Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting Maxwell's system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green's functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method. It can be generalized to 3D problems and to other classes of linear problems, including acoustics and elasticity.
Boundary equation methods have a long history, with practical applications dating back to the 1960s. An interesting historical account given by Cheng & Cheng [1] includes the work on wave scattering and radiation in 1962-1967 by Friedman & Shaw, Chen & Schweikert, Banaugh & Goldsmith, Mitzner, and others [2]- [8]. In eletromagnetics, boundary integral techniques became very popular due primarily to Harrington's work published in 1967-68 [9], [10] (see also [11]- [15]).
In traditional boundary integral methods, linear boundary value problems of field analysis are transformed into integral equations with respect to equivalent sources residing on the boundaries. In the simplest example of capacitance calculation [9], [10], the distributed charge density on conducting plates becomes the principal unknown variable. By equating the Coulomb potential of that charge to the given potential of the conductors, one obtains an integral equation. It can then be discretized using variational techniques (moment methods), collocation and Galerkin methods being particular cases of those.
As all numerical methods, boundary integral techniques do carry some trade-offs. Their key advantage is the lower dimensionality of the problem: 3D analysis is reduced to equivalent sources on 2D boundaries and 2D analysis -to 1D contours. Another advantage is a natural treatment of unbounded problems (e.g. wave scattering and radiation), without the artificial domain truncation unavoidable in differential methods such as finite difference (FD) schemes and the Finite Element Method (FEM).
Integral equation methods have, in general, two major disadvantages. First, the matrices of the discrete system are almost always full. This is due to the fact that a source at any point on the boundary contributes to the field at all other points. In contrast, FD and FE matrices are sparse, with very efficient system solvers available (iterative: multilevel methods, incomplete factorization and other effective preconditioners; direct: minimum degree, nested dissection and others; see e.g. [16] and references there). Cases where Green’s functions decay rapidly in space, giving rise to quasi-sparse integral equations, are exceptional (e.g. periodic structures in the electromagnetic band gap regime [17]).
Another disadvantage is that the integral kernels in field analysis are singular. At the surface points, the kernel singularity can usually be handled analytically, and the fields remain bounded as long as the surfaces are smooth. However, for points in the vicinity of the surface, the evaluation of the integral is problematic, as analytical expressions are usually unavailable and numerical quadratures require extreme care. The same is true for two adjacent surfaces with a narrow gap in between.
Significant progress in Fast Multipole Methods (FMM) [18], [19], [20], [21], [22] has helped to alleviate the first disadvantage of boundary methods. FMM accelerates the computation of fields due to distributed sources -or equivalently, matrixvector multiplications for the dense system matrices.
The second disadvantage is more difficult to overcome. Singular kernels are inherent in boundary integral methods because the fields of point sources are unbounded. However, a drastic change in the computational procedure leads to a singularity-free method; this is accomplished by reversing the sequence of stages in the boundary techniques. The standard sequence is
The alternative sequence is
Discretization of the differential problem is performed on a regular grid and yields an FD scheme. This scheme is converted -as explained in the remainder of the paper -to a boundary problem that involves discrete fundamental solutions (Green’s functions) on the grid. Discrete Green’s functions, unlike their continuous counterparts, are always nonsingular. This general idea is not new. In fact, there are two related but independently developed methodologies for boundary difference equations. The first one, put forward and thoroughly studied by Ryaben’kii, Reznik, Tsynkov and others [34], [35], [36], [37], is known as the method of difference potentials and can be viewed as a discrete analog of the Calderon projection operators in functional analysis [34].
The second methodology, called boundary algebraic equations by Martinsson & Rodin [23], is at least 50 years old (Saltzer [25]) and is a discrete analog of first-or second-order Fredholm boundary integral equations for potential problems [23].
In comparison with [23], the method of this paper has several novel features. First, the paper deals -to my knowledge, for the first time -with boundary difference equations for electromagnetic wave scattering. In [23], a simple model problem is considered: the Laplace equation (e.g. electrostatics or heat transfer) in a homogeneous domain with Dirichlet boundary conditions; the focus of [23] is on the mathematical analysis of the respective boundary difference operators, their spectral
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