We describe a set of best practices for scientific software development, based on research and experience, that will improve scientists' productivity and the reliability of their software.

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We consider the rapid simulation of three-dimensional electromagnetic problems in geophysical parameter regimes, where the conductivity may vary significantly and the range of frequencies is moderate. Toward developing a multigrid preconditioner, we present a Fourier analysis based on a finite-volume discretization of a vector potential formulation of time-harmonic Maxwell's equations on a staggered grid in three dimensions. We prove grid-independent bounds on the eigenvalue and singular value ranges of the system obtained using a preconditioner based on exact inversion of the dominant diagonal blocks of the non-Hermitian coefficient matrix. This result implies that a preconditioner that uses single multigrid cycles to effect inversion of the diagonal blocks also yields a preconditioned system with an 2 -condition number bounded independent of the grid size.We then present numerical examples for more realistic situations involving large variations in conductivity (i.e., jump discontinuities). Block-preconditioning with one multigrid cycle using Dendy's BOXMG solver is found to yield convergence in very few iterations, apparently independent of the grid size. The experiments show that the somewhat restrictive assumptions of the Fourier analysis do not prohibit it from describing the essential local behavior of the preconditioned operator under consideration. A very efficient, practical solver is obtained.

We present a solution method for solving electromagnetic problems in three dimensions in parameter regimes where the quasi-static approximation applies and the permeability is constant. Firstly, by using a potential formulation with a Coulomb gauge, we circumvent the ill-posed problem in regions of vanishing conductivity, obtaining an elliptic, weakly coupled system of differential equations. The system thus derived is strongly elliptic, which leads to reliable discretizations. Secondly, we derive a robust finitevolume discretization. Thirdly, we solve the resulting large, sparse algebraic systems using preconditioned Krylov-space methods. A particularly efficient algorithm results from the combination of BICGSTAB and a block preconditioner using an incomplete LU-decomposition of the dominant system blocks only. We demonstrate the efficacy of our method in several numerical experiments. § 1 Math. Models Methods Appl. Sci. 2001.11:1-21. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/18/15. For personal use only. 2 D. A. Aruliah et al.

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