Several ways to achieve robustness when solving wave propagation problems

Wave propagation problems are notoriously difficult to solve. Timeharmonic problems are especially challenging in mid and high frequency regimes. The main reason is the oscillatory nature of solutions, meaning that the number of degrees of freedom after discretisation increases drastically with the wave number, giving rise to large complex-valued problems to solve. Additional difficulties occur when the problem is defined in a highly heterogeneous medium, as is often the case in realistic physical applications. For time-discretised problems of Maxwell type, the main challenge remains the significant kernel in curl-conforming spaces, an issue that impacts on the design of robust preconditioners. This has already been addressed theoretically for a homogeneous medium but not yet in the presence of heterogeneities. In this review we provide a big-picture view of the main difficulties encountered when solving wave propagation problems, from the first step of their discretisation through to their parallel solution using two-level methods, by showing their limitations on a few realistic examples. We also propose a new preconditioner inspired by the idea of subspace decomposition, but based on spectral coarse spaces, for curl-conforming discretisations of Maxwell’s equations in heterogeneous media. Niall Bootland University of Strathclyde, Dept. of Maths and Stats, e-mail: niall.bootland@strath.ac.uk Victorita Dolean University of Strathclyde, Dept. of Maths and Stats and University Côte d’Azur, CNRS, LJAD e-mail: work@victoritadolean.com Pierre Jolivet University of Toulouse, CNRS, IRIT, e-mail: pierre.jolivet@enseeiht.fr Frédéric Nataf Sorbonne Université, CNRS, LJLL, e-mail: frederic.nataf@sorbonne-universite.fr Stéphane Operto University Côte d’Azur, CNRS, Géoazur, e-mail: stephane.operto@geoazur.unice.fr Pierre-Henri Tournier Sorbonne Université, CNRS, LJLL e-mail: tournier@ann.jussieu.fr 1 ar X iv :2 10 3. 06 02 5v 1 [ m at h. N A ] 1 0 M ar 2 02 1

[1]  Victorita Dolean,et al.  Large-scale frequency-domain seismic wave modeling on {\it{h}}-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners , 2020 .

[2]  Victorita Dolean,et al.  Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners , 2020, ArXiv.

[3]  Mark Ainsworth,et al.  Optimally Blended Spectral-Finite Element Scheme for Wave Propagation and NonStandard Reduced Integration , 2010, SIAM J. Numer. Anal..

[4]  Jinchao Xu,et al.  Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces , 2007, SIAM J. Numer. Anal..

[5]  Ivan G. Graham,et al.  Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption , 2017, Math. Comput..

[6]  S. Nepomnyaschikh.  Mesh theorems on traces, normalizations of function traces and their inversion , 1991 .

[7]  Martin J. Gander,et al.  A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods , 2016, SIAM Rev..

[8]  Roelof Versteeg,et al.  The Marmousi experience; velocity model determination on a synthetic complex data set , 1994 .

[9]  Jens Markus Melenk,et al.  Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation , 2011, SIAM J. Numer. Anal..

[10]  Hua Xiang,et al.  A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps , 2011, SIAM J. Sci. Comput..

[11]  Frédéric Nataf,et al.  A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator , 2014, J. Comput. Appl. Math..

[12]  Patrick R. Amestoy,et al.  Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea , 2016 .

[13]  Eero Vainikko,et al.  Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption , 2015, Math. Comput..

[14]  Frédéric Nataf,et al.  Spillane, N. and Dolean Maini, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R. (2013) Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps , 2018 .

[15]  Haijun Wu,et al.  Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number , 2014, SIAM J. Numer. Anal..

[16]  Designing an exploration scale OBN: Acquisition design for subsalt imaging and velocity determination , 2019, SEG Technical Program Expanded Abstracts 2019.

[17]  Martin J. Gander,et al.  Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods , 2012 .

[18]  E. A. Spence,et al.  Domain Decomposition with Local Impedance Conditions for the Helmholtz Equation with Absorption , 2018, SIAM J. Numer. Anal..

[19]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[20]  Théo Mary,et al.  Block Low-Rank multifrontal solvers: complexity, performance, and scalability. (Solveurs multifrontaux exploitant des blocs de rang faible: complexité, performance et parallélisme) , 2017 .

[21]  Frédéric Nataf,et al.  Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps , 2012, Comput. Methods Appl. Math..