We present an approach to solid-state electronic-structure calculations based
on the finite-element method. In this method, the basis functions are strictly
local, piecewise polynomials. Because the basis is composed of polynomials, the
method is completely general and its convergence can be controlled
systematically. Because the basis functions are strictly local in real space,
the method allows for variable resolution in real space; produces sparse,
structured matrices, enabling the effective use of iterative solution methods;
and is well suited to parallel implementation. The method thus combines the
significant advantages of both real-space-grid and basis-oriented approaches
and so promises to be particularly well suited for large, accurate ab initio
calculations. We develop the theory of our approach in detail, discuss
advantages and disadvantages, and report initial results, including the first
fully three-dimensional electronic band structures calculated by the method.Comment: replacement: single spaced, included figures, added journal referenc