Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

We present an approach to solid-state electronic-structure calculations bas ed on the finite-element method. In this method, the basis functions are st rictly local, piecewise polynomials. Because the basis is composed of polyn omials, the method is completely general and its convergence can be control led systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real space; produces sp arse, structured matrices, enabling the effective use of iterative solution methods; and is well suited to parallel implementation. The method thus co mbines the significant advantages of both real-space-grid and basis-oriente d approaches and so promises to be particularly well suited for large, accu rate nb initio calculations. We develop the theory of our approach in detai l, discuss advantages and disadvantages, and report initial results, includ ing electronic band structures and details of the convergence of the method . [S0163-1829(99)03515-8].