Abstract
We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.
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The data underlying this article can be shared upon reasonable request. Since the modelling code is to be embedded in a larger software package, it cannot be made freely available.
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Acknowledgements
We thank Paula Rulff for providing the results for the 3D model. This work was partly funded by Uppsala’s Center for Interdisciplinary Mathematics (CIM). The computations were enabled by resources in project SNIC 2021/22-883 provided by the Swedish National Infrastructure for Computing (SNIC) at UPPMAX, partially funded by the Swedish Research Council through grant agreement no. 2018-05973.
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Appendix A: Accuracy for the 3D model compared to finite element solutions
Appendix A: Accuracy for the 3D model compared to finite element solutions
In the following, we present results from numerical simulations obtained by our iterative framework and by a finite element code developed by [74], that uses a direct solver. In particular, we deploy twelve receivers across the target body of the 3D model (Problem 2) in cross-line configuration, meaning the receiver line is perpendicular to the transmission direction. The receiver line is offset by by 3.5 km in x-direction. The receivers range from − 3 to − 0.5 km and from 0.5 to 3 km in y-direction at intervals of 500 m. Figure 5 shows the electric and magnetic fields excited by line source transmitting at 10 Hz for both the iterative framework and the finite element code denoted by SEM and FEM respectively. The responses for both numerical schemes are in good agreement with each other. Further, Table 12 displays the mean relative deviations of the electric and magnetic field components across all receivers thus confirming the observed agreement. Note, that the relative deviations are given for frequencies of 10 and 100 Hz.
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Weiss, M., Neytcheva, M. & Kalscheuer, T. Iterative solution methods for 3D controlled-source electromagnetic forward modelling of geophysical exploration scenarios. Comput Geosci 27, 81–102 (2023). https://doi.org/10.1007/s10596-022-10182-2
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DOI: https://doi.org/10.1007/s10596-022-10182-2