Combined Use of BEM and FEM
The BEM divides only the boundary of the region under consideration. Thus, it reduces the dimensionality of the problem to be solved and the number of equations in the system. This advantage becomes more significant for 3-D problems, which are often encountered in geophysics. However, the BEM has drawbacks. For example, if the region has a complex distribution of physical properties, it is difficult to combine the boundary integral equations corresponding to each homogeneous medium and solve the resulting system.
In contrast, when the FEM deals with infinite region or 3-D problems, the number of nodes needed increases greatly; thus, a larger amount of computer memory is required. However, the FEM automatically satisfies the internal boundary condition, so it does not treat each body individually. Therefore, the FEM has a decided advantage in solving geophysical problems with complex distributions of physical properties
The combined use of the BEM and the FEM can be applied to better simulate a 3-D infinite region as well as complex distributions of physical properties. This chapter introduces the basic concept of the combined use. Here we use the problem of the 3-D electric field produced by a point source.
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The Boundary Element Method in Geophysics
The boundary element method (BEM) divides only the boundaries of the region under investigation into elements, so it diminishes the dimensionality of the problem, e.g., the 3D problem becomes a 2D problem, and the 2D problem becomes a 1D problem. This simplifies inputting the model into a computer and greatly reduces the number of algebraic equations. The advantage of this is even more evident for some 3D and infinite regional problems that often are encountered in geophysics. Originally published in China, this well-organized book is likely the most comprehensive work on the subject of solving applied geophysical problems. Basic mathematical principles are introduced in Chapter 1, followed by a general yet thorough discussion of the BEM in Chapter 2. Chapters 3 through 7 introduce the applications of BEM to solve problems of potential-field continuation and transformation, gravity and magnetic anomalies modeling, electric resistivity and induced polarization field modeling, magnetotelluric modeling, and various seismic modeling problems. Finally, in Chapter 8, a brief discussion is provided on how to incorporate the BEM and the finite-element method (FEM) together. In each chapter, detailed practical examples are given, and comparisons to both analytic and other numerical solutions are presented. This is an excellent book for numerically oriented geophysicists and for use as a textbook in numerical-analysis classes.