Finite element three-dimensional direct current resistivity modelling: Accuracy and efficiency considerations

The finite element method is a powerful tool for 3-D DC resistivity modelling and inversion. The solution accuracy and computational efficiency are critical factors in using the method in 3-D resistivity imaging. This paper investigates the solution accuracy and the computational efficiency of two common element-type schemes: Trilinear interpolation within a regular 8-node solid parallelepiped, and linear interpolations within six tetrahedral bricks within the same 8-node solid block. Four iterative solver's based on the pre-conditioned conjugate gradient method (SCG, TRIDCG, SORCG and ICCG), and one elimination solver called the banded Choleski factorization are employed for the solutions. The comparisons of the element schemes and solvers were made by means of numerical experiments using three synthetic models. The results show that the tetrahedron element scheme is far superior to the parallelepiped element scheme, both in accuracy and computational efficiency. The tetrahedron element scheme may save 43 per cent storage for an iterative solver, and achieve an accuracy of the maximum relative error of < 1 per cent with an appropriate element size. The two iterative solvers, SORCG and ICCG, are suitable options for 3-D resistivity computations on a PC, and both perform comparably in terms of convergence speed in the two element schemes. ICCG achieves the best convergence rate, but nearly doubles the total storage size of the computation. Simple programming codes for the two interative solvers are presented. We also show that a fine grid, which doubles the density of a coarse grid, will require at least 2⁷ = 128 times as much computing time when using the banded Choleski factorization. Such an increase, especially for 3-D resistivity inversion, should be compared with SORCG and ICCG solvers in order to find the computationally most efficient method when dealing with a large number of electrodes.