Implementation of the spectral element method and iterative solution techniques for 3D controlled-source electromagnetic modelling

Sammanfattning: Controlled-source electromagnetic methods are imaging techniques applied at/on the surface of the Earth which record the electromagnetic field in order to assess the electrical conductivity distribution of the Earth’s subsurface, along with anomalies in this material property, as well as to characterise and interpret structures in Earth’s crust. Extracting information from the recorded electromagnetic data requires inverse modelling algorithms. A key component of inverse modelling is the reverse process, namely, forward modelling, that is numerically simulating the electric and magnetic fields for a computerised model using discretisation techniques, appropriate boundary conditions and efficient solution methods. The forward problem for 3D models constitutes a computationally challenging and expensive task. Hence, cost effective forward modelling algorithms are crucial to limit the computational resources in terms of time and memory needed for realistic large-scale models. This thesis investigates numerical aspects of forward modelling problems encountered in frequency-domain land-based controlled-source electromagnetic methods. It presents the mathematical framework and the implementation of a 3D forward modelling algorithm based on the spectral element method which merges the high accuracy and the favourable convergence properties of the spectral method with the geometrical flexibility of the finite element method. Further, it introduces the total field formulation to the spectral element method. On the basis of the developed code, a cost-effective, efficient and robust iterative algorithm using the generalised conjugate residual method preconditioned with an efficient block-based PREconditioner for Square Blocks (PRESB) is implemented in order to reduce time and memory requirements of the modelling routine. Two core findings of this thesis, is that the efficiency of the iterative algorithm hinges on the efficiency of the solution of the two inner linear systems of equations which arise through applying preconditioner PRESB, and that the auxiliary-space Maxwell preconditioner AMS represents a highly efficient preconditioner for the aforementioned inner systems. To expand the scope of application of the iterative algorithm, this thesis investigates a block lower-triangular preconditioner based on either the Schur complement or its element-by-element approximation and finds that the AMS-preconditioned solver used to solve systems with the Schur complement or its approximation is not well suited for either. Only the Schur complement based preconditioner is shown to work for relevant problems.