**Efficient solver for Helmholtz problems**

Leading-edge sparse direct solvers allow one to tackle problems involving up to 100 millions of unknowns (

**Mary, 2017**). Beyond this limit, when considering sparse areal acquisitions with a reasonable number of right-hand sides (from hundreds to a thousand), iterative solvers with efficient preconditioners should be the best alternative. During WIND, we will investigate the GMRES solver equipped with the ORAS (Optimized Restricted Additive Schwarz) domain decomposition preconditioner to solve large-scale Helmholtz type problems (

**Dolean et al., 2015**). Compared to classical Additive Schwarz preconditioner, ORAS makes use of Robin conditions at the interface between subdomains. Local subproblems on each subdomain are processed with sparse direct solver to process efficiently multiple right-hand sides. Our goal is to assess the potential of this approach for both visco-acoustic and visco-elastic wave equation using a finite element discretisation on unstructured tetrahedral mesh.

Ongoing work deals with:

- strong (how the iteration count evolves with the number of subdomains) and weak (how the iteration count grows with frequency) scalabiltiy analysis of the solver.

- accuracy analysis (which polynomial order for FWI application?)

- Efficient multi-rhs management

- Multi-level preconditioner

- Visco-elastic solver

- Acoustic-elastic coupling

Preliminary simulations in the 3D SEG/EAGE Overthrust model at 5, 10 and 20Hz are illustrated in

**Figures 1-3** and in

**Table 1**. The simulations were performed on the

**Occigen** supercomputer of

**CINES**.

| | Cartesian | Tetrahedral |

F(Hz) | #core | #elts(M) | #dofs(M) | #it | T(s) | #elts(M) | #dofs(M) | #it | T(s) |

5 | 384 | 16 | 22 | 167 | 58 | 8 | 11 | 125 | 25 |

10 | 3072 | 131 | 176 | 340 | 121 | 63 | 95 | 253 | 59 |

20 | 12288 | - | - | - | - | 506 | 678 | 438 | 208 |

**Table 1:** Statistics of the simulation. F(Hz): Frequency. #core: number of cores. #elts: number of elements. #dofs: number of degrees of freedom. #it: iteration count. T(s): elapsed time (Courtesy P.-H. Tournier).

This work is supervised by V. Dolean (LJAD), P. Jolivet (ENSEEIHT) and P.-H. Tournier (LJLL).

**Figure 1:** *Cartesian and tetrahedral meshing of the SEG/EAGE Overthrust model (Courtesy P.-H. Tournier).*

**Figure 2:** *Comparison between solutions computed on the Cartesian and tetrahedral meshes(Courtesy P.-H. Tournier) .*

**Figure 3:** Wavefield solution at frequency 20Hz on the tetrahedral mesh (Courtesy P.-H. Tournier)._

**References:**

P.-H. Tournier, P. Jolivet, V. Dolean, H. Aghamiry, S. Operto and S. Riffo, Three-dimensional finite-difference & finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets, *Geophysics*, 87(5), T381--T402, 2022, https://doi.org/10.1190/geo2021-0702.1.