t u, t H v L (, ) = tu, t v L (, ) for u,v H 1,(, ) implies an elliptic operator A : H 1, (, ) [H1, (, )], unique solvability of the variational formulation (7) follows by using some compact perturbation argument, and injectivity, see [18, heorem 4.6]. Next, we consider a conforming finite element discretization for the variational formulation (7). For a time interval (, ) and a discretization parameter N N we define 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, the global mesh size h = maxh l and a related finite element space Sh 1(, ) = span{ϕ k} N k= of piecewise linear continuous functions, where the basis functions ϕ k are the usual hat functions. he Galerkin Bubnov finite element discretization of the variational formulation (7) is to find u h V h := Sh 1(, ) H1, (, ) = span{ϕ k} N k=1 such that t u h, t H v h L (, ) + µ u h,h v h L (, ) = f,h v h (, ) (8) is satisfied for all v h V h. It turns out that for a sufficiently small mesh size h 3 ( + µ )µ (9) there holds the discrete stability condition [18, heorem 4.9] c(µ, ) u h H 1 (, ) sup a(u h,h v h ) v h V h v h H 1 (, ) for all u h V h, implying the error estimate [18, heorem 4.1], when assuming u H (, ), u u h H 1 (, ) c(µ, )h u H (, ). When considering the stability of the finite element scheme (8) in the case of a uniform mesh, i.e. when analyzing the root condition, instead of (9) we conclude the weaker mesh assumption [18] 1 h µ. o overcome the mesh condition (9) we will stabilize the numerical scheme in (8) for which we need the following technical lemmata, where the trapezoidal rule is

9 A Stabilized Space ime Finite Element Method for the Wave Equation 5 used analogously as in [3, Chapter ]. In addition to Sh 1 (, ) we also use the finite element space Sh (, ) of piecewise constant functions. Lemma 1. For all f L (, ) there holds t t t I h f (s)ds = Q h f = ti h f (s)ds, (1) where I h : C[, ] Sh 1 (, ) is the piecewise linear nodal interpolation operator, and Q h : L (, ) Sh (, ) denotes the L projection on the piecewise constant finite element space Sh (, ). Proof. For t τ l = (t l 1,t l ), l = 1,...,N, we have t t I h f (s)ds = 1 [ tl tl 1 ] f (s)ds f (s)ds = 1 tl f (s)ds = Q h h l h f, l t l 1 and t t I h f (s)ds = 1 [ tl h l tl 1 ] f (s)ds f (s)ds = 1 tl f (s)ds = Q h h f. l t l 1 Lemma. For all u h Sh 1(, ) H1, (, ) and w h Sh 1(, ) H1,(, ) there holds the representation u h,w h L (, ) = 1 1 N h l tu h, t w h L (τ l ) + u h,q h w h L (, ), (11) where Q h : L (, ) Sh (, ) denotes the L projection on the piecewise constant finite element space Sh (, ). Proof. We consider the L projection Q h on the finite element space S h (, ), and with the error representation of the trapezoidal rule we obtain for each finite element τ l, l = 1,...,N, Q h t w h (s)ds = 1 h l tl t l 1 t w h (s)dsdt = 1 [ tl 1 w h (s)ds + = Q h I h t tl w h (s)ds h l 1 tw h τl. Using integration by parts and (1) we further have t u h (t)i h t w h (s)dsdt = = ] w h (s)ds h l 1 tw h τl t u h (t) t I h w h (s)dsdt u h (t)q h w h(t)dt.

10 6 Olaf Steinbach and Marco Zank With this we then conclude, by using integration by parts and the local definition of the L projection Q h, t u h,w h L (, ) = u h (t) t = = N tl N h tl l 1 N t l 1 t u h (t)q h w h (s)dsdt = t w h (s)dsdt t l 1 t u h (t) t w h (t)dt N tl t t u h (t) w h (s)dsdt tl t l 1 t u h (t)q h I h = 1 1 h N l tu h, t w h L (τ l ) t u h (t)i h t l 1 = 1 1 = 1 1 = 1 1 N N N t t w h (s)dsdt t h l tu h, t w h L (τ l ) t u h (t)i h w h (s)dsdt t h l tu h, t w h L (τ l ) + u h (t) t I h w h (s)dsdt h l tu h, t w h L (τ l ) + u h,q h w h L (, ), w h (s)dsdt i.e. the representation (11). Now we are in a position to find an alternative representation of the bilinear form a(, ). Corollary 1. For u h Sh 1(, ) H1, (, ) and w h Sh 1(, ) H1,(, ) we have a(u h,w h ) = t u h, t w h L (, ) + µ u h,w h L (, ) N = t u h, t w h L (, ) + µh l 1 tu h, t w h L (τ l ) + µ u h,q h w h L (, ) ( µh ) = l 1 1 t u h, t w h L (τ l ) + µ u h,q h w h L (, ). (1) N Motivated by the representation (1) we now define the perturbed bilinear form a h (u h,w h ) := t u h, t w h L (, ) + µ u h,q h w h L (, ) (13) for u h Sh 1(, ) H1, (, ) and w h Sh 1(, ) H1,(, ), and we consider the perturbed variational formulation to find ũ h Sh 1(, ) H1, (, ) such that is satisfied for all w h Sh 1(, ) H1,(, ). a h (ũ h,w h ) = f,w h (, ) (14)

11 A Stabilized Space ime Finite Element Method for the Wave Equation 7 Lemma 3. he perturbed bilinear form (13) is bounded, i.e. we have a h (u h,w h ) (1 + 1 ) µ u h H 1 (, ) w h H 1 (, ) for all u h Sh 1(, ) H1, (, ), w h Sh 1(, ) H1,(, ). Proof. With the Cauchy Schwarz inequality, the L stability of Q h, and with the Poincaré inequality we have a h (u h,w h ) u h H 1 (, ) w h H 1 (, ) + µ u h L (, ) Q h w h L (, ) (1 + 1 ) µ u h H 1 (, ) w h H 1 (, ) for u h Sh 1(, ) H1, (, ), w h Sh 1(, ) H1,(, ), and so the assertion. o prove a discrete stability condition for the perturbed bilinear form (13) we need the following lemma which is analogous to [3, heorem.1]. Lemma 4. For a given z h Sh 1(, ) H1,(, ) represented by z h (t) = N i= z i ϕ i (t) with z N = and a fixed index j {,...,N 1} there exists a function v j h S1 h (, ) H1, (, ) with the following properties: i. For t [,t j ] we have v j h (t) =. ii. For l = j + 1,...,N we have t v j h, tz h L (τ l ) = 1 ( ) z l z l 1 as well as v j h,q h z h L (τ l ) = 1 ( tl ) z h (s)ds 1 ( tl 1 iii.here holds the estimate t j v j h H 1 (, ) z h L (, ). t j ) z h (s)ds. Proof. For z h Sh 1(, ) H1,(, ) we consider the piecewise linear interpolation of the antiderivative, i.e. for t [, ] we define v h (t) := N k= ( tk ) z h (s)ds t ϕ k (t) = I h z h (s)ds, v h Sh 1 (, ) H1,(, ).

12 8 Olaf Steinbach and Marco Zank From (1) the relation t v h = Q h z h follows. For a fixed index j {,...,N 1} we now define z j N h (t) = z j i ϕ i(t), i= z j i = { ( 1) j i z j for i =,..., j, z i for i = j + 1,...,N. Note that z j h S1 h (, ) H1, (, ), and according to z j h we introduce v j h satisfying t v j h = Q h z j h. In particular for j > and t τ l for l = 1,..., j we then have t v j h (t) = Q h z j h (t) = 1 h l tl t l 1 z j h (s)ds = 1 ( ) z j l 1 + z j l =, and due to v j h () = we conclude v j h (t) = for t [,t j], i.e. i. o prove ii. we compute for l = j + 1,...,N as well as t v j h, tz h L (τ l ) = Q h z j h, tz h L (τ l ) = 1 ( ) z j l 1 + z j l )(z l z l 1 = 1 ) (z l 1 + z l )(z l z l 1 = 1 ( ) z l z l 1 t v j tl h,q h z h L (τ l ) = I h z j h (s)dsq h z h(t)dt t l 1 tl [ tl 1 = Q h z h τ l t l 1 z j tl h (s)dsϕ l 1(t) + ] z j h (s)dsϕ l(t) dt = 1 tl z h (s)ds 1 [ tl 1 h l t l 1 h l z j tl ] h (s)ds + z j h (s)ds = 1 tl [ tl 1 tl ] z h (s)ds z h (s)ds + z h (s)ds t l 1 t j t j = 1 ( tl ) tl tl 1 z h (s)ds + z h (s)ds z h (s)ds t l 1 t l 1 t j = 1 ( tl tl 1 ) z h (s)ds + z h (s)ds 1 ( tl 1 ) z h (s)ds t l 1 t j t j = 1 ( tl ) z h (s)ds 1 ( tl 1 ) z h (s)ds. t j From the L stability of Q h we finally conclude the third assertion, i.e. t j v j h H 1 (, ) = v j h H 1 (t j, ) = Q h z j h L (t j, ) = Q h z h L (t j, ) Q h z h L (, ) z h L (, ).

13 A Stabilized Space ime Finite Element Method for the Wave Equation 9 Lemma 5. he variational formulation to find z h Sh 1(, ) H1,(, ) such that a h (v h,z h ) = g, t v h L (, ) (15) is satisfied for all v h S 1 h (, ) H1, (, ) is uniquely solvable, where g L (, ) is given. Moreover, the stability estimate holds for any mesh with maximal mesh size h. z h L (, ) g L (, ) (16) Proof. he finite element stiffness matrix of the variational problem (15) is upper triangular with positive diagonal elements and hence, there exists a unique solution z h Sh 1(, ) H1,(, ) of (15). For the stability estimate we consider for an index j {,...,N 1} the function v j h S1 h (, ) H1, (, ) as given in Lemma 4. Plugging v j h into (15) and by using the properties of Lemma 4 this gives g, t v j h L (, ) = a h(v j h,z h) = t v j h, tz h L (, ) + µ v j h,q h z h L (, ) = = 1 N l= j+1 t v j N h, tz h L (τ l ) + µ v j h,q h z h L (τ l ) l= j+1 ( ( ) ( z l z l 1 + µ tl N l= j+1 = 1 z j + µ ( t j ) z h (s)ds. N l= j+1 t j ) ( tl 1 ) ) z h (s)ds z h (s)ds t j his result yields, with the Cauchy Schwarz inequality, the Poincaré inequality, and the use of the properties of Lemma 4, z h N L (, ) = z h N L (τ l ) = 1 N 1 j=1 N 1 j=1 N 1 j=1 h j z j + 1 N 1 h j+1 z j j= h ( ) l z l 3 + z lz l 1 + z l 1 h j g, t v j N 1 h L (, ) + h j+1 g, t v j h L (, ) j= 1 h j g L (, ) v j N 1 h H 1 (, ) + h j+1 g L (, ) v j h H 1 (, ) j= g L (, ) z h L (, ), N h l (z l + z l 1 ) i.e. the assertion.

14 1 Olaf Steinbach and Marco Zank Lemma 6. For each u h Sh 1(, ) H1, (, ) there holds the discrete inf sup condition µ u h H 1 (, ) sup a h (u h,w h ). w h Sh 1(, ) H1, (, ) w h H 1 (, ) Proof. For a fixed function u h S 1 h (, ) H1, (, ) let w h S 1 h (, ) H1, (, ) be the unique solution of (15) for g := t u h L (, ), i.e. we have a h (v h,w h ) = t u h, t v h L (, ) (17) for all v h Sh 1(, ) H1, (, ). For the particular choice v h(t) = w h () w h (t) with v h Sh 1(, ) H1, (, ) we obtain t w h, t w h L (, ) µ w h w h (),Q h w h L (, ) = tu h, t w h L (, ), and hence we conclude, by using the Cauchy Schwarz and Poincaré inequalities, and the L stability of the L projection Q h, w h H 1 (, ) = tu h, t w h L (, ) + µ w h w h (),Q h w h L (, ) u h H 1 (, ) w h H 1 (, ) + µ w h w h () L (, ) Q h w h L (, ) u h H 1 (, ) w h H 1 (, ) + 1 µ w h H 1 (, ) w h L (, ) ( 1 + µ ) u h H 1 (, ) w h H 1 (, ), where in the last step we used the stability estimate (16). he choice v h = u h Sh 1(, ) H1, (, ) in (17) and the estimate above yield a h (u h,w h ) = u h H 1 (, ) µ u h H 1 (, ) w h H 1 (, ) and hence the discrete inf sup condition follows. heorem 1. Let the unique solution u of (6) satisfy u H s (, ) for some s [1,]. here exists a unique solution ũ h Sh 1(, ) H1, (, ) of the Galerkin Petrov finite element discretization (14) satisfying ( u ũ h H 1 (, ) [ µ )( 1 + µ )] C 1 (s)h s 1 u H s (, ) µ ( 1 + µ ) h C u H 1 (, ), where the constants C 1,C > are coming from standard interpolation error and stability estimates. Proof. For any v h Sh 1(, ) H1, (, ) we first have

15 A Stabilized Space ime Finite Element Method for the Wave Equation 11 u ũ h H 1 (, ) u v h H 1 (, ) + ũ h v h H 1 (, ) and it remains to bound the second term. With the discrete inf sup condition in Lemma 6 and using the Galerkin orthogonality for the variational formulations (6) and (14), we first have µ ũ h v h H 1 (, ) sup a h (ũ h v h,w h ) w h Sh 1(, ) H1, (, ) w h H 1 (, ) a h (ũ h,w h ) a h (v h,w h ) = sup w h Sh 1(, ) H1, (, ) w h H 1 (, ) a(u,w h ) a h (v h,w h ) = sup w h Sh 1(, ) H1, (, ) w h H 1 (, ) a(u v h,w h ) + a(v h,w h ) a h (v h,w h ) = sup. w h Sh 1(, ) H1, (, ) w h H 1 (, ) Now, with the boundedness of the bilinear form a(, ) and the Poincaré inequality we further conclude a(u v h,w h ) = t (u v h ), t w h L (, ) + µ u v h,w h L (, ) u v h H 1 (, ) w h H 1 (, ) + µ u v h L (, ) w h L (, ) ( µ ) u v h H 1 (, ) w h H 1 (, ). Moreover, using the representation (1) we can estimate the consistency error by a(v h,w h ) a h (v h,w h ) = 1 N 1 µ h l tv h, t w h L (τ l ) Hence we have 1 1 µ h v h H 1 (, ) w h H 1 (, ). 1 ( 1 + µ ũ h v h H 1 (, ) µ ) u v h H 1 (, ) µ h v h H 1 (, ), and therefore ( u ũ h H 1 (, ) [ µ )( 1 + µ )] u v h H 1 (, ) µ ( 1 + µ ) h v h H 1 (, ) follows. In particular for the piecewise linear nodal interpolation v h = I h u we have

16 1 Olaf Steinbach and Marco Zank u I h u H 1 (, ) C 1(s)h s 1 u H s (, ), I h u H 1 (, ) C u H 1 (, ). As numerical example for the Galerkin finite element methods (8) and (14) we consider a uniform discretization of the time interval (, ) with = 1 and a mesh size h = /N. For µ = 1 we consider the solution u(t) = sin ( 5 4 πt ) and we compute the appearing integrals for the related right hand side in (8) and (14) by the usage of high order integration rules. In able 1 we present the results for the stabilized variational formulation (14) which is unconditionally stable, and where the error estimate in the energy norm of heorem 1 is confirmed. In addition we also present the error in L (, ) where we observe a second order convergence, as expected. But at this point we do not include any further discussion of error estimates in L (, ) since this is behind the scope of this contribution. N h u ũ h L (,1) eoc u ũ h H 1 (,1) eoc e e e e e e e e e e e e e e e e e e e e e e e e e e e e-3 1. able 1 Numerical results for the stabilized variational formulation (14), µ = 1, = 1. In able we present the related results for the variational formulation (8) without stabilization. We observe that we have convergence for a sufficiently small mesh size only. Note that 1/µ Wave equation Instead of the ordinary differential equation (5) we now consider the wave equation (1) and the related variational formulation (), and we aim to extend the results of Section. For u H 1,1 ;, (Q) and w H1,1 ;, (Q) we define the bilinear form a(u,w) := t u, t w L (Q) + xu, x w L (Q).

17 A Stabilized Space ime Finite Element Method for the Wave Equation 13 N h u u h L (,1) eoc u u h H 1 (,1) eoc e e e e e e e e e e e e e e e e e e e e e e e e e e e e-3 1. able Numerical results for the variational formulation (8), µ = 1, = 1. he Hilbert spaces H 1,1 ;, (Q) and H1,1 ;, (Q), defined in Section 1, are endowed with the inner product [ ] u,v 1,1 H ;, (Q) = u,v H 1,1 ;, (Q) := t u(x,t) t v(x,t)+ x u(x,t) x v(x,t) dxdt, where the induced norm u H 1 (Q) := [ t u(x,t) + x u(x,t) ] dxdt is well defined due to Poincaré inequalities with respect to space and time. As in [9] we have unique solvability of () when assuming f L (Q), in particular we have, see [18], u H 1 (Q) 1 f L (Q). Next, we examine a conforming finite element discretization for the variational formulation () in the case where = (,L) is an interval for d = 1, or is polygonal for d =, or is polyhedral for d = 3. For a tensor product ansatz we consider a sequence ( N ) N N of admissible decompositions N x Q = N = [, ] = ω i i=1 N t τ l with N := N x N t space time elements, where the time intervals τ l = (t l 1,t l ) with mesh size h t,l are defined via the decomposition = t < t 1 < t <... < t Nt 1 < t Nt =

18 14 Olaf Steinbach and Marco Zank of the time interval (, ). For the spatial domain we consider an admissible and globally quasi uniform decomposition into finite elements ω i with mesh size h x,i which can be represented by using standard maps with respect to related reference elements. Here, the spatial elements ω i are intervals for d = 1, triangles or quadrilaterals for d =, and tetrahedra or hexahedra for d = 3. Next, we consider the finite element space Q 1 h (Q) := V h x () S 1 h t (, ) of piecewise multilinear continuous functions, i.e. V hx () = span{ψ j } M x j=1 H1 (), S 1 h t (, ) = span{ϕ l } N t l= H1 (, ). In fact, V hx () is either the space Sh 1 x () of piecewise linear continuous functions on intervals (d = 1), triangles (d = ), and tetrahedra (d = 3), or V hx () is the space () of piecewise linear/bilinear/trilinear continuous functions on intervals (d = 1), quadrilaterals (d = ), and hexahedra (d = 3). A finite element function u h Q 1 h (Q) admits the representation u h (x,t) = N t M x l= j=1 u l jψ j (x)ϕ l (t) for (x,t) Q. (18) he Galerkin Petrov finite element discretization of the variational formulation () is to find u h Q 1 h (Q) H1,1 (Q) such that ;, a(u h,w h ) = t u h, t w h L (Q) + xu h, x w h L (Q) = f,w h Q (19) is satisfied for all w h Q 1 h (Q) H1,1 ;, (Q). After an appropriate ordering of the degrees of freedom, the discrete variational formulation (19) is equivalent to the linear system K h u = f with the system matrix K h := M hx A ht + A hx M ht R M x N t M x N t, () where M hx, A hx R M x are the mass and stiffness matrix with respect to space, and M ht, A ht R N t are the mass and stiffness matrix with respect to time, respectively. Remark 1. With the help of a von Neumann type stability analysis [5] for the matrix () of the Galerkin Petrov finite element method (19) for a uniform discretization in time with mesh size h t and a uniform discretization in space with mesh size h x for piecewise linear/bilinear/trilinear continuous functions on intervals (d = 1), squares (d = ), or cubes (d = 3) we can show stability of the corresponding numerical scheme, if the condition h t 1 d h x is satisfied, see [18].

19 A Stabilized Space ime Finite Element Method for the Wave Equation 15 From Remark 1 we conclude that we have only conditional stability of (19). o stabilize the numerical scheme in (19) we use as in (1) again Zlotnik s idea [3] and we prove the following representation. Lemma 7. For all u h Q 1 h (Q) H1,1 ;, (Q) and w h Q 1 h (Q) H1,1 ;, (Q) the bilinear form in (19) has the representation a(u h,w h ) = t u h, t w h L (Q) + d m=1 xm u h,q h t xm w h L (Q) (1) + d m=1 N t h t,l 1 t xm u h, t xm w h L ( τ l ), where Q h t : L (, ) S h t (, ) denotes the L projection with respect to time on the space S h t (, ) of piecewise constant functions. Proof. Let u h Q 1 h (Q) H1,1 ;, (Q) and w h Q 1 h (Q) H1,1 ;, (Q) be given. With the representation (18) we have for (x,t) Q u h (x,t) = as well as N t M x j=1 N t 1 w h (x,t) = l= M x j=1 u l jψ j (x)ϕ l (t) = w l jψ j (x)ϕ l (t) = M x j=1 M x j=1 Hence we have, for m = 1,...,d, and by using (11), xm u h, xm w h L (Q) = M x = M x M x i=1 j=1 [ 1 1 N t M x i=1 j=1 U j,h (t)ψ j (x), U j,h (t) = N t N t 1 W j,h (t)ψ j (x), W j,h (t) = l= u l jϕ l (t), w l jϕ l (t). U i,h (t)w j,h (t)dt xm ψ i (x) xm ψ j (x)dx ] h t,l tu i,h, t W j,h L (τ l ) + U i,h,q h t W j,h L (, ) xm ψ i (x) xm ψ j (x)dx = xm u h,q N t h t xm w h L (Q) + ht,l 1 t xm u h, t xm w h L ( τ l ). Due to the representation (1) we define for u h Q 1 h (Q) H1,1 ;, (Q) and w h (Q) H1,1 (Q) the perturbed bilinear form Q 1 h ;, a h (u h,w h ) := t u h, t w h L (Q) + d m=1 xm u h,q h t xm w h L (Q) ()

20 16 Olaf Steinbach and Marco Zank = t u h, t w h L (Q) + d m=1 Q h t xm u h, xm w h L (Q), (3) and we consider the perturbed variational problem to find ũ h Q 1 h (Q) H1,1 ;, (Q) such that a h (ũ h,w h ) = f,w h Q (4) is satisfied for all w h Q 1 h (Q) H1,1 ;, (Q). o prove the existence and uniqueness of a solution ũ h of (4) we show the following lemma, which is analogous to Lemma 4. Lemma 8. For a given v h Q 1 h (Q) H1,1 (Q) represented by ;, v h (x,t) = N t l= V l,h (x)ϕ l (t), V l,h (x) = M x j=1 v l jψ j (x) for (x,t) Q with V,h (x) = for x and V l,h V hx (), and for a fixed index j {1,...,N t } there exists a function z j h Q1 h (Q) H1,1 (Q) with the following properties: ;, i. For (x,t) [t j, ] we have z j h (x,t) =. ii. For l = 1,..., j and for x we have and for m = 1,...,d t z j h (x, ), tv h (x, ) L (τ l ) = 1 ( [V l 1,h (x)] [V l,h (x)] ), xm z j h (x, ),Q h t xm v h (x, ) L (τ l ) = = 1 ( t ) j xm v h (x,s)ds 1 ( t j t l 1 iii.for x there holds the estimate i= t z j h (x, ) L (, ) v h(x, ) L (, ). t l ) xm v h (x,s)ds. Proof. For v h Q 1 h (Q) H1,1 ;, (Q) we define { u j N t h (x,t) = U j i,h (x)ϕ i(t), U j V i,h (x) for i =,..., j, i,h (x) := ( 1) j i V j,h (x) for i = j + 1,...,N t, and for (x,t) Q we set z j h (x,t) := N t k= tk t u j h (x,s)dsϕ k(t) = I ht u j h (x,s)ds,

21 A Stabilized Space ime Finite Element Method for the Wave Equation 17 where I ht : C[, ] Sh 1 (, ) is the interpolation operator with respect to time. Note that z j h Q1 h (Q) H1,1 (Q). For x it follows from relation (1) that ;, t z j h (x, ) = Q h t u j h (x, ). In particular for j < N t, x, and for t τ l for l = j + 1,...,N t we then have t z j h (x,t) = Q h t u j h (x,t) = 1 tl u j h h (x,s)ds = 1 ( ) U j j t,l t l 1 l 1,h (x) +Ul,h (x) =, and due to z j h (x, ) = we conclude z j h (x,t) = for t [t j, ], i.e. i. o prove ii. we first compute for x and for l = 1,..., j t z j h (x, ), tv h (x, ) L (τ l ) = tl t z j h (x,t) tv h (x,t)dt t l 1 = 1 ( ) U j tl j l 1,h (x) +Ul,h (x) t v h (x,t)dt t l 1 = 1 ( ) U j j l 1,h (x) +Ul,h )(V (x) l,h (x) V l 1,h (x) = 1 ( Vl 1,h (x) +V l,h (x) )( V l 1,h (x) V l,h (x)) = 1 ( [V l 1,h (x)] [V l,h (x)] ). Moreover, for m = 1,...,d we have for x and for l = 1,..., j xm z j tl h (x, ),Q h t xm v h (x, ) L (τ l ) = Q h t xm v h (x, ) τl xm z j h (x,t)dt t l 1 tl [ tl 1 = Q h t xm v h (x, ) τl xm u j tl ] h (x,s)dsϕ l 1(t) + u j h (x,s)dsϕ l(t) dt t l 1 = 1 tl xm v h (x,t)dt 1 [ tl 1 h t,l t l 1 h t,l xm u j tl ] h (x,s)ds + xm u j h (x,s)ds = 1 [ tl tl 1 ] xm v h (x,t)dt xm v h (x,t)dt t j t j [ tl 1 xm v j tl ] h (x,s)ds + xm v j h (x,s)ds t j t j = 1 ( tl 1 ) xm v j h (x,s)ds 1 ( tl ) xm v j h (x,s)ds. t j From the L stability of Q h t and by using t j t z j h (x, ) L (, ) = tz j h (x, ) L (,t j ) = Q h t u j h (x, ) L (,t j )

22 18 Olaf Steinbach and Marco Zank = Q h t v h (x, ) L (,t j ) Q h t v h (x, ) L (, ) v h(x, ) L (, ) for x we finally conclude the third property. With the last lemma we are now able to prove existence, uniqueness, and stability of a solution ũ h of (4). Lemma 9. For f [H 1, (, ;L ())] and f 1, f L (Q) the variational formulation to find w h Q 1 h (Q) H1,1 (Q) such that ;, a h (w h,v h ) = f,v h Q + f 1, t v h L (Q) + N t h t,l f, t v h L ( τ l ) (5) is satisfied for all v h Q 1 h (Q) H1,1 ;, (Q) is uniquely solvable, and there holds the stability estimate } w h L (Q) { f [H 1, (, ;L ())] + f 1 L (Q) + h t f L (Q). (6) Proof. Let w h Q1 h (Q) H1,1 ;, (Q) be any solution of the homogeneous variational formulation (5) with f i, and with the representation (18), i.e. w h (x,t) = N t l= Wl,h (x)ϕ l(t) for (x,t) Q, Wl,h V h x (), W,h (x) =. For an index j {1,...,N t } we now consider an element z j h Q1 h (Q) H1,1 ;, (Q) as given in Lemma 8. Plugging z j h into (5) and by using the properties of Lemma 8 this gives = a h (w h,zj h ) = t w h, tz j h L (Q) + d m=1 Q h t xm w h, x m z j h L (Q) j j = t w h, tz j d h L ( τ l ) + Q h t xm w h, x m z j h L ( τ l ) m=1 j ( 1 = [W l 1,h (x)] 1 ) [W l,h (x)] dx ( d j ( 1 t ) j + xm w h m=1 (x,s)ds 1 ( t ) ) j xm w h t l 1 (x,s)ds dx t l = 1 [W j,h x (x)] dx + 1 d ( t ) j xm w h (x,s)ds dx. m=1

23 A Stabilized Space ime Finite Element Method for the Wave Equation 19 his result yields, with the Cauchy Schwarz inequality and the use of the properties of Lemma 8, w h L (Q) = N t = N t w h L ( τ l ) which implies w h. By using h ( t,l [Wl,h 3 (x)] +Wl,h (x)w l 1,h (x) + [Wl 1,h (x)]) dx N t h t, j j=1 [W j,h (x)] dx + N t 1 j=1 h t, j+1 [W j,h (x)] dx, dimq 1 h (Q) H1,1 ;, (Q) = dimq1 h (Q) H1,1 ;, (Q) we therefore conclude unique solvability of the variational formulation (5) for any right hand side f [H 1, (, ;L ())] and f 1, f L (Q). Following the approach as above we then obtain and w h L (Q) f,z j h Q + f 1, t z j h L (Q) + N t N t j=1 N t h t, j j=1 [W j,h(x)] dx + { h t, j N t 1 + j=1 N t j=1 h t,l f, t z j h L ( τ l ) = a h (w h,z j h ) 1 N t 1 j=1 f,z j h Q + f 1, t z j h L (Q) + N t h t, j+1 { h t, j+1 [W j,h (x)] dx f,z j h Q + f 1, t z j h L (Q) + N t [W j,hx (x)] dx h t,l f, t z j h L ( τ l ) } h t,l f, t z j h L ( τ l ) } h t, j { f [H 1, (, ;L ())] + f 1 L (Q) + h t f L (Q) t z j h L (Q) N t 1 + j=1 } h t, j+1 { f [H 1, (, ;L ())] + f 1 L (Q) + h t f L (Q) t z j h L (Q) { f [H 1, (, ;L ())] + f 1 L (Q) + h t f L (Q) } w h L (Q), } and hence the stability estimate is proven.

24 Olaf Steinbach and Marco Zank As a consequence of Lemma 9 we obtain unique solvability of the variational formulation (4), and the stability estimate ũ h L (Q) f [H 1, (, ;L ())]. o derive an estimate for the L error u ũ h L (Q) we introduce, as in [, Section ], a space time projection Q 1 h t v Q 1 h (Q) H1,1 ;, (Q) when v H1,1 ;, (Q) is given. First, the H 1 projection Q1 h x : L (, ;H 1()) V h x () L (, ) is defined by x v(x,t) x v hx (x,t)dxdt = x v(x,t) x v hx (x,t)dxdt (7) for all v hx V hx () L (, ). Note that we have the stability estimate x v L (Q) xv L (Q), and, when assuming v L (, ;H 1+s ()) for some s [,1], the standard error estimate v v L ( N ) ch1+s x v L (, ;H 1+s ()), (8) if is sufficiently regular. Second, we define the H, 1 projection Q1 h t : H, 1 (, ;L ()) L () Sh 1 t (, ) H, 1 (, ) by t Q 1 h t v(x,t) t v ht (x,t)dxdt = t v(x,t) t v ht (x,t)dxdt (9) for all v ht L () Sh 1 t (, ) H, 1 (, ). Again we have the stability estimate t Q 1 h t v L (Q) tv L (Q), and, when assuming v H 1+s (, ;L ()) for some s [,1], the standard error estimate v Q 1 h t v L (Q) ch1+s t v H 1+s (, ;L ()). (3) he next lemma shows that Q 1 h t v Q 1 h (Q) H1,1 ;, (Q) is well defined under some regularity assumptions on v, and that the projection operators in space and time commute, see also [, Lemma.1]. Lemma 1. For a given function v H 1,1 ;, (Q) satisfying tv L (, ;H 1 ()) and xm v H, 1 (, ;L ()), m = 1,...,d, there hold the following relations: i. t v = t v V hx () L (, ), ii. xm Q 1 h t v = Q 1 h t xm v L () Sh 1 t (, ) H, 1 (, ), m = 1,...,d, iii.q 1 h t v = Q 1 h t v Q 1 h (Q) H1,1 ;, (Q). In particular, the space time projections Q 1 h t v and Q 1 h t v are well defined.

25 A Stabilized Space ime Finite Element Method for the Wave Equation 1 Moreover, there holds the error estimate v Q 1 h t v L (Q) v Q1 h t v L (Q) + v Q1 h x v L (Q) + ch x h t t x v L (Q). Proof. For t v L (, ;H 1 ()) we consider Q1 h x t v V hx () L (, ) as the unique solution of the variational formulation x t v(x,t) x v hx (x,t)dxdt = x t v(x,t) x v hx (x,t)dxdt for all v hx V hx () C (, ). By using integration by parts in time twice this gives x t v(x,t) x v hx (x,t)dxdt = = = x v(x,t) x t v hx (x,t)dxdt x v(x,t) x t v hx (x,t)dxdt x t v(x,t) x v hx (x,t)dxdt for all v hx V hx () C (, ). Since C (, ) is dense in L (, ) this holds true for all v hx V hx () L (, ), i.e. i. he proof of ii. follows in the same manner. o prove iii. we first note that v V hx () H 1, (, ) H1, (, ;L ()), and so Q 1 h t v Q 1 h (Q) H1,1 ;, (Q) is well defined. Analogously we have that Q 1 h t v Q 1 h (Q) H1,1 ;, (Q) is well defined. Now, with i., ii., and the definitions (7), (9) we have that t x Q 1 h t v, t x v h L (Q) = tq 1 h t x v, t x v h L (Q) = t x v, t x v h L (Q) = t x v, t x v h L (Q) as well as t x Q 1 h t v, t x v h L (Q) = t x v, t x v h L (Q) for all v h Q 1 h ( N) H 1,1 ;, (Q), i.e. iii. he error estimate follows from the triangle inequality v Q 1 h t v L (Q) v Q1 h t v L (Q) + Q1 h t (v v) L (Q) v Q 1 h t v L (Q) + v Q h x v L (Q) + (I Q1 h t )(I )v L (Q) v Q 1 h t v L (Q) + v Q h x v L (Q) + ch t t (I )v L (Q) v Q 1 h t v L (Q) + v Q h x v L (Q) + ch t h x t x v L (Q).

26 Olaf Steinbach and Marco Zank Now we are in a position to prove an error estimate for the approximate solution ũ h. heorem. Let u H 1,1 ;, (Q) be the unique solution of the variational formulation () satisfying t u L (, ;H 1()) and x m u H, 1 (, ;L ()), m = 1,...,d, and x u H, 1 (, ;L ()). hen, the unique solution ũ h Q 1 h (Q) H1,1 ;, (Q) of the Galerkin Petrov finite element discretization (4) satisfies the error estimate u ũ h L (Q) u Q 1 h t u L (Q) + u Q1 h x u L (Q) + ch x h t t x u L (Q) { + x (u Q 1 h t u) [H 1, (, ;L ())] + t( u u) L (Q) + h } t 1 t x u L (Q). Proof. Since the solution u fulfills the assumptions of Lemma 1, the space time projection Q 1 h t u Q 1 h (Q) H1,1 ;, (Q) is well defined. When using the representation (1), the properties of Q 1 h t as given in Lemma 1, and applying integration by parts, we conclude for all w h Q 1 h (Q) H1,1 (Q) that ;, a h (ũ h Q 1 h t u,w h ) = a h (ũ h,w h ) a h (Q 1 h t u,w h ) = a(u,w h ) a h (Q 1 h t u,w h ) = a(u,w h ) a(q 1 h t u,w h ) + N t h t,l 1 t x Q 1 h t u, t x w h L ( τ l ) = t u, t w h L (Q) + xu, x w h L (Q) + tq 1 h t u, t w h L (Q) x Q 1 h t Q 1 N t h x u, x w h L (Q) + ht,l 1 t x Q 1 h t u, t x w h L ( τ l ) = t u, t w h L (Q) + xu, x w h L (Q) + t u, t w h L (Q) x Q 1 N t h t u, x w h L (Q) + ht,l 1 t x Q 1 h t u, t x w h L ( τ l ) = t ( u u), t w h L (Q) + x(u Q 1 h t u), x w h L (Q) + N t h t,l 1 t x Q 1 h t u, t x w h L ( τ l ) = t ( u u), t w h L (Q) + x(u Q 1 h t u),w h L (Q) N t h t,l 1 t x Q 1 h t u, t w h L ( τ l ). In particular we observe that ũ h Q 1 h t u is the unique solution of (5) in the case f = x (u Q 1 h t u), f 1 = t ( u u), f = 1 1 t x Q 1 h t u.

27 A Stabilized Space ime Finite Element Method for the Wave Equation 3 herefore, the stability estimate (6) and the stability of Q 1 h t in H, 1 (, ) give ũ h Q 1 h t u L (Q) { x (u Q 1 h t )u [H 1, (, ;L ())] + t ( u u) L (Q) + h } t 1 t x Q 1 h t u L (Q) { x (u Q 1 h t )u [H 1, (, ;L ())] + t ( u u) L (Q) + h } t 1 t x u L (Q). With the last estimate, the triangle inequality, and the error estimate of Lemma 1 we finally obtain u ũ h L (Q) u Q1 h t u L (Q) + ũ h Q 1 h t u L (Q) u Q 1 h t u L (Q) + u Q1 h x u L (Q) + ch x h t t x u L (Q) { + x (u Q 1 h t u) [H 1, (, ;L ())] + t( u u) L (Q) + h } t 1 t x u L (Q). By using the error estimates (3) for the H 1, projection Q 1 h t and (8) for the H 1 projection we now conclude from heorem the following error estimate. Corollary. Let the assumptions of heorem be satisfied. If in addition the unique solution u of () is sufficiently smooth and is sufficiently regular, we obtain the error estimate ( ) u ũ h L (Q) ch x +ch x h t t x u L (Q) + ch t u L (, ;H ()) + tu L (, ;H ()) ( tt u L (Q) + tt x u L (Q) + t x u L (Q) (31) ). As a numerical example for the Galerkin Petrov finite element method (4) we consider the one dimensional spatial domain = (, 1), i.e. we have the rectangular space time domain Q := (, ) := (,1) (,1). he discretization is done with respect to nonuniform meshes as shown in Fig. 1 where we apply a uniform refinement strategy. Note that these meshes do not fulfill the CFL condition (3). As exact solutions we choose for (x,t) Q u 1 (x,t) = sin(πx)sin ( 5 4 πt ), u (x,t) = sin(πx)t (1 t) 3/4. he appearing integrals to compute the related right hand side in (4) are calculated by using high order quadrature rules. he numerical results for the smooth solution u 1 are given in able 3 where we observe unconditional stability and quadratic convergence in L (Q), as predicted by the error estimate (31). Moreover we have linear convergence when measuring the error in H 1 (Q). Note that such an error

28 4 Olaf Steinbach and Marco Zank estimate can be shown by using the H 1 (Q) projection, an inverse inequality, and the error estimate (31). For the singular solution u the related results are given in able 4 where we observe a reduced order of convergence in L (Q) and in H 1 (Q), respectively. hese convergence rates correspond to the reduced Sobolev regularity u H 5/4 ε (Q), ε >. Remark. he Galerkin Petrov finite element method (4) seems to fulfill a kind of conservation of the total energy E(t) := 1 tu(,t) L () + 1 xu(,t) L (), t [, ]. As illustration we consider a solution of the homogeneous wave equation, i.e. u 3 (x,t) = (cos(πt) + sin(πt))sin(πx) for (x,t) Q := (,1) (,1) with the total energy E(t) = π for t [,1]. Here, the initial condition u 3 (x,) = sin(πx), x, is treated via homogenization, while the initial condition t u 3 (x,) = π sin(πx), x, is incorporated in a weak sense. For the solution u 3 and for the mesh as given in Fig. 1 we compute the discrete total energy In Fig. the difference E h (t) := 1 tũ h (,t) L () + 1 xũ h (,t) L (), t [, ]. E h (t) E(t) = E h (t) π for t [,1] is plotted pointwise for the refinement level with uniform mesh sizes h t = 1 and 6 1 h x = 1. Note that 6 1 t ũ h is piecewise constant in time. Probably due to the used space time approximation we observe some oscillations within the finite element accuracy, but no energy loss occurs.

29 A Stabilized Space ime Finite Element Method for the Wave Equation t 6 t x.5 1 x Fig. 1 Nonuniform meshes: Starting mesh and the mesh after one uniform refinement step. dof h x,max h x,min h t,max h t,min u 1 u 1,h L (Q) eoc u 1 u H 1,h 1 (Q) eoc e e e e e e e e e e e e e e e e e e e e e e- 1. able 3 Numerical results of (4) for Q = (,1) (,1) and for u 1. References 1. ANDREEV, R. Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33, 1 (13), AZIZ, A. K., AND MONK, P. Continuous finite elements in space and time for the heat equation. Math. Comp. 5, 186 (1989), BALES, L., AND LASIECKA, I. Continuous finite elements in space and time for the nonhomogeneous wave equation. Comput. Math. Appl. 7, 3 (1994), BANGERH, W., GEIGER, M., AND RANNACHER, R. Adaptive Galerkin finite element methods for the wave equation. Comput. Methods Appl. Math. 1, 1 (1), COHEN, G. C. Higher-order numerical methods for transient wave equations. Scientific Computation. Springer-Verlag, Berlin,. 6. COHEN, G. C., AND PERNE, S. Finite element and discontinuous Galerkin methods for transient wave equations. Scientific Computation. Springer, Dordrecht, 17.

30 E h (t) - E(t) 6 Olaf Steinbach and Marco Zank dof h x,max h x,min h t,max h t,min u u,h L (Q) eoc u u H,h 1 (Q) eoc e e e e e e e e e e e e e e e e e e e e e e+.6 able 4 Numerical results of (4) for Q = (,1) (,1) and for u t Fig. Difference of the total energy E(t) = π and E h(t) for the solution u 3 for a uniform mesh. 7. DÖRFLER, W., FINDEISEN, S., AND WIENERS, C. Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl. Math. 16, 3 (16), FRENCH, D. A., AND PEERSON,. E. A continuous space-time finite element method for the wave equation. Math. Comp. 65, 14 (1996), LADYZHENSKAYA, O. A. he boundary value problems of mathematical physics, vol. 49 of Applied Mathematical Sciences. Springer-Verlag, New York, LIONS, J.-L., AND MAGENES, E. Problèmes aux limites non homogènes et applications. Vol. 1. ravaux et Recherches Mathématiques, No. 17. Dunod, Paris, LIONS, J.-L., AND MAGENES, E. Problèmes aux limites non homogènes et applications. Vol.. ravaux et Recherches Mathématiques, No. 18. Dunod, Paris, MOLLE, C. Stability of Petrov-Galerkin discretizations: application to the space-time weak formulation for parabolic evolution problems. Comput. Methods Appl. Math. 14, (14), NEUMÜLLER, M. Space-time methods: fast solvers and applications. PhD thesis, U Graz, PAZY, A. Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York, PEERSEIM, D., AND SCHEDENSACK, M. Relaxing the CFL condition for the wave equation on adaptive meshes. J. Sci. Comput. 7 (17), SCHWAB, C., AND SEVENSON, R. Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comp. 78, 67 (9),

31 A Stabilized Space ime Finite Element Method for the Wave Equation SEINBACH, O. Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15, 4 (15), SEINBACH, O., AND ZANK, M. Coercive space time finite element methods for initial boundary value problems. 19. HOMÉE, V. Galerkin finite element methods for parabolic problems, second ed., vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 6.. URBAN, K., AND PAERA, A.. An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comp. 83, 88 (14), WLOKA, J. Partielle Differentialgleichungen. B. G. eubner, Stuttgart, ZEIDLER, E. Nonlinear functional analysis and its applications. II/A. Springer-Verlag, New York, ZLONIK, A. A. Convergence rate estimates of finite-element methods for second-order hyperbolic equations. In Numerical methods and applications. CRC, Boca Raton, FL, 1994, pp. 155.

32 Erschienene Preprints ab Nummer 15/1 15/1 O. Steinbach: Space-time finite element methods for parabolic problems 15/ O. Steinbach, G. Unger: Combined boundary integral equations for acoustic scattering-resonance problems problems. 15/3 C. Erath, G. Of, F. J. Sayas: A non-symmetric coupling of the finite volume method and the boundary element method 15/4 U. Langer, M. Schanz, O. Steinbach, W.L. Wendland (eds.): 13th Workshop on Fast Boundary Element Methods in Industrial Applications, Book of Abstracts 16/1 U. Langer, M. Schanz, O. Steinbach, W.L. Wendland (eds.): 14th Workshop on Fast Boundary Element Methods in Industrial Applications, Book of Abstracts 16/ O. Steinbach: Stability of the Laplace single layer boundary integral operator in Sobolev spaces 17/1 O. Steinbach, H. Yang: An algebraic multigrid method for an adaptive space-time finite element discretization 17/ G. Unger: Convergence analysis of a Galerkin boundary element method for electromagnetic eigenvalue problems 17/3 J. Zapletal, G. Of, M. Merta: Parallel and vectorized implementation of analytic evaluation of boundary integral operators 17/4 S. Dohr, O. Steinbach: Preconditioned space-time boundary element methods for the one-dimensional heat equation 17/5 O. Steinbach, H. Yang: Comparison of algebraic multigrid methods for an adaptive space time finite element discretization of the heat equation in 3D and 4D 17/6 S. Dohr, K. Niino, O. Steinbach: Preconditioned space-time boundary element methods for the heat equation 17/7 O. Steinbach, M. Zank: Coercive space-time finite element methods for initial boundary value problems 17/8 U. Langer, M. Schanz, O. Steinbach, W.L. Wendland (eds.): 15th Workshop on Fast Boundary Element Methods in Industrial Applications, Book of Abstracts 18/1 U. Langer, M. Schanz, O. Steinbach, W.L. Wendland (eds.): 16th Workshop on Fast Boundary Element Methods in Industrial Applications, Book of Abstracts 18/ S. Dohr, J. Zapletal, G. Of, M. Merta, M. Kravcenko: A parallel space-time boundary element method for the heat equation 18/3 S. Dohr, M. Merta, G. Of, O. Steinbach, J. Zapletal: A parallel solver for a preconditioned space-time boundary element method for the heat equation 18/4 S. Amstutz, P. Gangl: oplogical derivative for nonlinear magnetostatic problem