Abstract
This chapter discusses fast and efficient numerical solvers. Fast Elliptic Solvers (FES) are defined as solvers that produce the solution in s operations per unknown, where s is a constant or a very slowly logarithmically growing function of the number of unknowns. The capacitance matrix method (CMM) extends the usefulness of such solvers to nonrectangular regions. An iterative variant of CMM developed makes use of the fact that only a product of the capacitance matrix C and a given vector is required and this product is obtained essentially at the cost of a Fast Solver. The discrete potential theory guarantees that the conjugate gradient iteration with a properly constructed matrix C converges almost independently of the size of the mesh. In such implicit CMM, a solution is obtained at a cost proportional to several calls of a FES. Therefore, CMM is regarded as satisfying the definition of FES, with the reservation that the constant s is large, that is, of one order higher than for the solvers on rectangular regions.
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