# Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems?

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## Abstract

**:**

## 1. Introduction

- We present a range of numerical tests, on pollution-free meshes, comparing our proposed H-GenEO approach with another spectral coarse space applicable to the Helmholtz problem, namely the DtN method.
- We investigate the use of appropriate thresholding for the required generalised eigenproblems in both the DtN and H-GenEO coarse spaces.
- We consider robustness to non-uniform decomposition, heterogeneity, and increasing wave number as well as the scalability of the methods. We find that only the H-GenEO approach is scalable and robust to all of these factors for a 2D model problem.
- We provide both weak and strong scalability tests for H-GenEO applied to high wave number problems.

## 2. Materials and Methods

#### 2.1. Finite Element Discretisation

#### 2.2. Underlying Domain Decomposition Method

#### 2.3. Spectral Coarse Spaces

**Remark**

**1**

**.**We utilise the following notation for local Dirichlet, Robin, and Neumann matrices: for a variational problem that gives rise to a system matrix B, we denote by ${B}_{s}$ the corresponding local Dirichlet matrix on ${\mathsf{\Omega}}_{s}$. In the case that Robin conditions are used on internal subdomain interfaces, the local problem matrix is denoted by ${\widehat{B}}_{s}$. On the other hand, if Neumann conditions are used on such interfaces, we denote the local matrix by ${\tilde{B}}_{s}$.

#### 2.3.1. The DtN Coarse Space

#### 2.3.2. The GenEO Coarse Space

#### 2.3.3. H-GenEO: A GenEO-Type Coarse Space for Helmholtz Problems

#### 2.3.4. A Link between DtN and GenEO

## 3. Results and Discussion

`ffddm`, which handles the underlying domain decomposition data structures. As a model problem, we consider the case of a wave guide in 2D, defined on the unit square $\mathsf{\Omega}={(0,1)}^{2}$. We impose homogeneous Dirichlet conditions on two opposite sides, namely (2b) with ${u}_{{\mathsf{\Gamma}}_{D}}=0$ on ${\mathsf{\Gamma}}_{D}=\{0,1\}\times [0,1]$, and Robin conditions on the two remaining sides, that is (2c) on ${\mathsf{\Gamma}}_{R}=[0,1]\times \{0,1\}$. A point source is located in the centre of the domain at $(\frac{1}{2},\frac{1}{2})$ and provides the forcing function f. A schematic of this model problem is found in Figure 2.

#### 3.1. A Comparison of Methods for the Homogeneous Problem with Uniform Partitioning

#### 3.2. Scalability of DtN and H-GenEO for the Homogeneous Problem with Uniform Partitioning

#### 3.3. Robustness of DtN and H-GenEO for the Homogeneous Problem with METIS Decomposition

#### 3.4. The Effect of Heterogeneity

#### 3.5. Higher Order Finite Elements

#### 3.6. The Effect of Boundary Conditions within the H-GenEO Eigenproblem

#### 3.7. The Effect of More Overlap When Using H-GenEO

#### 3.8. Weak Scalability and Timing Results for H-GenEO

#### 3.9. High Wave Number Strong Scalability and Timing Results for H-GenEO

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Moiola, A.; Spence, E.A. Is the Helmholtz equation really sign-indefinite? SIAM Rev.
**2014**, 56, 274–312. [Google Scholar] [CrossRef] - Ernst, O.G.; Gander, M.J. Why it is difficult to solve Helmholtz problems with classical iterative methods. In Numerical Analysis of Multiscale Problems; Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R., Eds.; Springer: Berlin, Germany, 2012; pp. 325–363. [Google Scholar]
- Gander, M.J.; Zhang, H. A class of iterative solvers for the Helmholtz equation: Factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev.
**2019**, 60, 3–76. [Google Scholar] [CrossRef] [Green Version] - Gillman, A.; Barnett, A.H.; Martinsson, P.-G. A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media. BIT
**2015**, 55, 141–170. [Google Scholar] [CrossRef] [Green Version] - Wang, S.; de Hoop, M.V.; Xia, J. On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver. Geophys. Prospect.
**2011**, 59, 857–873. [Google Scholar] [CrossRef] - Calandra, H.; Gratton, S.; Pinel, X.; Vasseur, X. An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media. Numer. Linear Algebra Appl.
**2013**, 20, 663–688. [Google Scholar] [CrossRef] - Hu, Q.; Zhang, H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations. SIAM J. Sci. Comput.
**2016**, 38, A2232–A2261. [Google Scholar] [CrossRef] - Erlangga, Y.; Vuik, C.; Oosterlee, C.W. On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math.
**2004**, 50, 409–425. [Google Scholar] [CrossRef] [Green Version] - Erlangga, Y.; Oosterlee, C.W.; Vuik, C. A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput.
**2006**, 27, 1471–1492. [Google Scholar] [CrossRef] - Cocquet, P.-H.; Gander, M.J. How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid? SIAM J. Sci. Comput.
**2017**, 39, A438–A478. [Google Scholar] [CrossRef] - Dwarka, V.; Vuik, C. Scalable convergence using two-level deflation preconditioning for the Helmholtz equation. SIAM J. Sci. Comput.
**2020**, 42, A901–A928. [Google Scholar] [CrossRef] - Lahaye, D.; Vuik, C. How to choose the shift in the shifted Laplace preconditioner for the Helmholtz equation combined with deflation. In Modern Solvers for Helmholtz Problems; Lahaye, D., Tang, J., Vuik, K., Eds.; Birkhäuser: Cham, Switzerland, 2017; pp. 85–112. [Google Scholar]
- Engquist, B.; Ying, L. Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation. Comm. Pure Appl. Math.
**2011**, 64, 697–735. [Google Scholar] [CrossRef] [Green Version] - Engquist, B.; Ying, L. Sweeping preconditioner for the Helmholtz equation: Moving perfectly matched layers. Multiscale. Model. Simul.
**2011**, 9, 686–710. [Google Scholar] [CrossRef] [Green Version] - Taus, M.; Zepeda-Núñez, L.; Hewett, R.J.; Demanet, L. L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation. J. Comput. Phys.
**2020**, 420, 109706. [Google Scholar] [CrossRef] - Dai, R.; Modave, A.; Remacle, J.-F.; Geuzaine, C. Multidirectional sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems. J. Comput. Phys.
**2022**, 453, 110887. [Google Scholar] [CrossRef] - Farhat, C.; Macedo, A.; Lesoinne, M. A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer. Math.
**2000**, 85, 283–308. [Google Scholar] [CrossRef] - Farhat, C.; Avery, P.; Tezaur, R.; Li, J. FETI-DPH: A dual-primal domain decomposition method for acoustic scattering. J. Comput. Acoust.
**2005**, 13, 499–524. [Google Scholar] [CrossRef] - Després, B. Domain decomposition method for the Helmholtz problem. C R Math. Acad. Sci. Paris I Math.
**1990**, 311, 313–316. [Google Scholar] - Claeys, X.; Parolin, E. Robust treatment of cross points in Optimized Schwarz Methods. arXiv
**2020**, arXiv:2003.06657. [Google Scholar] - Gander, M.J.; Magoules, F.; Nataf, F. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput.
**2002**, 24, 38–60. [Google Scholar] [CrossRef] [Green Version] - Boubendir, Y.; Antoine, X.; Geuzaine, C. A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation. J. Comput. Phys.
**2012**, 231, 262–280. [Google Scholar] [CrossRef] [Green Version] - Collino, F.; Ghanemi, S.; Joly, P. Domain decomposition method for harmonic wave propagation: A general presentation. Comput. Methods Appl. Mech. Engrg.
**2000**, 184, 171–211. [Google Scholar] [CrossRef] [Green Version] - Cai, X.-C.; Casarin, M.A.; Elliott, F.W., Jr.; Widlund, O.B. Overlapping Schwarz algorithms for solving Helmholtz’s equation. Contemp. Math.
**1998**, 218, 391–399. [Google Scholar] - Gander, M.J.; Zhang, H. Optimized Schwarz methods with overlap for the Helmholtz equation. SIAM J. Sci. Comput.
**2016**, 38, A3195–A3219. [Google Scholar] [CrossRef] [Green Version] - Kimn, J.-H.; Sarkis, M. Restricted overlapping balancing domain decomposition methods and restricted coarse problems for the Helmholtz problem. Comput. Methods Appl. Mech. Engrg.
**2007**, 196, 1507–1514. [Google Scholar] - Graham, I.G.; Spence, E.A.; Vainikko, E. Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. In Modern Solvers for Helmholtz Problems; Lahaye, D., Tang, J., Vuik, K., Eds.; Birkhäuser: Cham, Switzerland, 2017; pp. 3–26. [Google Scholar]
- Kimn, J.-H.; Sarkis, M. Shifted Laplacian RAS solvers for the Helmholtz equation. In Domain Decomposition Methods in Science and Engineering XX; Bank, R., Holst, M., Widlund, O., Xu, J., Eds.; Springer: Berlin, Germany, 2013; pp. 151–158. [Google Scholar]
- Graham, I.G.; Spence, E.A.; Vainikko, E. Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Math. Comp.
**2017**, 86, 2089–2127. [Google Scholar] [CrossRef] [Green Version] - Graham, I.G.; Spence, E.A.; Zou, J. Domain Decomposition with local impedance conditions for the Helmholtz equation with absorption. SIAM J. Numer. Anal.
**2020**, 58, 2515–2543. [Google Scholar] [CrossRef] - Gong, S.; Graham, I.G.; Spence, E.A. Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation. IMA J. Numer. Anal.
**2021**, 41, 2139–2185. [Google Scholar] [CrossRef] - Gong, S.; Gander, M.J.; Graham, I.G.; Lafontaine, D.; Spence, E.A. Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation. arXiv
**2021**, arXiv:2106.05218. [Google Scholar] - Bootland, N.; Dolean, V.; Kyriakis, A.; Pestana, J. Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices. Electron. Trans. Numer. Anal.
**2022**, 55, 112–141. [Google Scholar] [CrossRef] - Bonazzoli, M.; Dolean, V.; Graham, I.G.; Spence, E.A.; Tournier, P.-H. Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption. Math. Comp.
**2019**, 88, 2559–2604. [Google Scholar] [CrossRef] [Green Version] - Spillane, N.; Dolean, V.; Hauret, P.; Nataf, F.; Pechstein, C.; Scheichl, R. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math.
**2014**, 126, 741–770. [Google Scholar] [CrossRef] [Green Version] - Nataf, F.; Xiang, H.; Dolean, V.; Spillane, N. A coarse space construction based on local Dirichlet-to-Neumann maps. SIAM J. Sci. Comput.
**2011**, 33, 1623–1642. [Google Scholar] [CrossRef] [Green Version] - Bootland, N.; Dolean, V. On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. In Numerical Mathematics and Advanced Applications ENUMATH 2019; Vermolen, F.J., Vuik, C., Eds.; Springer: Cham, Switzerland, 2021; pp. 175–184. [Google Scholar]
- Conen, L.; Dolean, V.; Krause, R.; Nataf, F. A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator. J. Comput. Appl. Math.
**2014**, 271, 83–99. [Google Scholar] [CrossRef] - Bootland, N.; Dolean, V.; Graham, I.G.; Ma, C.; Scheichl, R. Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems. arXiv
**2021**, arXiv:2110.13537. [Google Scholar] - Bootland, N.; Dolean, V.; Jolivet, P.; Tournier, P.-H. A comparison of coarse spaces for Helmholtz problems in the high frequency regime. Comput. Math. Appl.
**2021**, 98, 239–253. [Google Scholar] [CrossRef] - Zarmi, A.; Turkel, E. A general approach for high order absorbing boundary conditions for the Helmholtz equation. J. Comput. Phys.
**2013**, 242, 387–404. [Google Scholar] [CrossRef] - Beriot, H.; Modave, A. An automatic perfectly matched layer for acoustic finite element simulations in convex domains of general shape. Int. J. Numer. Methods Engergy
**2021**, 122, 1239–1261. [Google Scholar] [CrossRef] - Harari, I.; Slavutin, M.; Turkel, E. Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comput. Acoust.
**2000**, 8, 121–137. [Google Scholar] [CrossRef] - Babuska, I.M.; Sauter, S.A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal.
**1997**, 34, 2392–2423. [Google Scholar] [CrossRef] [Green Version] - Dolean, V.; Jolivet, P.; Nataf, F. An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 2015. [Google Scholar]
- Fish, J.; Qu, Y. Global-basis two-level method for indefinite systems. Part 1: Convergence studies. Int. J. Numer. Methods Engergy
**2000**, 49, 439–460. [Google Scholar] [CrossRef] - Nataf, F.; Xiang, H.; Dolean, V. A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps. C R Math. Acad. Sci. Paris I Math.
**2010**, 348, 1163–1167. [Google Scholar] [CrossRef] - Haferssas, R.; Jolivet, P.; Nataf, F. An additive Schwarz method type theory for Lions’s algorithm and a symmetrized optimized restricted additive Schwarz method. SIAM J. Sci. Comput.
**2017**, 39, A1345–A1365. [Google Scholar] [CrossRef] - Nataf, F.; Tournier, P.-H. A GenEO domain decomposition method for saddle point problems. arXiv
**2019**, arXiv:1911.01858. [Google Scholar] - Spillane, N. An abstract theory of domain decomposition methods with coarse spaces of the GenEO family. arXiv
**2021**, arXiv:2104.00280. [Google Scholar] - Bootland, N.; Dolean, V.; Graham, I.G.; Ma, C.; Scheichl, R. GenEO coarse spaces for heterogeneous indefinite elliptic problems. In Domain Decomposition Methods in Science and Engineering XXVI; Brenner, S., Chung, E.T.S., Klawonn, A., Kwok, F., Xu, J., Zou, J., Eds.; Springer: Cham, Switzerland, 2017; accepted. [Google Scholar]
- Hecht, F. New development in FreeFem++. J. Numer. Math.
**2012**, 20, 251–266. [Google Scholar] [CrossRef] - Lehoucq, R.B.; Sorensen, D.C.; Yang, C. ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 1998. [Google Scholar]
- Amestoy, P.R.; Duff, I.S.; L’Excellent, J.-Y.; Koster, J. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl.
**2001**, 23, 15–41. [Google Scholar] [CrossRef] [Green Version] - Karypis, G.; Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput.
**1998**, 20, 359–392. [Google Scholar] [CrossRef]

**Figure 1.**Local eigenfunctions for $k=46.5$.

**Top row:**Examples using DtN (14).

**Middle row:**Equivalent examples using H-GenEO (19).

**Bottom row:**Examples using H-GenEO which are not found amongst the DtN eigenfunctions.

**Figure 3.**The size of coarse space utilised for the homogeneous problem when using ORAS with the DtN and H-GenEO coarse spaces. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used. (

**a**) Varying the wave number k for $N=25$, (

**b**) Varying the number of subdomains N for $k=73.8$.

**Figure 4.**Piecewise constant layer profiles for the wave speed $c\left(\mathbf{x}\right)$. For the darkest shade $c\left(\mathit{x}\right)=1$, while for the lightest shade $c\left(\mathit{x}\right)=\rho $, with $\rho $ being the contrast factor.

**Figure 5.**Schematic of the growing 2D wave guide model problem used in a weak scaling test on $N=25L$ fixed size subdomains, with the underlying non-overlapping subdomains shown in grey.

**Figure 6.**Timings for the homogeneous problem when using ORAS with H-GenEO ($\frac{1}{2}$) and a varying number of subdomains for $k=186.0$ and ${h}^{-1}=3200$, giving a total of $10,\phantom{\rule{-0.166667em}{0ex}}246,\phantom{\rule{-0.166667em}{0ex}}401$ dofs. A non-uniform decomposition into N subdomains is used, given by METIS.

**Table 1.**Preconditioned GMRES iteration counts and size of coarse space (in parentheses) for the homogeneous problem when using ORAS and various coarse spaces. A uniform decomposition into $5\times 5$ square subdomains is used, giving 25 subdomains in total.

k | h^{−1} | One-Level | DtN | Δ-GenEO | H-GenEO |
---|---|---|---|---|---|

18.5 | 100 | 73 | 19 (147) | 53 (135) | 21 (164) |

29.3 | 200 | 97 | 26 (218) | 100 (271) | 18 (370) |

46.5 | 400 | 125 | 35 (303) | 148 (560) | 17 (779) |

73.8 | 800 | 156 | 42 (502) | 220 (1120) | 15 (1712) |

**Table 2.**Preconditioned GMRES iteration counts and size of coarse space (in parentheses) for the homogeneous problem when using ORAS and the DtN and H-GenEO coarse spaces with varying eigenvalue thresholds. A uniform decomposition into $5\times 5$ square subdomains is used, giving 25 subdomains in total.

k | h^{−1} | DtN (k) | DtN (k^{4/3}) | DtN (k^{3/2}) | H-GenEO ($\frac{1}{8}$) | H-GenEO ($\frac{1}{4}$) | H-GenEO ($\frac{1}{2}$) |
---|---|---|---|---|---|---|---|

18.5 | 100 | 19 (147) | 13 (260) | 11 (403) | 46 (80) | 31 (105) | 21 (164) |

29.3 | 200 | 26 (218) | 14 (483) | 13 (759) | 53 (139) | 33 (189) | 18 (370) |

46.5 | 400 | 35 (303) | 14 (868) | 12 (1479) | 56 (245) | 35 (378) | 17 (779) |

73.8 | 800 | 42 (502) | 16 (1588) | 15 (2925) | 40 (546) | 25 (800) | 15 (1712) |

**Table 3.**Preconditioned GMRES iteration counts (above), size of coarse space (middle), and average number of eigenvectors taken per subdomain (below) for the homogeneous problem when using ORAS with the DtN and H-GenEO coarse spaces and a varying number of subdomains N for $k=73.8$ and ${h}^{-1}=800$. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

N | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DtN (k) | 28 | 32 | 40 | 42 | 51 | 76 | 49 | 94 | 90 | 36 | 37 | 96 | 154 |

DtN (k^{4/3}) | 15 | 16 | 19 | 16 | 16 | 16 | 15 | 16 | 15 | 15 | 16 | 17 | 17 |

H-GenEO ($\frac{1}{8}$) | 26 | 31 | 36 | 40 | 71 | 70 | 65 | 127 | 81 | 116 | 247 | 194 | 138 |

H-GenEO ($\frac{1}{2}$) | 13 | 15 | 15 | 15 | 16 | 16 | 16 | 18 | 16 | 18 | 18 | 18 | 19 |

DtN (k) | 124 | 251 | 362 | 502 | 605 | 736 | 843 | 1000 | 946 | 1329 | 1554 | 1529 | 1327 |

DtN (k^{4/3}) | 392 | 790 | 1175 | 1588 | 1994 | 2366 | 2753 | 3176 | 3611 | 3976 | 4369 | 4955 | 5188 |

H-GenEO ($\frac{1}{8}$) | 200 | 305 | 408 | 546 | 536 | 600 | 788 | 733 | 936 | 927 | 780 | 974 | 1264 |

H-GenEO ($\frac{1}{2}$) | 852 | 1116 | 1428 | 1712 | 1903 | 2261 | 2444 | 2629 | 3120 | 3204 | 3482 | 3882 | 3816 |

DtN (k) | 31.0 | 27.9 | 22.6 | 20.1 | 16.8 | 15.0 | 13.2 | 12.3 | 9.5 | 11.0 | 10.8 | 9.0 | 6.8 |

DtN (k^{4/3}) | 98.0 | 87.8 | 73.4 | 63.5 | 55.4 | 48.3 | 43.0 | 39.2 | 36.1 | 32.9 | 30.3 | 29.3 | 26.1 |

H-GenEO ($\frac{1}{8}$) | 50.0 | 33.9 | 25.5 | 21.8 | 14.9 | 12.2 | 12.3 | 9.0 | 9.4 | 7.7 | 5.4 | 5.8 | 6.4 |

H-GenEO ($\frac{1}{2}$) | 213.0 | 124.0 | 89.3 | 68.5 | 52.9 | 46.1 | 38.2 | 32.5 | 31.2 | 26.5 | 24.2 | 23.0 | 19.5 |

**Table 4.**Preconditioned GMRES iteration counts (above), size of coarse space (middle), and average number of eigenvectors taken per subdomain (below) for the homogeneous problem when using ORAS with DtN(${k}^{4/3}$) or H-GenEO($\frac{1}{2}$) and a varying number of subdomains. A non-uniform decomposition into N subdomains is used, given by METIS.

Number of Subdomains N | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DtN (k^{4/3}) | H-GenEO ($\frac{1}{2}$) | ||||||||||||

k | h^{−1} | 20 | 40 | 80 | 120 | 160 | 200 | 20 | 40 | 80 | 120 | 160 | 200 |

18.5 | 100 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 17 | 19 | 22 | 27 | 27 |

29.3 | 200 | 12 | 15 | 11 | 12 | 12 | 12 | 15 | 17 | 19 | 20 | 22 | 23 |

46.5 | 400 | 12 | 13 | 15 | 13 | 13 | 13 | 15 | 16 | 16 | 18 | 20 | 20 |

73.8 | 800 | 15 | 15 | 14 | 16 | 14 | 16 | 15 | 16 | 17 | 17 | 17 | 19 |

117.2 | 1600 | 14 | 15 | 16 | 17 | 15 | 16 | 14 | 15 | 15 | 16 | 16 | 16 |

18.5 | 100 | 281 | 422 | 652 | 843 | 1005 | 1157 | 201 | 285 | 383 | 471 | 524 | 589 |

29.3 | 200 | 477 | 758 | 1130 | 1410 | 1693 | 1922 | 400 | 574 | 783 | 958 | 1097 | 1245 |

46.5 | 400 | 959 | 1466 | 2132 | 2677 | 3151 | 3553 | 869 | 1193 | 1670 | 2008 | 2253 | 2507 |

73.8 | 800 | 1695 | 2563 | 3751 | 4672 | 5486 | 6199 | 1863 | 2456 | 3433 | 4147 | 4749 | 5338 |

117.2 | 1600 | 3049 | 4695 | 6831 | 8486 | 9896 | 11,092 | 4238 | 5680 | 7575 | 9049 | 10,273 | 11,305 |

18.5 | 100 | 14.1 | 10.6 | 8.2 | 7.0 | 6.3 | 5.8 | 10.1 | 7.1 | 4.8 | 3.9 | 3.3 | 2.9 |

29.3 | 200 | 23.9 | 18.9 | 14.1 | 11.8 | 10.6 | 9.6 | 20.0 | 14.3 | 9.8 | 8.0 | 6.9 | 6.2 |

46.5 | 400 | 48.0 | 36.6 | 26.6 | 22.3 | 19.7 | 17.8 | 43.5 | 29.8 | 20.9 | 16.7 | 14.1 | 12.5 |

73.8 | 800 | 84.8 | 64.1 | 46.9 | 38.9 | 34.3 | 31.0 | 93.2 | 61.4 | 42.9 | 34.6 | 29.7 | 26.7 |

117.2 | 1600 | 152.4 | 117.4 | 85.4 | 70.7 | 61.9 | 55.5 | 211.9 | 142.0 | 94.7 | 75.4 | 164.2 | 56.5 |

**Table 5.**Preconditioned GMRES iteration counts (above) and size of coarse space (below) for the heterogeneous increasing layers problem when using ORAS with DtN(${k}^{4/3}$) or H-GenEO($\frac{1}{2}$) and a varying number of subdomains. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

Number of Subdomains N | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DtN (k^{4/3}) | H-GenEO ($\frac{1}{2}$) | |||||||||||||

ω | h^{−1} | ρ | 16 | 36 | 64 | 100 | 144 | 196 | 16 | 36 | 64 | 100 | 144 | 196 |

29.3 | 200 | 10 | 29 | 37 | 41 | 52 | 55 | 58 | 15 | 16 | 19 | 18 | 18 | 19 |

1000 | 44 | 44 | 50 | 58 | 52 | 52 | 15 | 15 | 17 | 18 | 17 | 17 | ||

46.5 | 400 | 10 | 32 | 38 | 41 | 66 | 65 | 73 | 15 | 16 | 16 | 19 | 18 | 18 |

1000 | 63 | 69 | 74 | 84 | 73 | 71 | 14 | 15 | 16 | 18 | 17 | 17 | ||

73.8 | 800 | 10 | 35 | 43 | 42 | 40 | 58 | 69 | 15 | 17 | 16 | 17 | 18 | 17 |

1000 | 89 | 93 | 107 | 111 | 114 | 109 | 14 | 15 | 15 | 16 | 17 | 16 | ||

29.3 | 200 | 10 | 116 | 173 | 234 | 363 | 399 | 467 | 224 | 354 | 452 | 662 | 679 | 754 |

1000 | 84 | 111 | 136 | 285 | 329 | 371 | 222 | 350 | 446 | 642 | 679 | 741 | ||

46.5 | 400 | 10 | 208 | 317 | 405 | 600 | 704 | 812 | 458 | 706 | 990 | 1234 | 1523 | 1678 |

1000 | 144 | 176 | 202 | 421 | 496 | 554 | 450 | 693 | 990 | 1216 | 1512 | 1666 | ||

73.8 | 800 | 10 | 379 | 557 | 693 | 1142 | 1217 | 1404 | 930 | 1425 | 2074 | 2584 | 3060 | 3553 |

1000 | 254 | 294 | 326 | 748 | 784 | 838 | 914 | 1409 | 2058 | 2572 | 3059 | 3534 |

**Table 6.**Preconditioned GMRES iteration counts (above) and size of coarse space (below) for the heterogeneous diagonal layers problem when using ORAS with DtN(${k}^{4/3}$) or H-GenEO($\frac{1}{2}$) and a varying number of subdomains. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

Number of Subdomains N | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DtN (k^{4/3}) | H-GenEO ($\frac{1}{2}$) | |||||||||||||

ω | h^{−1} | ρ | 16 | 36 | 64 | 100 | 144 | 196 | 16 | 36 | 64 | 100 | 144 | 196 |

29.3 | 200 | 10 | 13 | 14 | 13 | 14 | 21 | 25 | 16 | 18 | 20 | 18 | 23 | 25 |

1000 | 13 | 14 | 14 | 14 | 22 | 25 | 16 | 18 | 20 | 18 | 23 | 25 | ||

46.5 | 400 | 10 | 15 | 14 | 14 | 16 | 25 | 31 | 16 | 17 | 17 | 26 | 21 | 22 |

1000 | 15 | 14 | 15 | 16 | 25 | 34 | 16 | 17 | 18 | 27 | 22 | 22 | ||

73.8 | 800 | 10 | 14 | 18 | 16 | 15 | 20 | 26 | 16 | 17 | 17 | 17 | 19 | 20 |

1000 | 15 | 18 | 16 | 15 | 32 | 39 | 16 | 17 | 17 | 17 | 19 | 20 | ||

29.3 | 200 | 10 | 336 | 593 | 866 | 1090 | 1376 | 1390 | 260 | 376 | 499 | 689 | 737 | 828 |

1000 | 336 | 594 | 866 | 1090 | 1375 | 1390 | 259 | 375 | 499 | 687 | 737 | 826 | ||

46.5 | 400 | 10 | 621 | 1075 | 1540 | 1910 | 2370 | 2622 | 543 | 789 | 1095 | 1384 | 1599 | 1825 |

1000 | 621 | 1075 | 1539 | 1907 | 2368 | 2614 | 541 | 790 | 1093 | 1381 | 1596 | 1824 | ||

73.8 | 800 | 10 | 1164 | 1947 | 2692 | 3592 | 4145 | 4608 | 1145 | 1636 | 2243 | 2823 | 3233 | 3681 |

1000 | 1163 | 1946 | 2693 | 3592 | 4131 | 4569 | 1141 | 1633 | 2239 | 2822 | 3232 | 3671 |

**Table 7.**Preconditioned GMRES iteration counts for the heterogeneous alternating layers problem with $\rho =10/100/1000$ when using ORAS with H-GenEO($\frac{1}{2}$) and a varying number of subdomains. A non-uniform decomposition into N subdomains is used, given by METIS.

Number of Subdomains N with Sub-Columns for ρ = 10/100/1000 | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ω | h^{−1} | 20 | 40 | 80 | 120 | 160 | 200 | ||||||||||||

18.5 | 100 | 17 | 17 | 17 | 19 | 19 | 19 | 21 | 21 | 21 | 27 | 27 | 27 | 31 | 31 | 31 | 33 | 33 | 33 |

29.3 | 200 | 16 | 16 | 16 | 17 | 17 | 17 | 19 | 19 | 19 | 20 | 20 | 20 | 21 | 21 | 21 | 23 | 23 | 23 |

46.5 | 400 | 17 | 18 | 18 | 18 | 18 | 18 | 22 | 23 | 23 | 25 | 26 | 26 | 27 | 28 | 28 | 28 | 29 | 29 |

73.8 | 800 | 16 | 16 | 16 | 17 | 17 | 17 | 18 | 18 | 18 | 18 | 19 | 19 | 19 | 20 | 20 | 23 | 23 | 23 |

117.2 | 1600 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 |

**Table 8.**Preconditioned GMRES iteration counts (above) and size of coarse space (below) for P2 finite element discretisation of the heterogeneous diagonal layers problem with $\rho =10$ when using ORAS with DtN(${k}^{4/3}$) or H-GenEO($\frac{1}{2}$) and a varying number of subdomains. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

Number of Subdomains N | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DtN (k^{4/3}) | H-GenEO ($\frac{1}{2}$) | |||||||||||||

ω | h^{−1} | ρ | 16 | 36 | 64 | 100 | 144 | 196 | 16 | 36 | 64 | 100 | 144 | 196 |

18.5 | 100 | 10 | 13 | 11 | 10 | 10 | 10 | 18 | 15 | 16 | 19 | 18 | 23 | 24 |

29.3 | 200 | 10 | 14 | 12 | 12 | 13 | 19 | 25 | 15 | 17 | 18 | 18 | 23 | 25 |

46.5 | 400 | 10 | 15 | 12 | 12 | 15 | 23 | 30 | 15 | 16 | 17 | 20 | 21 | 22 |

73.8 | 800 | 10 | 17 | 16 | 14 | 13 | 18 | 25 | 15 | 16 | 16 | 17 | 18 | 20 |

18.5 | 100 | 10 | 151 | 326 | 510 | 706 | 898 | 937 | 125 | 186 | 231 | 346 | 398 | 519 |

29.3 | 200 | 10 | 300 | 608 | 892 | 1086 | 1444 | 1516 | 260 | 377 | 507 | 686 | 733 | 824 |

46.5 | 400 | 10 | 589 | 1108 | 1572 | 2069 | 2491 | 2638 | 540 | 794 | 1100 | 1403 | 1594 | 1820 |

73.8 | 800 | 10 | 919 | 1916 | 2862 | 3614 | 4409 | 4748 | 1144 | 1645 | 2239 | 2805 | 3325 | 3695 |

**Table 9.**Preconditioned GMRES iteration counts (above) and size of coarse space (below) for the homogeneous problem when using ORAS with impedance-H-GenEO (the eigenproblem (19) is altered to have impedance as opposed to Neumann boundary conditions on the left-hand side) and a varying number of subdomains. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

Number of Subdomains N | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

k | h^{−1} | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 |

18.5 | 100 | 17 | 19 | 23 | 27 | 30 | 36 | 42 | 45 | 43 | 58 | 61 | 61 | 67 |

29.3 | 200 | 17 | 19 | 22 | 25 | 34 | 33 | 41 | 38 | 35 | 49 | 60 | 62 | 65 |

46.5 | 400 | 15 | 18 | 19 | 22 | 26 | 25 | 26 | 39 | 43 | 47 | 52 | 51 | 54 |

73.8 | 800 | 15 | 19 | 19 | 20 | 25 | 27 | 26 | 35 | 33 | 39 | 43 | 44 | 51 |

18.5 | 100 | 68 | 102 | 140 | 158 | 204 | 217 | 236 | 287 | 334 | 341 | 404 | 477 | 504 |

29.3 | 200 | 148 | 215 | 296 | 370 | 392 | 521 | 504 | 576 | 720 | 768 | 725 | 793 | 908 |

46.5 | 400 | 360 | 492 | 628 | 754 | 917 | 988 | 1236 | 1176 | 1468 | 1550 | 1740 | 1807 | 1930 |

73.8 | 800 | 848 | 1106 | 1420 | 1696 | 1877 | 2218 | 2432 | 2574 | 2960 | 3180 | 3443 | 3834 | 3732 |

**Table 10.**Preconditioned GMRES iteration counts (above) and size of coarse space (below) for the homogeneous problem when using ORAS with H-GenEO, varying the amount of overlap (in terms of element width, with 2 representing minimal overlap) and number of subdomains for $k=46.5$ and ${h}^{-1}=400$. A uniform decomposition into $\sqrt{N}\times \sqrt{N}$ square subdomains is used.

Number of Subdomains N | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Overlap | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 |

2 | 14 | 15 | 15 | 17 | 16 | 16 | 16 | 20 | 26 | 19 | 19 | 22 | 21 |

4 | 10 | 11 | 11 | 12 | 12 | 13 | 12 | 17 | 14 | 16 | 17 | 21 | 20 |

8 | 8 | 10 | 10 | 10 | 13 | 13 | 12 | 20 | 23 | 22 | 26 | 31 | 27 |

16 | 16 | 21 | 26 | 26 | 37 | 77 | 61 | 75 | 86 | 109 | 178 | 157 | 164 |

2 | 368 | 492 | 644 | 779 | 938 | 1030 | 1248 | 1195 | 1476 | 1558 | 1758 | 1845 | 2016 |

4 | 352 | 472 | 600 | 699 | 871 | 947 | 1088 | 1124 | 1296 | 1449 | 1689 | 1697 | 1690 |

8 | 336 | 436 | 538 | 650 | 799 | 863 | 1024 | 981 | 1132 | 1395 | 1511 | 1425 | 1512 |

16 | 316 | 417 | 500 | 610 | 733 | 717 | 920 | 942 | 1108 | 1239 | 1086 | 1212 | 1280 |

**Table 11.**Weak scaling results and timings for the alternating layers problem when using ORAS with H-GenEO ($\frac{1}{2}$) and a varying number of subdomains for $k=73.8$, ${h}^{-1}=800$ and $\rho =100$. A uniform decomposition into $N=25L$ subdomains is used, as depicted in Figure 5. Note that setup refers to the initial decomposition and partitioning, which is performed sequentially, while the local problems and eigensolves are carried out in parallel.

N | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 |
---|---|---|---|---|---|---|---|---|

Iteration count | 17 | 18 | 18 | 19 | 19 | 20 | 21 | 21 |

Coarse space size | 3010 | 6150 | 9290 | 12,430 | 15,570 | 18,710 | 21,850 | 24,990 |

Total run time (s) | 45.8 | 48.6 | 53.0 | 58.7 | 63.5 | 70.0 | 79.7 | 88.1 |

Weak scaling efficiency | − | 94.2% | 86.4% | 78.0% | 72.1% | 65.4% | 57.5% | 52.0% |

Eigensolve time (s) | 37.1 | 37.9 | 37.9 | 38.3 | 37.8 | 37.9 | 37.9 | 37.7 |

Setup time (s) | 5.5 | 7.7 | 12.9 | 16.5 | 19.9 | 23.8 | 27.4 | 30.8 |

Efficiency without setup | - | 98.5% | 100.5% | 95.5% | 92.4% | 87.2% | 77.1% | 70.3% |

**Table 12.**Strong scaling results and timings for the homogeneous problem when using ORAS with H-GenEO ($\frac{1}{2}$) and a varying number of subdomains for $k=186.0$ and ${h}^{-1}=3200$, giving a total of $10,\phantom{\rule{-0.166667em}{0ex}}246,\phantom{\rule{-0.166667em}{0ex}}401$ dofs. A non-uniform decomposition into N subdomains is used, given by METIS. The average local eigenproblem size is given approximately as the number of dofs divided by N.

N | 80 | 120 | 160 | 200 | 240 | 280 | 320 | 360 | 400 |
---|---|---|---|---|---|---|---|---|---|

Iteration count | 14 | 16 | 15 | 16 | 17 | 17 | 18 | 19 | 19 |

Coarse space size | 16,014 | 19,018 | 21,348 | 23,747 | 25,560 | 27,270 | 28,793 | 30,357 | 31,773 |

Total run time (s) | 1214.4 | 614.6 | 404.4 | 279.3 | 217.3 | 195.0 | 159.4 | 154.0 | 147.6 |

Parallel efficiency | − | 132% | 150% | 174% | 186% | 178% | 190% | 175% | 165% |

Eigenproblem size | 128,080 | 85,387 | 64,040 | 51,232 | 42,693 | 36,594 | 32,020 | 28,462 | 25,616 |

Average no. of eigenvectors | 200.2 | 158.5 | 133.4 | 118.7 | 106.5 | 97.4 | 90.0 | 84.3 | 79.4 |

Eigensolve time | 68.6% | 70.1% | 66.8% | 62.7% | 53.7% | 52.3% | 41.6% | 39.3% | 37.3% |

Coarse factorisation time | 29.6% | 27.8% | 30.8% | 34.4% | 42.8% | 44.3% | 54.0% | 56.3% | 58.3% |

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## Share and Cite

**MDPI and ACS Style**

Bootland, N.; Dolean, V.
Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems? *Math. Comput. Appl.* **2022**, *27*, 35.
https://doi.org/10.3390/mca27030035

**AMA Style**

Bootland N, Dolean V.
Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems? *Mathematical and Computational Applications*. 2022; 27(3):35.
https://doi.org/10.3390/mca27030035

**Chicago/Turabian Style**

Bootland, Niall, and Victorita Dolean.
2022. "Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems?" *Mathematical and Computational Applications* 27, no. 3: 35.
https://doi.org/10.3390/mca27030035