Finite element modelling of geophysical electromagnetic data with goal-oriented hr-adaptivity

• Published in: Computational geosciences Vol. 24; no. 3; pp. 1257 - 1283
• Main Authors: Jahandari, Hormoz, MacLachlan, Scott, Haynes, Ronald D., Madden, Niall
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2020; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Coefficients; Computation; Differential equations; Differential media; Earth and Environmental Science; Earth Sciences; Errors; Estimates; Finite element method; Geophysics; Geotechnical Engineering & Applied Earth Sciences; Grid refinement (mathematics); Helmholtz equations; Hydrogeology; Inhomogeneous media; Interfaces; Iterative methods; Mathematical Modeling and Industrial Mathematics; Mathematical models; Modelling; Original Paper; Partial differential equations; Soil Science & Conservation; Surveys; Two dimensional models; Variational methods; Vector potentials; refinement; Mesh adaptivity; Electromagnetics; Finite element
• ISSN: 1420-0597; 1573-1499
• DOI: 10.1007/s10596-020-09944-7

Large discontinuities (jumps) in coefficients often appear when modelling physical problems involving inhomogeneous media. An example of this is the geophysical electromagnetic (EM) problem, where these jumps occur at interfaces which separate regions of (high) conductivity contrasts. These interfaces, along with other problem features, such as singular EM sources, motivate the use of adaptive mesh refinement to improve the efficiency of the solution of forward problems. Also, pointwise observations are made in geophysical EM surveys, which motivates the use of goal-oriented mesh adaptivity. In this work, we study the combined application of
h
- and
r
-refinements for the modelling of geophysical EM data. The proposed
hr
-adaptivity algorithm is thoroughly investigated in 1D, and then extended, and numerically validated, in 2D, using simple examples as well as a realistic model with irregular interfaces. In 1D, the steady-state diffusion and Helmholtz equations, which are commonly solved for the EM scalar and vector potentials, respectively [
16
], constitute the physical partial differential equations (PPDEs). In 2D, the Helmholtz equation is used as the PPDE. Additionally, a real 2D problem with a benchmark model is considered where the transverse electric (TE) mode of Maxwell’s equations is used as the PPDE. The
r
-refinements are based on an equidistribution principle and variational methods, for the 1D and 2D cases, respectively, which lead to mesh PDEs (MPDEs). The coupled PPDE and MPDE are solved in an iterative manner to enhance the accuracy of the PPDE solution by improving the equidistribution of a monitor function based on
a posteriori
error estimates. The results, in 1D, with two hierarchical error–based monitor functions and a residual error–based monitor function display the similarity between these functions in terms of the accuracy that could be achieved. In both 1D and 2D, comparisons are made between global and goal-oriented adaptivity which show the advantage of goal-oriented error estimates in gaining higher accuracy at a target, compared to global error estimates. In 2D, the results also demonstrate the higher efficiency of
hr
-adaptivity compared with pure
h
-refinement, in terms of computation time.