# 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

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**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