Iterative solver with folded preconditioner for finite element simulation of magnetotelluric fields

Abstract

We propose an iterative solver with a folded preconditioner for large sparse systems of linear algebraic equations (SLAEs), which are obtained as a result of a vector finite element approximation of electromagnetic fields in magnetotelluric problems. The solver is based on the computational scheme of the Conjugate A-Orthogonal Conjugate Residual method (COCR), diagonal preconditioner, and residual smoothing. To build a finite element approximation, the primary-secondary field approach is used. To construct a preconditioner, we use a matrix, the elements of which are the weights of the representation of gradients of nodal basis functions as a linear combination of edge basis functions. For Dublin Test Model 1 in a wide frequency range from 0.0001 Hz to 10 Hz, we compare the accuracy and computational efficiency of our code with the results of other authors, as well as compare the computational efficiency of the proposed diagonal folded preconditioned (DFP) COCR solver with other solvers. It is shown that the apparent resistivities obtained on the coarsest mesh with our code differ by no more than a few percent from the solution on a finer mesh and from some solutions obtained by other authors using the finite element, finite difference, and integral equation methods. The computational time required for our code is 8 times less than the minimum time presented by other authors (with a corresponding recalculation of the performance of the computers used). It is also shown that the DFP COCR solver practically does not increase the number of iterations at low frequencies and its computational time is about 20 times less than the time required for the COCR solver with a standard diagonal preconditioner. A comparison of the DFP COCR solver with the Parallel Direct Sparse solver (PARDISO) shows that even at low and medium frequencies, the computational time of the DFP COCR solver is about 20 times less than the computational time of the PARDISO, and at high frequencies this difference reaches already 50 times.