JOHANNES KEPLER UNIVERSITY LINZ. Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems

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1 JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems Martin Jakob Gander Section de Mathématiques -4 rue du Lièvre CP 64 Genève 4 Switzerland Martin Neumüller Institute of Computational Mathematics Johannes Kepler University Altenberger Str Linz Austria NuMa-Report No. 4-8 November 4 A 44 LINZ Altenbergerstraße 69 Austria

2 Technical Reports before 998: Hedwig Brandstetter Was ist neu in Fortran 9? March G. Haase B. Heise M. Kuhn U. Langer Adaptive Domain Decomposition Methods for Finite and Boundary Element August 995 Equations Joachim Schöberl An Automatic Mesh Generator Using Geometric Rules for Two and Three Space August 995 Dimensions Ferdinand Kickinger Automatic Mesh Generation for 3D Objects. February Mario Goppold Gundolf Haase Bodo Heise und Michael Kuhn Preprocessing in BE/FE Domain Decomposition Methods. February Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element February 996 Discretisation Bodo Heise und Michael Jung Robust Parallel Newton-Multilevel Methods. February Ferdinand Kickinger Algebraic Multigrid for Discrete Elliptic Second Order Problems. February Bodo Heise A Mixed Variational Formulation for 3D Magnetostatics and its Finite Element Discretisation. May Michael Kuhn Benchmarking for Boundary Element Methods. June Bodo Heise Michael Kuhn and Ulrich Langer A Mixed Variational Formulation for 3D Magnetostatics in the Space Hrot) February 997 Hdiv) 97- Joachim Schöberl Robust Multigrid Preconditioning for Parameter Dependent Problems I: The June 997 Stokes-type Case Ferdinand Kickinger Sergei V. Nepomnyaschikh Ralf Pfau Joachim Schöberl Numerical Estimates of Inequalities in H. August Joachim Schöberl Programmbeschreibung NAOMI D und Algebraic Multigrid. September 997 From 998 to 8 technical reports were published by SFB3. Please see From 4 on reports were also published by RICAM. Please see For a complete list of NuMa reports see

3 ANALYSIS OF A NEW SPACE-TIME PARALLEL MULTIGRID ALGORITHM FOR PARABOLIC PROBLEMS MARTIN J. GANDER AND MARTIN NEUMÜLLER Abstract. We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi-coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties. Key words. Space-time parallel methods multigrid in space-time DG-discretizations strong and weak scalability parabolic problems AMS subject classifications. 65N55 65F 65L6. Introduction. About ten years ago clock speeds of processors have stopped increasing and the only way to obtain more performance is by using more processing cores. This has led to new generations of supercomputers with millions of computing cores and even today s small devices are multicore. In order to exploit these new architectures for high performance computing algorithms must be developed that can use these large numbers of cores efficiently. When solving evolution partial differential equations the time direction offers itself as a further direction for parallelization in addition to the spatial directions and the parareal algorithm [ ] has sparked renewed interest in the area of time parallelization a field that is now just over fifty years old see the historical overview [8]. We are interested here in space-time parallel methods which can be based on the two fundamental paradigms of domain decomposition or multigrid. Domain decomposition methods in spacetime lead to waveform relaxation type methods see [7 7 9] for classical Schwarz waveform relaxation [ 3 ] for optimal and optimized variants and [8 33 5] for Dirichlet-Neumann and Neumann-Neumann waveform relaxation. The spatial decompositions can be combined with parareal to obtain algorithms that run on arbitrary decompositions of the space-time domain into space-time subdomains see [3 4]. Space-time multigrid methods were developed in [ ] and reached good F-cycle convergence behavior when appropriate semi-coarsening and extension operators are used. For a variant for non-linear problems see [ ]. We present and analyze here a new space-time parallel multigrid algorithm that has excellent strong and weak scalability properties on large scale parallel computers. As a model problem we consider the heat equation in a bounded domain Ω R d d = 3 with boundary Γ := Ω on the bounded time interval [ T ].) t ux t) ux t) = fx t) for x t) Q := Ω T ) ux t) = for x t) Σ := Γ T ) ux ) = u x) for x t) Σ := Ω {}. Section de Mathématiques -4 rue du Lièvre CP 64 CH- Genève martin.gander@unige.ch) Inst. of Comp. Mathematics Altenbergerstr Linz Austria martin.neumueller@jku.at)

4 We divide the time interval [ T ] into subintervals = t < t <... < t N < t N = T with t n = n τ and τ = T N and use a standard finite element discretization in space and a discontinuous Galerkin approximation in time which leads to the large linear system in space-time.) [K τ M h + M τ K h ] u n+ = f n+ + N τ M h u n n =... N. Here M h is the standard mass matrix and K h is the standard stiffness matrix in space obtained by using the nodal basis functions {φ i } N x i= H Ω) i.e. M h [i j] := φ j x)φ i x)dx K h [i j] := φ j x) φ i x)dx i j =... N x. Ω The matrices for the time discretization where a discontinuous Galerkin approximation with polynomials of order p t N is used are given by K τ [k l] := M τ [k l] := tn tn Ω t n ψ n l t) t ψ n k t)dt + ψ n l t n )ψ n k t n ) k l =... N t t n ψ n l t)ψ n k t)dt N τ [k l] := ψ n l t n )ψ n k t n ). Here the basis functions for one time interval t n t n ) are given by P pt t n t n ) = span{ψl n}n t l= N t = p t + and for p t = we would for example get a Backward Euler scheme. The right hand side is given by tn f n+ [ln x + j] := fx t)φ j x)ψ l t)dxdt j =... N x l =... N t. t n Ω On the time interval t n t n+ ) we can therefore define the approximation u n+ h x t) = N t N x l= j= u n+ lj φ j x) ψ l t) with u n+ lj := u n+ [ln x + j] where u n+ is the solution of the linear system.). We thus have to solve the block triangular system.3) A τh B τh A τh B τh A τh u u. u N = with A τh := K τ M h + M τ K h and B τh := N τ M h. To solve the linear system.3) one can simply apply a forward substitution with respect to the blocks corresponding to the time steps. Hence one has to invert the matrix A τh for each time step where for example a multigrid solver can be applied. This is the usual way how time dependent problems are solved when implicit schemes are used [38 3] but this process is entirely sequential. We want to apply a parallelizable space-time multigrid scheme to solve the global linear system.3) at once. We present our method in Section and study its properties in Section 3 using local Fourier mode analysis. Numerical examples are given in Section 4 and the parallel implementation is discussed in Section 5 where we also show scalability studies. We give an outlook on further developments in Section 6. f f.. f N

5 . Multigrid method. We present now our new space-time multigrid method to solve the linear space-time system.3) which we rewrite in compact form as 3.) L τh u = f. For an introduction to multigrid methods see [ ]. We need a hierarchical sequence of space-time meshes T NL for L =... M L which has to be chosen in an appropriate way see Section 3.. For each space-time mesh T NL we compute the system matrix L τl h L for L =... M L. On the last finest) level M L we have to solve the original system.) i.e. L τml h ML = L τh. We denote by Sτ ν L h L the damped block Jacobi smoother with ν N steps.) u k+ = u k + ω t D τl h L ) [ f L τl h L u k]. Here D τl h L D τ L h L denotes an approximation of the inverse of the block diagonal matrix := diag{a τl h L } N L n= where a block A τ L h L := M hl K τl + K hl M τl corresponds to one time step. We will consider in particular approximating D τl h L ) by applying one multigrid V-cycle in space at each time step using a standard tensor product multigrid like in [3]. For the prolongation operator P L we use the standard interpolation from coarse space-time grids to the next finer space-time grids. The prolongation operator will thus depend on the space-time hierarchy chosen. The restriction operator is the adjoint of the prolongation operator R L = P L ). With ν ν N we denote the number of pre- and post smoothing steps and γ N defines the cycle index where typical choices are γ = V-cycle) and γ = W-cycle). On the coarsest level L = we solve the linear system which consists of only one time step exactly by using an LU-factorization for the system matrix L τh. For a given initial guess we apply this space-time multigrid cycle several times until we have reached a given relative error reduction ε MG. To study the convergence behavior of our space-time multigrid method we use local Fourier mode analysis. This type of analysis was used in [6] to study a twogrid cycle for an ODE model problem and we will need the following definitions and results whose proof can be found in [6]. Theorem. Discrete Fourier transform). For m N let u R m. Then u = m k= m with the coefficients û k φθ k ) φθ k )[l] := e ilθ k l =... m θ k := kπ m û k := m u φ θ k)) l = m m l= u[l]φ θ k )[l] for k = m... m. Definition. Fourier modes Fourier frequencies). Let N L N. Then the vector valued function φθ k )[l] := e ilθ k l =... N L is called Fourier mode with frequency { kπ θ k Θ L := : k = N L... N } L π π]. N L

6 4 The frequencies Θ L are further separated into low and high frequencies Θ low L := Θ L π π ] := Θ L π π ) ] π π] Θ high L = Θ L \ Θ low L. Definition.3 Fourier space). For N L N t N let the vector Φ L θ k ) C NtN L be defined as in Lemma.5 with frequency θ k Θ L. Then we define the linear space of Fourier modes with frequency θ k as Ψ L θ k ) := span { Φ L θ k ) } = { ψ L θ k ) C N tn L : ψ L n θ k ) = UΦ L nθ k ) n =... N L and U C N t N t }. Definition.4 Space of harmonics). For N L N t N and for a low frequency θ k Θ low L let the vector ΦL θ k ) C NtN L be defined as in Lemma.5. Then the linear space of harmonics with frequency θ k is given by E L θ k ) := span { Φ L θ k ) Φ L γθ k )) } = { ψ L θ k ) C NtN L : ψ L n θ k ) = U Φ L nθ k ) + U Φ L nγθ k )) n =... N L and U U C Nt Nt}. Lemma.5. The vector u = u u... u NL ) R N L N t N L = N L can be written as for N L N t N and u = N L k= N L + ψ L θ k U) = θ k Θ L ψ L θ k U) with the vectors ψn L θ k U) := UΦ L nθ k ) and Φ L nθ k )[l] := φθ k )[n] for n =... N L and l =... N t and the coefficient matrix U = diagû k []... û k [N t ]) C N t N t with the coefficients û k [l] := NL N L i= u i[l]φ θ k )[i] for k = N L... N L. Lemma.6. For λ C the eigenvalues of the matrix K τl + λm τl ) N τl C N t N t are given by σk τl + λm τl ) N τl ) = { Rλτ L )} where Rz) is the A-stability function of the given discontinuous Galerkin time stepping scheme. In particular the A-stability function Rz) is given by the p t p t + ) subdiagonal Padé approximation of the exponential function e z. Lemma.7. The mapping γ : Θ low L one to one mapping. Θhigh L with γθ k ) := θ k signθ k )π is a Lemma.8. Let θ k Θ low L. Then the restriction operator RL as defined in 3.5) has the mapping property R L : E L θ k ) Ψ L θ k ) with the mapping U U ) ˆRθk ) ˆRγθk )) ) ) U C N t N t U

7 5 and the Fourier symbol ˆRθ k ) := e iθ k R + R. Lemma.9. Let θ k Θ low L. Then the the prolongation operator PL as defined in 3.5) has the mapping property P L : Ψ L θ k ) E L θ k ) with the mapping U ) ˆPθk ) U C ˆPγθ N t N t k )) and the Fourier symbol ˆPθ k ) := [ e iθ k R + R ]. Lemma.. The frequency mapping is a one to one mapping. β : Θ low L Θ L with θ k θ k 3. Local Fourier mode analysis. For simplicity we assume that Ω = ) is a one-dimensional domain which is divided into uniform elements with mesh size h. The analysis for higher dimensions is more technical but the tools stay the same as for the one dimensional case. The standard one dimensional mass and stiffness matrices are M h = h K h = h Smoothing analysis. The iteration matrix of damped block Jacobi is S ν τ L h L = [ I ω t D τl h L ) L τl h L ] ν where D τl h L is a block diagonal matrix with blocks A τl h L. We first use the exact inverse of the diagonal matrix D τl h L in our analysis the V-cycle approximation is studied later see Remark 3.6. We denote by N Lt N the number of time steps and by N Lx N the degrees of freedom in space for level L N. Using Theorem. we can prove Lemma 3.. Let u = u u... u NLt ) R N tn L x N L t for Nt N Lx N Lt N where we assume that N Lx and N Lt are even numbers and assume that u n R NtN Lx and u nr R Nt for n =... N Lt and r =... N Lx. Then the vector u can be written as u = ψ LxLt θ x θ t ) θ t Θ Lt θ x Θ Lx

8 6 with the vectors ψnr LxLt θ x θ t ) := UΦ LxLt for n =... N Lt r =... N Lx nr θ x θ t ) Φ LxLt nr θ x θ t ) := Φ Lt n θ t )φ Lx θ x )[r] and with the coefficient matrix U := diag û xt []... û xt [N t ]) C N t N t with the coefficients for θ x Θ Lx and θ t Θ Lt û xt [l] := N L x N Lx N Lt N Lt u nr [l]φ θ x )[r]φ θ t )[n]. r= n= Proof. For u = u u... u NLt ) R N tn L x N L t we define for s =... NLx the vector w s R N L t N t as wn[l] s := u ns [l]. Applying Lemma.5 to the vector w s results in u is [l] = wi s [l] = ψ Lt θ t ) = U t [l l]φθ t )[i] θ t Θ Lt θ t Θ Lt with U t [l l] = ŵt s [l] = N Lt u ns [l]φ θ t )[n]. N L Next we define for a fixed n {... N Lt } and a fixed l {... N t } the vector z nl R N Lx as z nl [s] := u ns [l]. Applying Theorem. to the vector z nl we get for s =... N Lx n= u ns [l] = z nl [s] = ẑx nl φθ x )[s] with ẑx nl = θ x Θ Lx N Lx N Lx u nr [l]φ θ x )[r]. Combining the results above we obtain the statement of this lemma with u is [l] = θ x Θ Lx = θ x Θ L x = θ x Θ Lx = θ x Θ L x φθ x )[s]φθ t )[i] N Lx θ t Θ Lt û xt [l]φθ x )[s]φθ t )[i] θ t Θ Lt U[l l]φ LxLt is θ x θ t )[l] θ t Θ Lt θ x θ t )[l]. ψ LxLt is θ t Θ Lt N Lt N Lx N Lt r= n= r= u nr [l]φ θ x )[r]φ θ t )[n] Definition 3. Fourier space). For N t N Lx N Lt N and the frequency θ x Θ Lx and θ t Θ Lt let the vector Φ LxLt θ x θ t ) C NtN Lx N L t be as in Lemma 3.. Then we define the linear space of Fourier modes with frequencies θ x θ t ) as Ψ LxL t θ x θ t ) := span { Φ L xl t θ x θ t ) } = { ψ L xl t θ x θ t ) C N tn L x N L t : ψ nr θ x θ t ) := UΦ L xl t nr θ x θ t ) n =... N Lt r =... N Lx and U C N t N t }.

9 7 Lemma 3.3 Shifting equality). For N t N Lx N Lt N and the frequencies θ x Θ Lx θ t Θ Lt let ψ L xl t θ x θ t ) Ψ Lx L t θ x θ t ). Then we have the shifting equalities ψ L xl t n r θ x θ t ) = e iθt ψ LxLt nr θ x θ t ) ψ L xl t nr θ x θ t ) = e iθx ψ LxLt nr θ x θ t ) for n =... N Lt and r =... N Lx. Proof. The result follows from the fact that φθ)[n ] = e in )θ = e iθ e inθ = e iθ φθ)[n] which can be applied for the frequencies in space θ x Θ Lx and the frequencies in time θ t Θ Lt. Lemma 3.4 Fourier symbol of L τl h L ). For the frequencies θ x Θ Lx and θ t Θ Lt we consider the vector ψ LxLt θ x θ t ) Ψ LxL t θ x θ t ). Then for n =... N L t and r =... N Lx we have LτL h L ψ L xl t θ x θ t ) ) nr = ˆL τl h L θ x θ t )ψ L xl t nr θ x θ t ) where the Fourier symbol is given by ˆL τl h L θ x θ t ) := h L 3 + cosθ x)) [ K τl + h L βθ x)m τl e iθ t N τl ] C N t N t with the function βθ x ) := 6 cosθ x) +cosθ x) [ ]. Proof. Let ψ L xl t θ x θ t ) Ψ Lx L t θ x θ t ). Then we have for n =... N Lt using Lemma 3.3 and LτL h L ψ L xl t θ x θ t ) ) = B n τ L h L ψ LxLt n θ x θ t ) + A τl h L ψ L xl t n θ x θ t ) = e iθt B τl h L + A τl h L ) ψ L xl t n θ x θ t ). Hence we have to study the action of A τh and B τh on the local vector ψ L xl t n θ x θ t ). By using the definition of B τh we obtain for r =... N Lx and l =... N t using Lemma 3.3 BτL h L ψ L xl t n N t h L = 6 = h L 6 k= N t k= θ x θ t ) ) NLx r [l] = ψ LxLt N t s= k= nr θ x θ t )[k] + 4ψ L xl t nr M hl [r s]n τl [l k]ψ L xl t ns θ x θ t )[k] N τl [l k] e iθ x e iθ ) x ψ L xl t nr θ x θ t )[k] = h L 3 + cosθ N t x)) N τl [l k]ψ L xl t nr θ x θ t )[k] k= = h L 3 + cosθ x)) N τl ψnr LxLt θ x θ t ) ) [l]. ) θ x θ t )[k] + ψ LxLt nr+ θ x θ t )[k] N τl [l k]

10 8 Next we study the action of the matrix A τh on the local vector ψ L xl t n θ x θ t ): AτL h L ψ L xl t n θ x θ t ) ) NLx r [l] = k= N t s= k= N L x + N t s= k= = h L 3 + cosθ N t x)) K τl [l k]ψns LxLt θ x θ t )[k] N t + ψ L xl t h L k= nr θ x θ t )[k] + ψnr LxLt M hl [r s]k τl [l k]ψ L xl t ns θ x θ t )[k] K hl [r s]m τl [l k]ψns LxLt θ x θ t )[k] ) θ x θ t )[k] ψ L xl t nr+ θ x θ t )[k] M τl [l k] = h L 3 + cosθ x)) K τl ψnr LxLt θ x θ t ) ) [l] + N t cosθ x )) M τl [l k]ψ L xl t nr θ x θ t )[k] h L k= [ hl = 3 + cosθ x)) K τl + ] ) cosθ x ))M τl ψ L xl t nr θ x θ t ) [l] h L where we used Lemma 3.3. Hence we conclude the proof with LτL h L ψ L xl t θ x θ t ) ) = h L nr 3 + cosθ x)) K τl e iθ t ) N τl ψ nr θ x θ t ) + cosθ x ))M τl ψnr LxLt θ x θ t ) h L = h L 3 + cosθ x)) K τl + 6h L cosθ x ) + cosθ x ) M τ L e iθt N τl ) ψ LxLt nr θ x θ t ) = h L 3 + cosθ x)) K τl + h L βθ x)m τl e iθt N τl ) ψ L xl t nr θ x θ t ). If we assume periodic boundary conditions in space-time i.e. 3.) ut ) = ut ) for t T ) u x) = ut x) for x Ω = ) we obtain from Lemma 3.4 the mapping property 3.) L τl h L : Ψ Lx L t θ x θ t ) Ψ Lx L t θ x θ t ) U ˆL τl h L θ x θ t )U. Lemma 3.5 Mapping property of Sτ ν L h L ). For the frequencies θ x Θ Lx and θ t Θ Lt we consider the vector ψ L xl t θ x θ t ) Ψ Lx L t θ x θ t ). Then under the assumption of periodic boundary conditions 3.) we have for n =... N Lt and r =... N Lx S ν τl h L ψ L xl t θ x θ t ) ) ] ν [ŜτL = nr h L θ x θ t ) ψ nr θ x θ t )

11 9 where the Fourier symbol is given by Ŝ τl h L θ x θ t ) := ω t )I Nt + ω t e iθt K τl + h L βθ x)m τl ) NτL C Nt Nt with the function βθ x ) as defined in Lemma 3.4. Proof. Let ψ L xl t θ x θ t ) Ψ Lx L t θ x θ t ) then for a fixed n =... N Lt fixed r =... N Lx we have that and a S τl h L ψ L xl t θ x θ t ) ) = I nr NtN Lx N ω Lt td τl h L ) ) L τl h L ψ θ x θ t ) ) nr ) = I Nt ω t ÂτL h L θ x )) ˆLτL h L θ x θ t ) ψ L xl t nr θ x θ t ) with Further calculations give =: Ŝτ L h L θ x θ t )ψ LxLt nr θ x θ t ) Â τl h L θ x ) : = h L 3 + cosθ x)) K τl + h L cosθ x ))M τl = h L 3 + cosθ x)) [ K τl + h L βθ x)m τl ]. ) ) [KτL ÂτL h L θ x ) ˆLτL h L θ x θ t ) = ÂτL h L θ x ) + h L βθ x)m τl e iθ t ] N τl Hence we have = I Nt e iθ t [ K τl + h L βθ x)m τl ] NτL. Ŝ τl h L θ x θ t ) = I Nt ω t I Nt e iθ t K τl + h L βθ x)m τl ) NτL ) = ω t )I Nt + ω t e iθt K τl + h L βθ x)m τl ) NτL. By induction this completes the proof. In view of Lemma 3.5 the following mapping property holds when periodic boundary conditions are assumed: 3.3) S ν τ L h L : Ψ LxL t θ x θ t ) Ψ LxL t θ x θ t ) U Ŝτ L h L θ x θ t )) ν U. Next we will analyze the smoothing behavior for the high frequencies. To do so we consider two coarsening strategies: semi coarsening in time and full space-time coarsening. Definition 3.6 High and low frequency ranges). Let N Lt N Lx N. We define the set of frequencies Θ Lx L t := { kπ lπ ) : k = N L x N Lx N Lt... N L x and l = N L t... N L t and the sets of low and high frequencies with respect to semi coarsening in time Θ lows L xl t := Θ Lx L t π π] π π ] Θhighs L xl t := Θ Lx L t \ Θ lows L xl t } π π]

12 θ t θ t π π π Θ highs π Θ highf π Θ lows π θ x π π Θ lowf π π θ x Θ highs π π π π a) Semi coarsening. b) Full space-time coarsening. Fig. : Low and high frequencies θ x and θ t for semi coarsening and full space-time coarsening. and full space-time coarsening Θ lowf := Θ LxL t π π ] Θ highf := Θ LxL t \ Θ lowf. In Figure the high and low frequencies are illustrated for the two coarsening strategies. Definition 3.7 Asymptotic smoothing factors). Let Ŝτ L h L θ x θ t ) be the symbol of the block Jacobi smoother. Then the smoothing factor for semi-coarsening in time is { } µ s S := max ϱŝτ L h L θ x θ t )) : θ x θ t ) Θ highs and the smoothing factor for full space-time coarsening is { } µ f S := max ϱŝτ L h L θ x θ t )) : θ x θ t ) Θ highf. To study the smoothing behavior we need the eigenvalues of the Fourier symbol Ŝ τl h L θ x θ t ): Lemma 3.8. The spectral radius of the Fourier symbol Ŝ τl h L θ x θ t ) is given by with ) { } ρ ŜτL h L θ x θ t ) = max ω t Ŝω t αθ x µ) θ t ) ) Ŝωt α θ t ) := ωt ) + ω t ω t )α cosθ t ) + α ωt where αθ x µ) := R µβθ x )) and Rz) is the p t p t + ) subdiagonal Padé approximation of the exponential function e z and µ := τ L h L is a discretization parameter.

13 Proof. The eigenvalues of the Fourier symbol are given by Ŝ τl h L θ x θ t ) = ω t )I Nt + ω t e iθ t K τl + h L βθ ) x)m τl NτL σŝτ L h L θ x θ t )) = ω t + e iθ k ω t σk τl + h L βθ x)m τl ) N τl ). With Lemma.6 and using the definition of αθ x µ) we are now able to compute the spectrum as σŝτ L h L θ x θ t )) = { ω t ω t + e iθ k ω t αθ x µ) }. Hence we obtain the spectral radius ϱŝτ L h L θ x θ t )) = max { ω t ω t + e iθ k ω t αθ x µ) }. Direct calculations lead to ω t + e iθ k ω t αθ x µ) = ω t ) + ω t ω t )αθ x µ) cosθ k ) + αθ x µ)) ω t which completes the proof. Next we study the smoothing behavior of the damped block Jacobi iteration for the case when semi-coarsening in time is applied. Lemma 3.9. For the function ) Ŝωt α θ t ) := ωt ) + ω t ω t )α cosθ t ) + α ωt with α = αθ x µ) as defined in Lemma 3.8 and even polynomial degrees p t the minmax principle inf ω t ] sup θ t [ π π] θ x [π] Ŝω t αθ x µ) θ t ) = holds for any discretization parameter µ with the optimal parameters ω t = θ t = π and θ x =. Proof. Since we consider even polynomial degrees p t the p t p t + ) subdiagonal Padé approximation Rz) of the exponential function e z is positive for all z. Hence we also have that αθ x µ) = R µβθ x )) is positive for all µ and θ x [ π]. Since ω t ] we obtain θ t := argsup θ t [ π π] Ŝω t αθ x µ) θ t ) = π. Since α µ) = and αθ x µ) for all θ x [ π] and µ we get θx := argsup Ŝω t αθ x µ) θ ) =. θ x [π]

14 Hence we have to find the infimum of ) Ŝωt αθx µ) θt ) = ωt ) + ωt which is obtained for ωt =. This implies that ) Ŝω t αθx µ) θt ) = which completes the proof. Lemma 3. Asymptotic smoothing factor for semi-coarsening). For the function ) Ŝωt αθ x µ) θ t ) = ωt ) + ω t ω t )αθ x µ) cosθ t ) + αθ x µ)) ωt where α = αθ x µ) is defined as in Lemma 3.8 and the choice ωt = polynomial degree p t N we have the bound sup θ t [ π π] θ x [π] Ŝω t αθ x µ) θ t ). and any Proof. For even polynomial degrees p t we can apply Lemma 3.9 to get the bound stated. For odd polynomial degrees the p t p t + ) subdiagonal Padé approximation Rz) of the exponential function e z is negative for large negative values of z. If the value of αθ x µ) = R µβθ x)) for the optimal parameter θ x [ π] is positive we get directly the bound of Lemma 3.. Otherwise we obtain For a negative αθ x µ) this implies that sup θ t [ π π] θ x [π] θ t := argsup θ t [ π π] Ŝω t αθ x µ) θ t ) = π. Ŝω t αθ x µ) θ t ) + αθ x µ) ) ) < since any subdiagonal p t p t + ) Padé approximation Rz) is bounded from below by Rz) 5 3 3) for all z <. Lemma 3. shows that the asymptotic smoothing factor for semi-coarsening in time is bounded by µ s S. Hence by applying the damped block Jacobi smoother with the optimal damping parameter ωt frequencies Θ highs are asymptotically damped by a factor of at least. = the error components in the high Lemma 3. Asymptotic smoothing factor for full space-time coarsening). For the optimal choice of the damping parameter ωt = we have sup Ŝωt α θ t ) = + α ) θ t [π] with the optimal parameter θ t = { α π α >.

15 3 Proof. Let α R. For the optimal damping parameter ω t = we have ) Ŝωt α θ t ) = + α cosθt ) + α ). 4 First we study the case α where we get For the case α < we obtain This implies that θt := argsup Ŝωt α θ t ) =. θ t [π] θt := argsup Ŝωt α θ t ) = π. θ t [π] ) Ŝω t α θ t ) = + α + α ) = α ) which completes the proof. Lemma 3. shows that we obtain good smoothing behavior for the high frequencies with respect to the space discretization i.e. θ x Θ high L x if α = αθ x µ) is sufficiently small for any frequency θ x [ π π]. Hence combining Lemma 3. with Lemma 3. we see that good smoothing behavior can be obtained for all frequencies θ x θ t ) Θ highf if the function α = αθ x µ) is sufficiently small. This results in a restriction on the discretization parameter µ. With the next lemma we will analyze the behavior of the smoothing factor µ f S with respect to the discretization parameter µ for even polynomial degrees p t N. Lemma 3.. Let p t N be even. Then for the optimal choice of the damping parameter ωt = we have sup Ŝω t αθ x µ) θ t ) = + R 3µ)) θ t [π] θ x [ π π] where Rz) is the p t p t +) subdiagonal Padé approximation of the exponential function e z. Proof. In view of Lemma 3. it remains to compute the supremum sup θ x [ π π] + αθ x µ) ). Since for even polynomial degrees p t the function αθ x µ) = R µβθ x )) is monotonically decreasing in βθ x ) the supremum is obtained for βθ x ) = 3 since βθ x ) [3 ] for θ x [ π π]. This implies that θ x = π and we obtain the statement of the lemma with sup Ŝωt αθ x µ)) θ t ) = Ŝω t αθx µ) θ t ) = θ t [π] + αθ x µ) ) = + R 3µ)). θ x [ π π]

16 Ŝ) Ŝ) Ŝ) θ t 3 3 a) µ =. θ x θ t 3 3 θ x b) µ =. θ t 3 3 θ x c) µ =.. Fig. : Smoothing factor Ŝω t αθ x µ) θ t ) for θ x θ t [ π] for p t = and different discretization parameters µ. The proof of Lemma 3. only holds for even polynomial degrees but the result is also true for odd polynomial degrees p t only the proof gets more involved since the Padé approximation Rz) z is not monotonically decreasing for odd polynomial degrees. Remark 3.3. In view of Lemma 3. we obtain a good smoothing behavior for the high frequencies in space θ x Θ high L x i.e. µ f S if the discretization parameter µ is large enough i.e. 3.4) µ µ p t with R 3µ p t ) =. Hence we are able to compute the critical discretization parameter µ p t the polynomial degree p t with respect to µ = µ = ) µ µ µ To compute the critical discretization parameter µ we used the fact that the p t p t + ) subdiagonal Padé approximation Rz) converges to the exponential function e z for z as p t. Remark 3.4. Lemma 3. shows that for all frequencies θ x θ t ) Θ LxL t we have the bound Ŝω t αθ x µ)) θ t ) + R βθ x)µ) ). Only for θ x = we have that βθ x ) = which implies R βθ x )µ)) =. Hence if the discretization parameter µ = τ L h L is large enough we have that R βθ x )µ) for almost all frequencies θ x Θ Lx which implies a good smoothing behavior for almost all frequencies see Figures a c. Only the frequencies θ x Θ Lx which are close to zero imply Ŝω t αθ x µ)) θ t ). Hence for a large discretization parameter

17 µ the smoother itself is a good iterative solver for most frequencies only the frequencies θ x Θ Lx which are close to zero i.e. very few low frequencies θ x Θ low L x do not converge well. To obtain also a perfect solver for a large discretization parameter µ we can simply apply a correction step after one damped block Jacobi iteration by restricting the defect in space several times until we arrive at a very coarse problem. For this small problem one can solve the coarse correction exactly by solving these small problems forward in time. Afterward we correct the solution by prolongating the coarse corrections back to the fine space-grids. 3.. Two-grid analysis. The iteration matrices for the two-grid cycles with semi-coarsening and full space-time coarsening are M s τ L h L [ ] := S ν τ L h L I P L xl t s L τl h L ) R L xl t s L τl h L S ν τ L h L [ ] M f τ L h L := S ν τ L h L I P L xl t f L τl h L ) R L xl t f L τl h L S ν τ L h L with the restriction and prolongation matrices R L xl t s := I NL RL t R LxLt x f := R L x x R L t Ps LxLt := I NLx P Lt P L xl t f := Px Lx P Lt. The restriction and prolongation matrices in time i.e. R L t and P L t are given by see [6]) R R R L R R 3.5) := RNtN L N tn L P L := R L ) R R with the local prolongation matrices R := Mτ L M τl and R := Mτ L M τl where for { } basis functions {ψ k } N t k= Nt Pp t τ L ) and ψk k= Pp t τ L ) the local projection matrices from coarse to fine grids are defined for k l =... N t by M τ L [k l] := τl ψ l t)ψ k t)dt and M τl [k l] := τl τ L ψl t)ψ k t + τ)dt. The restriction and prolongation matrices in space for the one dimensional case are R L x x := R N Lx N Lx 3.6) 3.7) P Lx x := R Lx x ) R N Lx N Lx. To analyze the two-grid iteration matrices M s τ L h L and M f τ L h L we need Lemma 3.5. Let u = u u... u NLt ) R N tn L x N L t for Nt N Lx N Lt N where we assume that N Lx and N Lt are even numbers and assume that u n R N tn L x and u nr R N t 5

18 6 for n =... N Lt u = and r =... N Lx. Then the vector u can be written as θ x θ t ) Θ lowf LxL t [ ψ L xl t θ x θ t ) + ψ LxLt γθ x ) θ t ) + ψ L xl t θ x γθ t )) + ψ L xl t γθ x ) γθ t )) ] with the shifting operator γθ) := θ signθ)π and the vector ψ L xl t θ x θ t ) C N tn Lx N Lt as in Lemma 3.. Proof. Using Lemma 3. and Lemma.7 leads to the desired result with u = ψ LxLt θ x θ t ) θ x Θ Lx θ t Θ Lt = = θ x Θ low Lx + θ x Θ low Lx θ t Θ low θ xθ t) Θ lowf LxL t ψ LxLt θ x θ t ) + L t L ψ xl t θ x θ t ) + θ t Θ high L t θ x Θ high Lx θ x Θ high Lx θ t Θ low ψ LxLt θ x θ t ) L t θ t Θ high L t [ ψ θ x θ t ) + ψ L xl t γθ x ) θ t ) L ψ xl t θ x θ t ) + ψ L xl t θ x γθ t )) + ψ L xl t γθ x ) γθ t )) ]. Definition 3.6 Space of harmonics). For N t N Lx N Lt N and the frequencies θ x θ t ) Θ lowf let the vector Φ L xl t θ x θ t ) C N tn L x N L t be as in Lemma 3.. Then we define the linear space of harmonics with frequencies θ x θ t ) as E Lx L t θ x θ t ) := span { Φ L xl t θ x θ t ) Φ L xl t γθ x ) θ t ) Φ L xl t θ x γθ t )) Φ L xl t γθ x ) γθ t )) } = { ψ L xl t θ x θ t ) C N tn Lx N Lt : ψ L xl t nr θ x θ t ) = U Φ L xl t nr θ x θ t ) + U Φ L xl t nr γθ x ) θ t ) + U 3 Φ LxLt nr θ x γθ t )) + U 4 Φ LxLt nr γθ x ) γθ t )) for all n =... N Lt r =... N Lx and U U U 3 U 4 C N t N t }. With the assumption of periodic boundary conditions see 3.) Lemma 3.4 implies for the system matrix L τl h L for all frequencies θ x θ t ) Θ lowf the mapping property: 3.8) L τl h L : E LxL t θ x θ t ) E LxL t θ x θ t ) U ˆL τl h L θ x θ t )U U U U 3 ˆL τl h L γθ x ) θ t )U ˆL τl h L θ x γθ t ))U 3 =: L τl h L θ x θ t ) U U 3 U 4 ˆL τl h L γθ x ) γθ t ))U 4 U 4 where L τl h L θ x θ t ) C 4N t 4N t is a block diagonal matrix. With the same arguments we obtain with Lemma 3.5 for the smoother for all frequencies θ x θ t ) Θ lowf

19 the mapping property 3.9) Sτ ν L h L : E Lx L t θ x θ t ) E Lx L t θ x θ t ) U U U 3 U 4 Ŝτ L h L θ x θ t )) ν U Ŝτ L h L γθ x ) θ t )) ν U Ŝτ L h L θ x γθ t ))) ν U 3 Ŝτ L h L γθ x ) γθ t ))) ν U 4 =: SτL h L θ x θ t ) ) ν with the block diagonal matrix S τl h L θ x θ t ) C 4Nt 4Nt. To analyze the two-grid cycle on the space of harmonics E Lx L t θ x θ t ) for frequencies θ x θ t ) Θ lowf we further have to investigate the mapping properties of the restriction and prolongation operators for the two different coarsening strategies R L xl t s R LxLt f and P L xl t s P LxLt f. Lemma 3.7. Let R Lx x and Px Lx be the restriction and prolongation matrices as defined in 3.6). For θ x Θ low L x let φ L x θ x ) C N Lx and φ L x θ x ) C N Lx be defined as in Theorem.. Then R L x x φ L x θ x ) ) [r] = ˆR x θ x )φ Lx θ x )[r] for r =... N Lx with the Fourier symbol ˆRx θ x ) := + cosθ x ). For the prolongation operator we further have P L x x φ Lx θ x ) ) [s] = ˆPx θ x )φ L x θ x ) + ˆP ) x γθ x ))φ L x γθ x )) [s] for s =... N Lx with the Fourier symbol ˆPx θ x ) := ˆR x θ x ). Proof. The prove is classical see [39] for example. Definition 3.8. For N t N Lx N Lt N and the frequencies θ x θ t ) Θ lowf L xl t let the vector Φ LxLt θ x θ t ) C NtN Lx N L t be defined as in Lemma 3.. Then we define the linear space with frequencies θ x θ t ) as Ψ Lx L t θ x θ t ) : = span { Φ L xl t θ x θ t ) Φ L xl t γθ x ) θ t ) } = { ψ LxLt θ x θ t ) C NtN Lx N L t : ψ LxLt nr U U U 3 U 4 7 θ x θ t ) = U Φ LxLt nr θ x θ t ) + U Φ LxLt nr γθ x ) θ t ) for all n =... N Lt r =... N Lx and U U C N t N t }. For semi-coarsening the next lemma shows the mapping property for the restriction operator R L xl t s. Lemma 3.9. The restriction operator R L xl t s satisfies the mapping property R L xl t s : E Lx L t θ x θ t ) Ψ Lx L t θ x θ t ) with the mapping U U U U 3 R s θ t ) U U 3 U 4 U 4

20 8 and the matrix ) ˆRθt ) ˆRγθt )) R s θ t ) := ˆRθt ) ˆRγθt )) with the Fourier symbol ˆRθt ) C N t N t as defined in Lemma.8. Proof. Let Φ LxLt θ x θ t ) Ψ LxL t θ x θ t ) and Φ LxLt θ x θ t ) Ψ LxL t θ x θ t ) be defined as in Lemma 3.. Then for n =... N Lt and r =... N Lx we have using Lemma.8 R s Φ L xl t θ x θ t ) ) NLx = nr N Lt s= m= = φ Lx θ x )[r] I NL x [r s]rl t [n m]φ L xl t ms θ x θ t ) N Lt m= R Lt [n m]φ Lt m θ t ) = φ L x θ x )[r] R L t Φ L t θ t ) ) n = ˆRθ t )Φ Lt n θ t )φ Lx θ x )[r] = ˆRθ t )Φ LxLt nr θ x θ t ). Applying this result to the vector ψ LxLt θ x θ t ) E LxL t θ x θ t ) with θ x θ t ) Θ f L xl t results in R L xl t s ψ LxLt θ x θ t ) ) = ˆRθ nr t )U Φ LxLt nr θ x θ t ) Since Φ L xl t nr + ˆRθ t )U Φ L xl t nr γθ x ) θ t ) + ˆRγθ t ))U 3 Φ L xl t nr θ x γθ t )) + ˆRγθ t ))U 4 Φ L xl t nr γθ x ) γθ t )). θ x γθ t )) = Φ L xl t nr θ x θ t ) we further obtain [ = ˆRθt )U + ˆRγθ ] t ))U 3 Φ L xl t nr θ x θ t ) [ + ˆRθt )U + ˆRγθ ] t ))U 4 Φ L xl t nr γθ x ) θ t ) which completes the proof. Lemma 3.. With the assumptions of periodic boundary conditions 3.) the following mapping property for the restriction operator holds: with the mapping R L xl t f : E Lx L t θ x θ t ) Ψ Lx L t θ x θ t ) U U U U 3 R f θ x θ t ) U U 3 U 4 U 4 and the matrix R f θ x θ t ) := ˆRθx θ t ) ˆRγθx ) θ t ) ˆRθx γθ t )) ˆRγθx ) γθ t )) )

21 9 with the Fourier symbol ˆRθ x θ t ) := ˆR x θ x ) ˆRθ t ) C N t N t where ˆR x θ x ) C is defined as in Lemma 3.7. Proof. For the frequencies θ x θ t ) Θ low L xl t let Φ LxLt θ x θ t ) Ψ LxL t θ x θ t ) and Φ L x L t θ x θ t ) Ψ Lx L t θ x θ t ) be defined as in Lemma 3.. Then for n =... N Lt and r =... N Lx we have ) N R LxLt f Φ L xl t θ x θ t ) = L x nr N Lt s= m= N L x = s= R L x x R L x x [r s]r L t [n m]φ L xl t ms θ x θ t ) N Lt [r s]φ L x θ x )[r] R L t [n m]φ L t m θ t ) m= = R Lx x φ Lx θ x ) ) [r] R Lt Φ Lt θ t ) ) n. Applying Lemma 3.7 and Lemma.8 leads to = ˆR x θ x ) ˆRθ t )Φ Lt n θ t )φ Lx θ x )[r] = ˆRθ x θ t )Φ L x L t nr θ x θ t ). Using this result for the vector ψ L xl t θ x θ t ) E Lx L t θ x θ t ) with θ x θ t ) Θ f results in ) R L xl t f ψ LxLt θ x θ t ) = ˆRθ x θ t )U Φ Lx Lt nr θ x θ t ) nr With the relations + ˆRγθ x ) θ t )U Φ L x L t nr γθ x ) θ t ) + ˆRθ x γθ t ))U 3 Φ L x L t nr θ x γθ t )) + ˆRγθ x ) γθ t ))U 4 Φ L x L t nr γθ x ) γθ t )). Φ L x L t θ x θ t ) = Φ L x L t γθ x ) θ t ) = Φ L x L t nr θ x γθ t )) = Φ Lx Lt nr γθ x ) γθ t )) we obtain the statement of this lemma with ) R L xl t f ψ LxLt θ x θ t ) = [ ˆRθx θ t )U + ˆRγθ x ) θ t )U + ˆRθ x γθ t ))U 3 nr + ˆRγθ x ) γθ t ))U 4 ] Φ L x L t nr θ x θ t ). With the next lemmas we will analyze the mapping properties of the prolongation operators P L xl t s and P L xl t f. Lemma 3.. For θ x θ t ) Θ f the prolongation operator P L xl t s satisfies the mapping property P L xl t s : Ψ Lx L t θ x θ t ) E Lx L t θ x θ t )

22 with the mapping ) U U ˆPθ t ) ˆPθt ) ˆPγθ t )) ˆPγθt )) U U ) ) =: P U s θ t ) U and the Fourier symbol ˆPθt ) C Nt Nt defined as in Lemma.9. Proof. Let ψ L xl t θ x θ t ) Ψ Lx L t θ x θ t ) for θ x θ t ) Θ f. Then we have for n =... N Lt and r =... N Lx that P L xl t s ψ LxLt θ x θ t ) ) NLx = nr = = N Lt s= m= N Lt m= N Lt m= I NLx [r s]p Lt [n m]ψ LxLt ms θ x θ t ) P L t [n m]ψ L xl t mr θ x θ t ) [ P Lt [n m] φ Lx θ x )[r]u )Φ Lt m θ t ) ] + φ L x γθ x ))[r]u )Φ L t m θ t ). Since φ Lx θ x )[r]u C Nt Nt and φ Lx γθ x ))[r]u C Nt Nt we further obtain by applying Lemma.9 that = ˆPθ t )φ Lx θ x )[r]u )Φ Lt n θ t ) + ˆPγθ t ))φ Lx θ x )[r]u )Φ Lt n γθ t )) + ˆPθ t )φ Lx γθ x ))[r]u )Φ Lt n θ t ) + ˆPγθ t ))φ L x γθ x ))[r]u )Φ L t n γθ t )). With the definition of the Fourier mode Φ L xl t nr θ x θ t ) we get = ˆPθ t )U Φ LxLt nr θ x θ t ) + ˆPγθ t ))U Φ LxLt nr θ x γθ t )) + ˆPθ t )U Φ LxLt nr γθ x ) θ t ) + ˆPγθ t ))U Φ LxLt nr γθ x ) γθ t )) which completes the proof. Lemma 3.. With the assumptions of periodic boundary conditions 3.) the following mapping property for the prolongation operator holds: with the mapping P L xl t f : Ψ Lx L t θ x θ t ) E LxL t θ x θ t ) U ˆPθ x θ t ) ˆPγθ x ) θ t ) ˆPθ x γθ t )) ˆPγθ x ) γθ t )) U =: P f θ x θ t )U

23 and the Fourier symbol ˆPθ x θ t ) := ˆP x θ x ) ˆPθ t ) where ˆPθ t ) is defined as in Lemma.9. Proof. Let ψ L x L t θ x θ t ) Ψ Lx L t θ x θ t ) for θ x θ t ) Θ f. Then we have for n =... N Lt and r =... N Lx that ) N L x P LxLt f ψ L x L t θ x θ t ) = nr s= N L x = s= P L x x Using Lemma.9 gives N Lt m= P L x x [r s]p L t [n m]ψ L x L t ms θ x θ t ) N Lt [r s]φ Lx θ x )[s] P L t [n m]φ L t m θ t ). m= = ˆPx θ x )φ L x θ x )[r] + ˆP ) x γθ x ))φ L x γθ x ))[r] ˆPθt )UΦ Lt n θ t ) + ˆPγθ t ))UΦ Lt n γθ t )) Using now the definition of the Fourier mode Φ L xl t nr θ x θ t ) leads to ). = ˆPθ x θ t )UΦ L xl t nr θ x θ t ) + ˆPγθ x ) θ t )UΦ L xl t nr γθ x ) θ t ) + ˆPθ x γθ t ))UΦ LxLt nr θ x γθ t )) + ˆPγθ x ) γθ t ))UΦ LxLt nr γθ x ) γθ t )) which completes the proof. For periodic boundary conditions 3.) we further obtain with Lemma. the mapping property for the coarse grid correction when semi coarsening in time is applied 3.) L τl h L ) : Ψ LxL t θ x θ t ) Ψ LxL t θ x θ t ) ) U ) ) Ls τl h θ U L x θ t ) C N t N t U U with the matrix ) ) Ls θ ˆLτL h L θ x θ t ) τlhl x θ t ) := C ˆLτL h L γθ x ) θ t )) Nt Nt. For full space-time coarsening we have the mapping property 3.) L τl h L ) : Ψ Lx L t θ x θ t ) Ψ Lx L t θ x θ t ) U Lf τl h L θ x θ t )) U C N t N t ) ) with Lf θ τlhl x θ t ) := ˆLτL h L θ x θ t ) C N t N t. We can now prove the following two theorems:

24 Theorem 3.3. Let θ x θ t ) Θ lowf. With the assumption of periodic boundary conditions 3.) the following mapping property holds for the two-grid operator M s τ L h L with semi coarsening in time: with the mapping and the iteration matrix with M s τ L h L : E LxL t θ x θ t ) E LxL t θ x θ t ) U U U U 3 M s µθ k θ t ) U U 3 U 4 U 4 M s µθ k θ t ) := SτL h L θ x θ t )) ν Ks θ x θ t ) SτL h L θ x θ t )) ν C 4N t 4N t K s θ x θ t ) := I 4Nt P s θ t ) Ls τl h L θ x θ t )) Rs θ t ) L τl h L θ x θ t ). Proof. The statement of this theorem follows by using Lemma 3.9 Lemma 3. and the mapping properties 3.8) 3.9) and 3.). Theorem 3.4. Let θ x θ t ) Θ lowf. With the assumption of periodic boundary conditions 3.) the following mapping property holds for the two-grid operator M f τ L h L with full space-time coarsening: with the mapping and the iteration matrix with M f τ L h L : E Lx L t θ x θ t ) E Lx L t θ x θ t ) U U U U 3 M f µθ k θ t ) U U 3 U 4 U 4 M f µθ k θ t ) := SτL h L θ x θ t )) ν Kf θ x θ t ) SτL h L θ x θ t )) ν C 4N t 4N t K f θ x θ t ) := I 4Nt P f θ x θ t ) Lf τl h L θ x θ t )) Rf θ x θ t ) L τl h L θ x θ t ). Proof. The statement of this theorem is a direct consequence of Lemma 3. Lemma 3. and the mapping properties 3.8) 3.9) and 3.). In view of Lemma 3.5 we can now represent the initial error e = u u as e [ = ψ θ x θ t ) + ψ L xl t γθ x ) θ t ) =: θ x θ t ) Θ lowf LxL t θ x θ t ) Θ lowf LxL t ψθx θ t ) + ψ LxLt θ x γθ t )) + ψ LxLt γθ x ) γθ t )) ]

25 with ψθ x θ t ) E Lx L t θ x θ t ) for all θ x θ t ) Θ lowf. Using Theorem 3.3 and Theorem 3.4 we now can analyze the asymptotic convergence behavior of the twogrid cycle by simply computing the largest spectral radius of M s µθ k θ t ) or M f µθ k θ t ) with respect to the frequencies θ x θ t ) Θ lowf. This motivates Definition 3.5 Asymptotic two-grid convergence factors). For the two-grid iteration matrices M s τ L h L and M f τ L h L we define the asymptotic convergence factors ϱ ˆM { s µ) := max ϱ M } s µθ k θ t )) : θ x θ t ) Θ lowf with θ x ϱ ˆM { f µ) := max ϱ M } f µθ k θ t )) : θ x θ t ) Θ lowf L xl t with θ x. 3 Note that in the definition of the two-grid convergence factors we have neglected all frequencies θ t ) Θ lowf since the Fourier symbol with respect to the Laplacian is zero for θ x = see also the remarks in [39 chapter 4]. To derive the assymptotic convergence factors ϱ ˆM s µ) and ϱ ˆM f µ) for a given discretization parameter µ R + and a given polynomial degree p t N we have to compute the eigenvalues of 3.) Ms µ θ k θ t ) C 4Nt 4Nt and Mf µ θ k θ t ) C 4Nt 4Nt with N t = p t + for each low frequency θ x θ t ) Θ lowf. Since it is difficult to find closed form expressions for the eigenvalues of the iteration matrices 3.) we will compute the eigenvalues numerically. In particular we will compute the average convergence factors for the domain Ω = ) with a decomposition into 4 uniform sub intervals i.e. N Lx = 3. Furthermore we will analyze the two-grid cycles for N Lt = 56 time steps. We plot as solid lines in the Figures 3a 3b the theoretical convergence factors ϱ ˆM s µ) as functions of the discretization parameter µ = τ L h L [ 6 6 ] for different polynomial degrees p t { } and different number of smoothing steps ν = ν = ν { 5}. We observe that the theoretical convergence factors are always bounded by ϱ ˆM s µ). For the case when semi coarsening in time is applied we see that the two-grid cycle converges for any discretization parameter µ. We also see for polynomial degree p t = that the theoretical convergence factors are much smaller than the theoretical convergence factors for the lowest order case p t =. We also plot in Figures 3a 3b using dots triangles and squares the numerically measured convergence factors for solving the equation L τl h L u = f with the two-grid cycle when semi coarsening in time is applied. For the numerical test we use a zero right hand side i.e. f = and a random initial vector u with values between zero and one. The convergence factor of the two-grid cycle is measured by max k=...n iter r k+ r k with r k := f L τl h L u k where N iter N N iter 5 is the number of two-grid iterations used until we have reached a given relative error reduction of ε MG = 4. We observe that the numerical results agree very well with the theoretical results even though the local

26 4 convergence factor ˆM s µ ) ν = analysis ν = analysis ν = 5 analysis ν = experiment ν = experiment ν = 5 experiment convergence factor ˆM s µ ) ν = analysis ν = analysis ν = 5 analysis ν = experiment ν = experiment ν = 5 experiment discretization parameter µ a) p t = discretization parameter µ b) p t =. Fig. 3: Asymptotic convergence factor ϱ ˆM s µ) for different discretization parameters µ and numerical convergence factors for N t = 56 time steps and N x = 3. convergence factor ˆM f µ ) µ = 3 ν = analysis ν = analysis ν = 5 analysis ν = experiment ν = experiment ν = 5 experiment convergence factor ˆM f µ ) µ.95 ν = analysis ν = analysis ν = 5 analysis ν = experiment ν = experiment ν = 5 experiment discretization parameter µ a) p t = discretization parameter µ b) p t =. Fig. 4: Average convergence factor ϱ ˆM f µ) for different discretization parameters µ and numerical convergence factors for N t = 56 time steps and N x = 3. Fourier mode analysis is not rigorous for the numerical simulation that does not use periodic boundary conditions. In Figures 4a 4b we plot the theoretical convergence factors ϱ ˆM s µ) for the twogrid cycle M f τ L h L with full space-time coarsening as function of the discretization parameter µ [ 6 6 ] for different polynomial degrees p t { }. We observe that the theoretical convergence factors are bounded by ϱ ˆM f µ) if the discretization parameter µ is large enough i.e. for µ µ. In Remark 3.3 we already computed these critical values µ for several polynomial degrees p t. As before we compared the theoretical results with the numerical results when full space-time coarsening is applied. In Figures 4a 4b the measured numerical convergence factors are plotted as dots triangles and squares. We see that the theoretical results agree very well with the numerical results. Overall we conclude that the two-grid cycle always converges to the exact solution of the linear system.3) when semi coarsening in time is applied. Furthermore if the discretization parameter µ is large enough we can also apply full space-time coarsening which leads to a smaller coarse problem compared to the semi coarsening case. Remark 3.6. For the two-grid analysis above we used for the block Jacobi

27 5 smoother 3.3) S ν τ L h L = [ I ω t D τl h L ) L τl h L ] ν the exact inverse of the diagonal matrix D τl h L = diag {A τl h L } N L t n=. In practice it is more efficient to use an approximation D τ L h L by applying one multigrid iteration in space for the blocks A τl h L see also [34] where such an approximate block Jacobi method is used directly to precondition GMRES. Hence the smoother 3.3) changes to 3.4) S ν τ L h L := [ I ω t I M τl h L ) D τl h L ) L τl h L ] ν { } with the matrix M τl h L := diag M x NLt τ L h L where M x τ n= L h L is the iteration matrix of the multigrid scheme for the matrix A τl h L. In the case that the iteration matrix M x τ L h L is given by a two-grid cycle we further obtain the representation [ ] M x τ L h L = S xνx τ L h L I P L x x A τ L h L R L x x A τl h L S xνx τ L h L with a damped Jacobi smoother in space ) ] S xνx τ L h L := [I ω x D x ν x τl hl AτL h L Dτ x L h L := diag { h 3 K τ L + h M } NLx τ L r= and the restriction and prolongation operators R L x x := R Lx x I Nt and P L x x := Px Lx I Nt. With the different smoother 3.4) we also have to analyze the two different two-grid iteration matrices 3.5) 3.6) [ ] M s τ L h L := S ν τ L h L I Ps LxLt L τl h L ) R LxLt s L τl h L S ν τ L h L [ ] M f τ L h L := S ν τ L h L I P L xl t f L τl h L ) R L xl t f L τl h L S ν τ L h L. Hence it remains to analyze the mapping property of the operator M τl h L on the space of harmonics E LxL t θ x θ t ). By several computations we find under the assumptions of periodic boundary conditions 3.) that M τl h L : E Lx L t θ x θ t ) E Lx L t θ x θ t ) with the mapping 3.7) U U U 3 U 4 MτL h L θ x ) MτL h L θ x ) ) U U U 3 U 4 and the iteration matrix M τl h L θ x ) := S xνx τ L h L θ x )K x τ L h L θ x ) S xνx τ L h L θ x ) C Nt Nt K x τ L h L θ x θ t ) := I Nt P x θ x )Ã τ L h L θ x ) R x θ x )Ãτ L h L θ x ) C Nt Nt

28 6 with the matrices à à τl h L θ x ) := ) ÂτL h L θ x ) C N t N t  τl h L γθ x )) τ L h L θ x ) := ÂτL h L θ x )) C N t N t ν x ŜτL S xνx h L ω x θ x )) τ L h L θ x ) := ν x C ŜτL h L ω x γθ x ))) N t N t R x θ x ) := ) ˆRx θ x )I Nt ˆRx γθ x ))I Nt C N t N t ) ˆPx θ P x θ x ) := x )I Nt C ˆP N t N t x γθ x ))I Nt and the Fourier symbols  τl h L θ x ) := h L 3 + cosθ x)) K τl + cosθ x )) M τl C Nt Nt h L Ŝ τl h L ω x θ x ) := I Nt ω x hl 3 K τ L + h L M τl ) ÂτLhL θ x) C Nt Nt. Hence we can analyze the modified two-grid iteration matrices 3.5) by taking the additional approximation with the mapping 3.7) into account. For the smoothing steps ν x = ν x = and the damping parameter ω x = 3 for the spatial multigrid component the theoretical convergence factors with semi coarsening in time are plotted in Figures 5a 5b for the discretization parameter µ [ 6 6 ] with respect to the polynomial degrees p t { }. We observe that the theoretical convergence factors are always bounded by ϱm s µ). We also notice that the theoretical convergence factors are a little bit larger for small discretization parameters µ compared to the case when the exact inverse of the diagonal matrix D τl h L is used. The numerical factors are plotted as dots triangles and squares in Figures 5a 5b. We observe that the theoretical convergence factors coincide with the numerical results. In Figures 6a 6b the convergence of the two-grid cycle for the full space-time coarsening case is studied. Here we see that the computed convergence factors are very close to the results which we obtained for the case when the exact inverse of the diagonal matrix D τl h L is used. 4. Numerical examples. We present now numerical results for the multigrid version of our algorithm for which we analyzed the two-grid cycle in Section 3.. Following our two-grid analysis we also apply full space-time coarsening only if µ L µ in the multigrid version. If µ L < µ we only apply semi coarsening in time. In that case we will have for the next coarser level µ L = τ L h L = µ L. This implies that the discretization parameter µ L gets larger when semi coarsening in time is used. Hence if µ L k µ for k < L we can apply full space-time coarsening to reduce the computational costs. If full space-time coarsening is applied we have µ L = τ L h L ) = µ L which results in a smaller discretization parameter µ L. We therefore will combine semi coarsening in time or full space-time coarsening in the right way to get to the next coarser space-time level. For different discretization parameters µ = cµ c { } this coarsening strategy is shown in Figure 7 for 8 time and 4 space levels. The restriction and prolongation operators for the space-time multigrid scheme are then defined by the given coarsening strategy.