Boundary element methods (BEM) are often used for complex 3-D capacitance extraction because of their efficiency, ease of data preparation, and automatic handling of open regions. BEM capacitance extraction, however, yields a dense set of linear equations that makes solving via direct matrix methods such as Gaussian elimination prohibitive for large problem sizes. Although iterative, multipole-accelerated techniques have produced dramatic improvements in BEM capacitance extraction, accurate sparse approximations of the electrostatic potential matrix are still desirable for the following reasons. First, the corresponding capacitance models are sufficient for a large number of analysis and design applications. Moreover, even when the utmost accuracy is required, sparse approximations can be used to precondition iterative solution methods. In this paper, we propose a definition of electrostatic potential that can be used to formulate sparse approximations of the electrostatic potential matrix in both uniform and multilayered planar dielectrics. Any degree of sparsity can be obtained, and unlike conventional techniques which discard the smallest matrix terms, these approximations are provably positive definite for the troublesome cases with a uniform dielectric and without a groundplane.
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