Geophysical modelling of 3D electromagnetic diffusion with multigrid

Abstract

Abstract. The performance of a multigrid solver for time-harmonic electromagnetic problems in geophysical settings was investigated. With the low frequencies used in geophysical surveys for deeper targets, the light-speed waves in the earth can be neglected. Diffusion of induced currents is the dominant physical effect. The governing equations were discretised by the Finite-Integration Technique. The resulting set of discrete equation was solved by a multigrid method.The multigrid method provided excellent convergence with constant grid spacings, but not on stretched grids. The slower convergence rate of the multigrid method could be compensated by using bicgstab2, in which case multigrid acted as a preconditioner. Still, the overall performance was less than satisfactory with substantial grid stretching.

Controlled-source electromagnetics for reservoir monitoring on land

“…Next, we review the multigrid solver (Mulder, 2006(Mulder, , 2008. Because the standard approach breaks down on stretched grids, a variant based on semi-coarsening and line relaxation was designed (Jönsthövel et al, 2006;Mulder, 2007).…”

“…Mulder (2006Mulder ( , 2008 presented a numerical method for solving the system of equations 2.1 in the frequency domain. The Finite-Integration Technique (Weiland, 1977) provided a finite-volume discretization of the equations.…”

“…For an assessment of the method's performance, we refer to earlier papers (Mulder, 2006(Mulder, , 2008Mulder et al, 2008). The method is matrix-free: we never explicitly form the large sparse linear matrix that describes the discretised problem, but only evaluate its action on the latest estimate of the solution, thereby reducing storage requirements.…”

“…We repeated the artificial test problem of Mulder (2008). It is based on eigenfunctions that obey the PEC boundary conditions.…”

“…For general resistivity models, a finite-difference, finite-volume, or finite-element discretization of the governing equations requires an efficient solver. The multigrid method (Mulder, 2006(Mulder, , 2008 allows for a reasonably fast solution of the discretized equations when used as a preconditioner for BiCGStab2 (van der Vorst, 1992;Gutknecht, 1993), a conjugate-gradient iterative method. With stronger grid stretching, we can apply a more robust multigrid variant based on semicoarsening and line relaxation (Mulder, 2007).…”