Finite‐difference solution of the three‐dimensional electromagnetic problem using divergence‐free preconditioners

2006 ◽  
Author(s):  
Mike Zaslavsky ◽  
Sofia Davydycheva ◽  
Vladimir Druskin ◽  
Aria Abubakar ◽  
Tarek Habashy ◽  
...  
Keyword(s):  
Finite Difference ◽  
Three Dimensional ◽  
Finite Difference Solution ◽  
Divergence Free ◽  
Difference Solution

Three‐dimensional induction logging problems, Part 2: A finite‐difference solution

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 484-491 ◽  
Author(s):  
Gregory A. Newman ◽  
David L. Alumbaugh
Keyword(s):  
Finite Difference ◽  
Electric Field ◽  
Krylov Subspace ◽  
Inverse Operator ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Linear System Of Equations ◽  
Order Of Magnitude ◽  
Finite Difference Solution ◽  
Difference Solution

A 3‐D finite‐difference solution is implemented for simulating induction log responses in the quasi‐static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite‐difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl‐free and divergence‐free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.

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