Computers and Mathematics with Applications 49 (2005) 1285-1294. doi:10.1016/j.camwa.2004.02.012Received by publisher: 0000-01-01Harvest Date: 2016-01-04 12:22:56DOI: 10.1016/j.camwa.2004.02.012Page Range: 1285-129

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SUMMARY We present a parallel and high-order Nédélec finite element solution for the marine controlled-source electromagnetic (CSEM) forward problem in 3-D media with isotropic conductivity. Our parallel Python code is implemented on unstructured tetrahedral meshes, which support multiple-scale structures and bathymetry for general marine 3-D CSEM modelling applications. Based on a primary/secondary field approach, we solve the diffusive form of Maxwell’s equations in the low-frequency domain. We investigate the accuracy and performance advantages of our new high-order algorithm against a low-order implementation proposed in our previous work. The numerical precision of our high-order method has been successfully verified by comparisons against previously published results that are relevant in terms of scale and geological properties. A convergence study confirms that high-order polynomials offer a better trade-off between accuracy and computation time. However, the optimum choice of the polynomial order depends on both the input model and the required accuracy as revealed by our tests. Also, we extend our adaptive-meshing strategy to high-order tetrahedral elements. Using adapted meshes to both physical parameters and high-order schemes, we are able to achieve a significant reduction in computational cost without sacrificing accuracy in the modelling. Furthermore, we demonstrate the excellent performance and quasi-linear scaling of our implementation in a state-of-the-art high-performance computing architecture.

The objective of this paper is to present the subject of wavelets from a filter-theory perspective, which is quite familiar to electrical engineers. Such a presentation provides both physical and mathematical insights into the problem. It is shown that taking the discrete wavelet transform of a function is equivalent to filtering it by a bank of constant-Q filters, the non-overlapping bandwidths of which differ by an octave. The discrete wavelets are presented, and a recipe is provided for generating such entities. One of the goals of this tutorial is to illustrate how the wavelet decomposition is carried out, starting from the fundamentals, and how the scaling functions and wavelets are generated from the filter-theory perspective. Examples are presented to illustrate the class of problems for which the discrete wavelet techniques are ideally suited. It is interesting to note that it is not necessary to generate the wavelets or the scaling functions in order to implement the discrete wavelet transform. Finally, it is shown how wavelet techniques can be used to solve operatodmatrix equations. It is shown that the "orthogonal-transform property" of the discrete wavelet techniques does not hold in numerical computations.'This is part 1 of a two-part article. Part 2, which treats the continuous case, will appear in the December issue. IntroductionMany books and numerous papers have been published describing wavelets. It is not possible for us to include all the references. Selected references [ 1-31 have been chosen to illustrate where additional materials are available. No attempt has been made to provide the earliest reference material.Wavelets are a set of functions that can be used effectively in a number of situations, to represent natural, highly transient phenomena that result from a dilation and shift of the original waveform. For example, when a pulse propagates through a layered medium, due to dispersion and for different electrical properties of the layers, the pulse is dilated and delayed, due to the finite velocity of propagation. The application of wavelets (which literally translates from ondellets in French into English as small waves) was first made in the area of geophysics [4], in 1980, by the French geophysicist J. Morlet, of Elf-Aquitane. A good history fi-om the mathematical perspective is available in the special issue of the IEEE Proceedings [5].In electrical engineering [6-81, however, wavelets have been popular for some time, under the various names of multirate sampling, quadrature-mirror filters, and so on. Since the majority of the readers of this article are assumed to have an electrical-engineering background, it will perhaps be useful to describe the methodology

A convergence study of a non-standard Schwarz domain decomposition method for finite element mesh truncation in electromagnetics has been carried out. The original infinite domain is divided into two overlapping domains. The interior finite domain is modeled by finite elements and the exterior infinite domain by an integral equation representation of the field. A numerical study of the spectrum of the iteration matrix for nonconvex mesh truncation boundaries is performed. The projection of the error between two consecutive iterations onto the eigenvector space of the iteration matrix is performed. The numerical results explain the convergence behavior of the Schwarz iterations.