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Fast solvers for time-harmonic Maxwell’s equations in 3D Aruliah, Dhavide Arjunan

Abstract

The speed of iterative solvers for discretizations of partial differential equations (PDEs) is a significant bottleneck in the performance of codes designed to solve large-scale electromagnetic inverse problems. A single data inversion requires solving Maxwell's equations dozens if not hundreds of times. An inherent difficulty in geophysical contexts is that the conductivity and permeability coefficients may exhibit discontinuities spanning several orders of magnitude. Furthermore, in the air, the conductivity effectively vanishes. In standard formulations of Maxwell's equations, the curl operator that dominates the PDE operator leads to strong mixing of field components and illconditioning of linear systems resulting from standard discretizations. The primary objective of this research is to build fast iterative solvers for the forward-modeling problem associated with electromagnetic inverse problems in the frequency domain. Toward this goal, a Helmholtz decomposition of the electric field using a Coulomb gauge condition recasts the PDE problem in terms of scalar and vector potentials. The resulting indefinite system is then stabilized by addition of a vanishing term that lies in the kernel of the dominant curl operator. Finally, an extra differentiation recasts the PDE system in a diagonally-dominant form reminiscent of a "pressure-Poisson" formulation for incompressible fluid flow. The continuous PDE problem obtained is equivalent to the original Maxwell's system but has a structure that is amenable to reliable solution techniques. Using a finite-volume scheme, the PDE is discretized on a staggered grid in three dimensions. The discretization obtained possesses conservation properties typical of finite-volume methods. Furthermore, interface conditions imposed by discontinuities in the material coefficients are sensibly accounted for in deriving the discretization. Although the simple representation of the media on a Cartesian tensor-product grid uses staircase approximations of surfaces of discontinuity of the material coefficients, some analysis and a numerical study demonstrate the suitability of such coarse approximations for diffusive problems. The discretization yields a non-Hermitian sparse linear system of algebraic equations; various preconditioners for Krylov-subspace methods are described, analyzed, implemented, and tested. Of particular interest is a multigrid preconditioner that exploits both the structure of the PDE problem and the availability of well-established solvers for elliptic PDE problems (in particular, Dendy's BOXMG solver). The end result is a robust solver for the forward-modeling equations that can be incorporated within a competitive inverse problem code.

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