BDDC for a Space-time Finite Element Discretization of Parabolic Problems

Ulrich Langer and Huidong Yang

1 Introduction

Continuous space-time finite element methods for parabolic problems have been recently studied, e.g., in [1, 9, 10, 13]. The main common features of these methods are very different from those of time-stepping methods. Time is considered to be just another spatial coordinate. The variational formulations are studied in the full space- time cylinder that is then decomposed into arbitrary admissible simplex elements. In this work, we follow the space-time finite element discretization scheme proposed in [10] for a model initial-boundary value problem, using continuous and piecewise linear finite elements in space and time simultaneously. It is a challenging task to efficiently solve the large-scale linear system of alge- braic equations arising from the space-time finite element discretization of parabolic problems. In this work, as a preliminary study, we use the balancing domain de- composition by constraints (BDDC [2, 11, 12]) preconditioned GMRES method to solve this system efficiently. We mention that robust preconditioning for space-time isogeometric analysis schemes for parabolic evolution problems has been reported in [3, 4]. The remainder of the paper is organized as follows: Sect. 2 deals with the space- time finite element discretization for a parabolic model problem. In Sect. 3, we discuss BDDC preconditioners that are used to solve the linear system of algebraic equations. Numerical results are shown and discussed in Sect. 4. Finally, some conclusions are drawn in Sect. 5.

Ulrich Langer Johann Radon Institute, Altenberg Strasse 69, 4040 Linz, Austria, e-mail: ulrich.langer@ricam. oeaw.ac.at Huidong Yang Johann Radon Institute, Altenberg Strasse 69, 4040 Linz, Austria, e-mail: huidong.yang@oeaw. ac.at

367 368 Ulrich Langer and Huidong Yang 2 The space-time finite element discretization

The following parabolic initial-boundary value problem is considered as our model problem: Find D : & R such that →

mC D ΔGD = 5 in &,D = 0 on Σ,D = D on Σ , (1) − 0 0 where & := Ω 0,) , Ω R2 is a sufficiently smooth and bounded spatial × ( ) ⊂ computational domain, Σ := mΩ 0,) , Σ := Ω 0 , Σ) := Ω ) . × ( ) 0 × { } × { } Let us now introduce the following Sobolev spaces:

1,0 2  & = D ! & : GD ! & ,D = 0 on Σ , 0 ( ) { ∈ 2 ( ) ∇ ∈ [ 2 ( )] } 1,1 2  & = D !2 & : GD !2 & , mC D !2 & and D Σ Σ) = 0 , 0,0¯ ( ) { ∈ ( ) ∇ ∈ [ ( )] ∈ ( ) | ∪ } 1,1 2  & = D !2 & : GD !2 & , mC D !2 & and D Σ Σ = 0 . 0,0 ( ) { ∈ ( ) ∇ ∈ [ ( )] ∈ ( ) | ∪ 0 } Using the classical approach [7, 8], the variational formulation for the parabolic model problem (1) reads as follows: Find D 1,0 & such that ∈ 0 ( ) 0 D,E = ; E , E 1,1 & , (2) ( ) ( ) ∀ ∈ 0,0¯ ( ) where ¹ ¹ 0 D,E = D G,C mC E G,C 3 G,C GD G,C GD G,C 3 G,C , ( ) − & ( ) ( ) ( ) + & ∇ ( ) · ∇ ( ) ( ) ¹ ¹ ; E = 5 G,C E G,C 3 G,C D0 G E G, 0 3G. ( ) & ( ) ( ) ( ) + Ω ( ) ( )

Remark 1 (Parabolic solvability and regularity [7, 8]) If 5 !2,1 & := E : ∫ ) ∈ ( ) { E ,C ! Ω 3C < and D0 !2 Ω , then there exists a unique general- 0 k (· )k 2 ( ) ∞} ∈ ( ) ized solution D 1,0 & +1,0 & of (2), where +1,0 & := D 1,0 & : ∈ 0 ( ) ∩ 2 ( ) 2 ( ) { ∈ ( ) D & < and lim D ,C ΔC D ,C ! Ω = 0, uniformly on 0,) , and | | ∞ ΔC 0 k (· + ) − (· )k 2 ( ) [ ]} → 1 D & := max D , g ! Ω GD ! Ω 0,) . If 5 !2 & and D0  Ω , then | | 0 g ) k (· )k 2 ( ) +k∇ k 2 ( ×( )) ∈ ( ) ∈ 0 ( ) ≤ ≤ Δ,1 1,1 the generalized solution D belongs to  & := E  & : ΔGD ! & 0 ( ) { ∈ 0 ( ) ∈ 2 ( )} and continuously depends on C in the norm of the space 1 Ω . 0 ( ) To derive the space-time finite element scheme, we mainly follow the approach proposed in [10]. Let +ℎ = span i8 be the span of continuous and piecewise linear { } basis functions i8 on shape regular finite elements of an admissible triangulation ℎ. 1,1 T Then we define +0ℎ = +ℎ  & = Eℎ +ℎ : Eℎ Σ Σ = 0 . For convenience, ∩ 0,0 ( ) { ∈ | ∪ 0 } we consider homogeneous initial conditions, i.e., D0 = 0 on Ω. Multiplying the PDE mC D ΔGD = 5 on ℎ by an element-wise time-upwind test function − ∈ T Eℎ \ ℎ mC Eℎ, Eℎ + ℎ, we get + ∈ 0 BDDC for STFEM of Parabolic Problems 369 ¹ mC DEℎ \ ℎ mC DmC Eℎ ΔGD Eℎ \ ℎ mC Eℎ 3 G,C = ( + − ( + )) ( ) ¹ 5 Eℎ \ ℎ mEℎ 3 G,C , ( + ) ( ) where ℎ refers to the diameter of an element in the space-time triangulation ℎ T of &. Further, \ denotes a stabilization parameter [10]; see Remark 3. In the space- time finite element scheme [10], the time is considered as another spatial coordinate, and the partial derivative w.r.t. time is viewed as a convection term in the time direction. Therefore, as in the classical SUPG (streamline upwind Petrov-Galerkin) scheme, we use time-upwind test functions elementwise. Integration by parts (the first part) with respect to the space and summation yields Õ ¹ mC DEℎ \ ℎ mC DmC Eℎ GD GEℎ \ ℎ ΔGDmC Eℎ 3 G,C ( + + ∇ · ∇ − ) ( ) ℎ ∈T Õ ¹ Õ ¹ =G GDEℎ 3B = 5 Eℎ \ ℎ mC Eℎ 3 G,C . − m · ∇ ( + ) ( ) ℎ ℎ ∈T ∈T

Since =G GD is continuous across the inner boundary m of , =G = 0 on Σ0 Σ) , E =· ∇ Σ Í ∫ = DE 3B ∪ and ℎ 0 on , the term ℎ m G G ℎ vanishes. − ∈T Δ·, ∇1 1,1 If the solution D of (2) belongs to  ℎ := E  & : ΔGE 0,0 (T ) { ∈ 0,0 ( ) | ∈ ! , ℎ , cf. Remark 1, then the consistency identity 2 ( ) ∀ ∈ T }

0ℎ D,Eℎ = ;ℎ Eℎ ,Eℎ + ℎ, (3) ( ) ( ) ∈ 0 holds, where Õ ¹ 0ℎ D,Eℎ := mC DEℎ \ ℎ mC DmC Eℎ GD GEℎ \ ℎ ΔGDmC Eℎ 3 G,C , ( ) ( + + ∇ · ∇ − ) ( ) ℎ ∈T Õ ¹ ;ℎ Eℎ := 5 Eℎ \ ℎ mC Eℎ 3 G,C . ( ) ( + ) ( ) ℎ ∈T

With the restriction of the solution to the finite-dimensional subspace +0ℎ, the space- time finite element scheme reads as follows: Find Dℎ + ℎ such that ∈ 0

0ℎ Dℎ,Eℎ = ;ℎ Eℎ ,Eℎ + ℎ. (4) ( ) ( ) ∈ 0

Thus, we have the Galerkin orthogonality: 0ℎ D Dℎ,Eℎ = 0, Eℎ + ℎ. ( − ) ∀ ∈ 0 Remark 2 Since we use continuous and piecewise linear trial functions, the integrand \ ℎ ΔGDℎ mC Eℎ vanishes element-wise, which simplifies the implementation. − Remark 3 On fully unstructured meshes, \: = $ ℎ: [10]; on uniform meshes, ( ) \: = \ = $ 1 [9]. In this work, we have used \ = 0.5 and \ = 2.5 on uniform ( ) meshes for testing robustness of the BDDC preconditioners. The detailed results for \ = 2.5 are presented in Table 1. 370 Ulrich Langer and Huidong Yang

It was shown in [10] that the bilinear form 0ℎ , is +0ℎ-coercive: 0ℎ Eℎ,Eℎ ` E 2 E + (· E·) 2 = Í E( 2 ) ≥ 2 ℎ ℎ, ℎ 0ℎ with respect to the norm ℎ ℎ ℎ G ℎ ! k k ∀ ∈ k k ∈T (k∇ k 2 + 2 1 2 ( ) \ ℎ mC Eℎ Eℎ . Furthermore, the bilinear form is bounded on ! 2 ! Σ) k k 2 ( ) ) + k k 2 ( ) +0ℎ, +0ℎ: 0ℎ D,Eℎ `1 D 0ℎ, Eℎ ℎ, D +0ℎ, , Eℎ +0ℎ, where +0ℎ, = ∗ × | ( )| ≤ k k ∗ k k ∀ ∈ ∗ ∀ ∈ ∗ Δ,1 + E 2 = E 2 Í \ ℎ 1 E 2 , ℎ 0ℎ equipped with the norm ℎ, ℎ ℎ − ! 0 0 (T ) + k k0 k k + ∈T ( ) k k 2 Í 2 ∗ ( ) \: ℎ: ΔGE . Let ; and : be positive reals such that ; : > 3 2. We ℎ !2 + ∈T k k ( ) B ≥ B / now define the broken Sobolev space  ℎ := E !2 & : E  (T ) { ∈E 2 ( ) =| Í∈ ( E)2 ∀ ∈ ℎ equipped with the broken Sobolev semi-norm  B : ℎ  B . T } | | ℎ ∈T | | (T1,)1 : ( ) Using the Lagrangian interpolation operator Πℎ mapping  &  & to + ℎ, 0,0 ( ) ∩ ( ) 0 we obtain D Dℎ ℎ D ΠℎD ℎ ΠℎD Dℎ ℎ. The term D ΠℎD ℎ can be k − k ≤ k − k + k − k k − k bounded by means of the interpolation error estimate, and the term ΠℎD Dℎ ℎ k − k by using ellipticity, Galerkin orthogonality and boundedness of the bilinear form. 2 ; 1 Í ( − ) 2 1 2 The discretization error estimate D Dℎ ℎ  ℎ ℎ D ; / holds k − k ≤ ( ∈T | | ) 1,1 : ; ( ) for the solution D provided that D belongs to  &  &  ℎ , and the 0,0 ( ) ∩ ( ) ∩ (T ) finite element solution Dℎ + ℎ, where  > 0, independent of mesh size; see [10]. ∈ 0

3 Two-level BDDC preconditioners

After the space-time finite element discretization of the model problem (1), the linear system of algebraic equations reads as follows:

G = 5 , (5)          Γ G 5  1 #  with := , G := , 5 := ,   = diag   , ...,   , where # Γ ΓΓ GΓ 5Γ denotes the number of polyhedral subdomains &8 from a non-overlapping domain decomposition of &. In system (5), we have decomposed the degrees of freedom into the ones associated with the internal () and interface (Γ) nodes, respectively. We aim to solve the Schur-complement system living on the interface:

(GΓ = 6Γ, (6)

1 1 with ( := ΓΓ Γ  Γ and 6 := 5Γ Γ 5 . − −  − −  The bilinear form 0ℎ , is coercive on the space-time finite element space + ℎ (· ·) 0 like in the corresponding elliptic case. There are efficient domain decomposition preconditioners for such elliptic problems [14]. This motivated us to use such pre- conditioners for solving positive definite space-time finite element equations too. Following [12] (see also details in [5]), Dohrmann’s (two-level) BDDC precondi- tioners % for the interface Schur complement equation (6), originally proposed for symmetric and positive definite systems in [2, 11], can be written in the form

1 ) %− = ' )BD1 ) ',Γ, (7)  ,Γ ( + 0) BDDC for STFEM of Parabolic Problems 371

8 where the scaled operator ',Γ is the direct sum of restriction operators ',Γ mapping the global interface vector to its component on local interface Γ8 := m&8 Γ, ∩ with a proper scaling factor. Here the coarse level correction operator )0 is constructed as

) 1 ) ) = Φ Φ (Φ − Φ (8) 0 ( ) ) with the coarse level basis function matrix Φ =  Φ1 ) , , Φ# )  , where the ( ) ··· ( ) basis function matrix Φ8 on each subdomain interface is obtained by solving the following augmented system:

(8 8 )  Φ8   0  8 8 = 8 . (9)  0 Λ 'ΠGΓ

8 with the given primal constraints  of the subdomain &8 and the vector of Lagrange multipliers on each column of Λ8. The number of columns of each Φ8 equals to the number of global coarse level degrees of freedom, typically living on the subdomain 8 corners, and/or interface edges, and/or faces. Here the restriction operator 'Π maps the global interface vector in the continuous primal variable space on the coarse level to its component on Γ8. The subdomain correction operator )BD1 is defined as

# 1  8 8 ) −  8  Õ  8 )  (  'Γ )BD1 = ' 0 , (10) ( Γ) 8 0 0 8=1 with vanishing primal variables on all the coarse levels. Here the restriction operator 8 'Γ maps global interface vectors to their components on Γ8.

4 Numerical experiments

We use D G,H,C = sin cG sin cH sin cC as exact solution of (1) in & = 0, 1 3; see ( ) ( ) ( ) ( ) ( ) the left plot in Fig. 1. We perform uniform mesh refinements of & using tetrahedral elements. By using Metis [6], the domain is decomposed into # = 2: , : = 3, 4,..., 9, non-overlapping subdomains &8 with their own tetrahedral elements; see the right plot in Fig. 1. The total number of degrees of freedom is 2: 1 3, : = 4, 5, 6, 7. ( + ) We run BDDC preconditioned GMRES iterations until the relative residual error 9 reaches 10− . The experiments are performed on 64 nodes each with 8-core In- tel Haswell processors (Xeon E5-2630v3, 2.4Ghz) and 128 GB of memory. Three variants of BDDC preconditioners are used with corner (), corner/edge (), and corner/edge/face () constraints, respectively. The number of BDDC precondi- tioned GMRES iterations and the computational time measured in seconds [s] with respect to the number of subdomains (row-wise) and number of degrees of freedom (column-wise) are given in Table 1 for \ = 2.5. Since the system is unsymmetric but 372 Ulrich Langer and Huidong Yang

Fig. 1 Solution (left), space- time domain decomposition (right) with 1293 degrees of freedom and 512 subdomains. positive definite, the BDDC preconditioners do not show the same typical robustness and efficiency behavior when applied to the symmetric and positive definite system [14]. Nevertheless, we still observe certain scalability with respect to the number subdomains (up to 128), in particular, with corner/edge and corner/edge/face con- straints. Increasing \ will improve the preformance of the BDDC preconditioners with respect to the number of GMRES iterations, computational time, and scala- bility with respect to the number of subdomains as well as number of degrees of freedom, whereas decreasing \ leads to a worse performance. For instance, in the case of \ = 0.5, the last row of Table 1 reads as follows: 1293 OoM OoM /(−) /(−) 173 126.93B 171 109.94B 185 45.05B > 500 206 33.13B . This /( ) /( ) /( ) /(−) /( ) behaviour is expected since larger \ makes the problem more elliptic. However, we note that \ also affects the norm ℎ in which we measure the discretization error. k · k

5 Conclusions

In this work, we have applied two-level BDDC preconditioned GMRES methods to the solution of finite element equations arising from the space-time discretiza- tion of a parabolic model problem. We have compared the performance of BDDC preconditioners with different coarse level constraints for such an unsymmetric, but positive definite system. The preconditioners show certain scalability provided that \ is sufficiently large. Future work will concentrate on improvement of coarse-level corrections in order to achieve robustness with respect to different choices of \.

References

1. Bank, R.E., Vassilevski, P.S., Zikatanov, L.T.: Arbitrary dimension convection-diffusion schemes for space-time discretizations. J. Comput. Appl. Math. 310, 19–31 (2017) 2. Dohrmann, C.R.: A for substructuring based on constrained energy minimiza- tion. SIAM J. Sci. Comput. 25(1), 246–258 (2003) 3. Hofer, C., Langer, U., Neumüller, M., Schneckenleitner, R.: Parallel and robust preconditioning for space-time isogeometric analysis of parabolic evolution problems. SIAM J. Sci. Comput. 41(3), A1793–A1821 (2019) BDDC for STFEM of Parabolic Problems 373

Table 1: \ = 2.5. BDDC performance using different coarse level constraints (//), with respect to the number of subdomains (row-wise) and degrees of freedoms (column-wise).

No preconditioner 8 16 32 64 128 256 512 173 46 51 55 61 66 > 500 > 500 (0.02B)(0.02B)(0.02B)(0.03B)(0.04B) − − 333 72 79 87 99 108 > 500 > 500 (0.64B)(0.32B)(0.15B)(0.12B)(0.13B) − − 653 116 126 145 163 176 191 > 500 (27.59B)(9.17B)(4.07B)(1.92B)(1.06B)(2.02B) − 1293 OoM OoM 240 271 304 > 500 382 ( )( )(145.64B)(58.51B)(24.3B)( )(12.41B) − − −  (corner) preconditioner 173 23 28 27 32 36 51 110 (0.02B)(0.02B)(0.03B)(0.03B)(0.07B)(0.72B)(2.48B) 333 30 33 39 50 50 206 182 (0.60B)(0.26B)(0.14B)(0.10B)(0.09B)(3.86B)(6.2B) 653 35 47 61 64 69 77 287 (19.85B)(7.42B)(3.47B)(1.44B)(0.68B)(1.05B)(9.00B) 1293 OoM OoM 94 104 107 340 112 ( )( )(124.30B)(46.45B)(16.90B)(33.45B)(5.89B) − −  (corner+edge) preconditioner 173 21 22 22 25 27 42 80 (0.03B)(0.02B)(0.01B)(0.03B)(1.78B)(0.78B)(1.65B) 333 27 26 32 38 32 117 132 (0.54B)(0.23B)(0.11B)(0.12B)(0.15B)(2.52B)(5.98B) 653 33 44 51 54 54 54 235 (19.00B)(7.27B)(2.98B)(1.33B)(0.75B)(1.32B)(15.13B) 1293 OoM OoM 82 83 90 366 94 ( )( )(109.19B)(38.59B)(15.21B)(37.00B)(7.91B) − −  (corner+edge+face) preconditioner 173 21 21 22 22 22 38 74 (0.02B)(0.01B)(0.02B)(0.04B)(0.13B)(0.74B)(1.74B) 333 27 26 30 35 31 115 123 (0.68B)(0.28B)(0.14B)(0.12B)(0.25B)(3.79B)(7.08B) 653 34 44 49 52 53 51 226 (26.64B)(9.29B)(3.86B)(1.61B)(1.11B)(1.88B) 21.30B 1293 OoM OoM 82 83 88 369 92 ( )( )(145.39B)(53.10B)(23.04B)(52.33)(13.8B) − − 4. Hofer, C., Langer, U., Neumüller, M., Toulopoulos, I.: Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems. Electron. Trans. Numer. Anal. 49, 126–150 (2018) 5. Jing, L., Widlund, O.B.: FETI-DP, BDDC, and block Cholesky methods. Int. J. Numer. Meth. Engng. 66(2), 250–271 (2006) 6. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing 20(1), 359–392 (1998) 7. Ladyžhenskaya, O.A.: The boundary value problems of mathematical physics, Applied Math- ematical Sciences, vol. 49. Springer-Verlag, New York (1985) 8. Ladyžhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and quasilinear equations of parabolic type. AMS, Providence, RI (1968) 9. Langer, U., Moore, S.E., Neumüller, M.: Space-time isogeometric analysis of parabolic evolu- tion problems. Comput. Methods Appl. Mech. Eng. 306, 342–363 (2016) 10. Langer, U., Neumüller, M., Schafelner, A.: Space-time finite element methods for parabolic evolution problems with variable coefficients. In: T. Apel, U. Langer, A. Meyer, O. Steinbach (eds.) Advanced finite element methods with applications - Proceedings of the 30th Chemnitz FEM symposium 2017, Lecture Notes in Computational Science and Engineering (LNCSE), vol. 128, pp. 247–275. Springer, Berlin, Heidelberg, New York (2019) 11. Mandel, J., Dohrmann, C.R.: Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10(7), 639–659 (2003) 12. Mandel, J., Dohrmann, C.R., Tezaur, R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2), 167–193 (2005) 13. Steinbach, O.: Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15, 551–566 (2015) 14. Toselli, A., Widlund, O.B.: Domain decomposition methods - Algorithms and theory, Springer Series in Computational Mathematics, vol. 34. Springer, Berlin, Heidelberg, New York (2004)