Abstract—Some important problems of numerical modeling of electromagnetic fields are examined. A modified decomposition solution of geoelectric problems based on the Shwartz alternating method is proposed, which ensures a fast convergence of the iterative process and decreases the number of subproblems to be solved. Moreover, this algorithm minimizes the possible dimensions of the grid region in which one of the subproblems is solved numerically. Simple models are used for examining the convergence of the iterative process described in the paper. Decomposition algorithms for problems of arbitrary dimension arc constructed, and efficient methods for matching numerical and analytical solutions of the subproblems are developed. Results of a semi-analytical solution of a magnetotelluric sounding 2-D problem and data on the convergence rate of the iterative process are presented

 

INTRODUCTION

A typical model of the medium in geoelectrics is a local heterogeneity embedded in a structurally rather simple, unbounded medium with a regular distribution of its properties. Typically, this medium is horizontally layered (a.c. or d.c. electromagnetic sounding) or cylin- drically layered (logging or surface electrical explora­tion problems with the field source in a borehole). Solu­tions in terms of such relatively simple models can be found analytically and are represented as improper (Fourier or Fourier-Bessel) integrals. The problem is to study the influence of a structurally complex local het­erogeneity on the field in a layered medium.

The application of general numerical (finite-differ­ence or finite-element) methods in unbounded regions requires the solution of several problems, the most important of which are the following:

(1)   The unbounded region should be replaced by a finite region. How is it to be chosen?

(2)    Boundary conditions should be specified not at infinity but at the boundary of the grid region. How can this be done if the boundary values will be known only after the problem is solved?

Both problems can be solved via algorithms based on the Shwartz alternating method. Decomposition of complex problems into several simpler subproblems and matching of their solutions by the Shwartz alternat­ing method are discussed in [Kantorovich and Krylov, 1962; Zavadskii, 1972; Yudin, 1982; Vanyan etal., 1984]. A modification of the Shwartz algorithm appli­cable to geoelectric problems is known as the decom­position alternating method (DAM) [Yudin, 1985]. One of the numerous variants of decomposition is the global decomposition algorithm (GDAM). It essentially con­sists in successive solution of external and internal boundary-value problems interconnected via boundary conditions in the iterative process. In order that the problems can be connected, the regions in which the problems involved in the Shwartz iterative process are solved should have a nonempty intersection. In this intersection, the solution of the problem and the condi­tions at the boundary of the grid region are constructed simultaneously. The convergence rate of successive approximations to the sought-for solution depends on the extent to which the regions overlap.

Our decomposition approaches to the solution of forward problems of mathematical physics based on the Shwartz iterative process found application in the numerical modeling of contamination dissemination in the atmosphere [Filatov etal., 2001; Alexandrov and Filatov, 2002].

In conjunction with the numerical solution of boundary-value problems by the Galerkin wavelet method [Yudin et at., 2001], the GDAM makes it pos­sible to handle problems that arise when boundary con­ditions are approximated in the wavelet basis with a large number of coefficients of the fast discrete wavelet transformation filter [Yudin et al., 2002].

One of the shortcomings of the global decomposi­tion algorithm is the need to solve several external boundary-value problems [Yudin, 1985]. The purpose of the GDAM modification discussed in this paper is to achieve a high convergence rate of the Shwartz iterative process by maximizing the overlapping area of the regions, decreasing the number of subproblems to be solved, and minimizing the size of the region in which the problem is solved numerically. Basic ideas underly­ing this approach are explained in [Yudin and Yudin, 2002].