A novel hybrid finite element – Finite difference approach for 3D Magnetotelluric modeling

Abstract

A new method is developed to solve the Magnetotelluric (MT) forward problem in three dimensions (3D). The finite-differences (FD) and the finite-elements (FE) methods are considered to be popular numerical techniques to solve the partial differential equations in electromagnetics. The FD approach is known to be easily applicable to a given problem while the FE method is much more flexible when a topography is present, however, it is also computationally more expensive to solve the linear matrix equations for the FE due to the number of non-zero entries it populates when compared to the FD’s. For that reason, a new algorithm is offered to mix the FE and FD in a single mesh using hexahedra and rectangular prisms in 3D. By doing that, it is possible to take advantage of the beneficial aspects of these two methods. It was found out that the newly developed hybrid approach (HYB) is almost as fast as the FD method while being very memory efficient which is an important aspect due to the limited memory space of a GPU which is employed to solve the systems iteratively. It was found that the HYB method is as accurate as the FE method while it is a more reasonable choice for larger models due to the solution speed of the FE method as it does not scale well with the increased model size. On average, a speed-up of 1.6 is found between the FE and HYB methods with the three models tested and it increases with the increasing model size. Using a GPU during the iterative solution made it possible to solve the linear systems in seconds and with the help of the HYB method, the memory consumption was found to be less than half of what the FE method needed. This ultimately means that much larger models can be simulated on the limited memory that a GPU has. Finally, the technique shown to implement the HYB approach makes the deployment of it to existing FE codes so much easier.