An efficient hybrid direct-iterative solver for three-dimensional higher-order edge-based finite element simulation for magnetotelluric data in anisotropic media

This work investigates the use of the high-order finite element method (FEM) for 3D magnetotelluric (MT) modeling in anisotropic media. The high-order FE is more accurate but also more difficult than the linear elements. Numerically, the resulting linear system is larger and denser than the linear case with the same grid. In 3D cases, finite element discretization of Maxwell's equations often leads to large but sparse linear systems of equations, requiring effective methods to solve them. However, for the domain-frequency Maxwell's equations, the discrete rotor operator with a large kernel hampers the convergence of common iterative solvers. Robust and effective solvers are needed to solve 3D electromagnetic (EM) problems. In this study, we presented a robust direct-iterative solver based on a block-diagonal preconditioner. The proposed solver is efficient for high-order FE modeling with arbitrary anisotropy. We implemented the algorithm in a high-level Julia language, which is easy to read, maintain, and extend. Numerical examples show that the high-order elements produce a more accurate solution on a coarse grid than the linear case with the same grid. Compared with the direct solver, the hybrid direct-iterative solver reduces the computation time and memory usage. In addition, the solver only needs a few iterations to converge compared with the traditional BCISGTAB solver with an SSOR preconditioner. Our modeling results suggested the FGMRES solver with the block-diagonal preconditioner is quite effective for 3D MT high-order FE modeling.