Comparison of Iterative and Direct Solvers for Finite Element Linear Systems in 3D Induction Logging

Abstract

Numerical modeling of three-dimensional time-harmonic electro magnetics in the problems of high-frequency induction logging by vector finite element method is considered. Electromagnetic field in the frequency range from 0.875 to 14 MHz is excited by an ungrounded loop of the VIKIZ induction logging tool placed in a horizontally layered medium with three-dimensional inhomogeneities. The curl-curl vector partial differential equation with displacement currents taken into account is used. Edge-based vector basis functions on parallelepipedal elements are used. The obtained finite element systems contain from 47804 to 612122 complex-valued degrees of freedom. Both direct (PARDISO) and iterative complex-valued solvers (COCG, COCR, BICGCR) are used to solve the corresponding systems of linear algebraic equations. The advantage of iterative methods over direct methods is shown for the obtained finite element matrices at the frequency range from 1.75 to 14 MHz. The computational time was decreased significantly for the matrices corresponding to the high frequencies. COCR and BICGCR converged at approximately equal numbers of iterations but COCG was a bit slower. The PARDISO solver has shown no benefit over all considered iterative solvers.