Finite element methods in ab initio electronic structure calculations

We review the application of the finite element (FE) method to ab initio electronic structure calculations in solids. The FE method is a general approach for the solution of differential and integral equations which uses a strictly local, piecewise-polynormal basis. Because the basis is composed of polynomials, the method is completely general and its accuracy is systematically improvable. Because the basis is 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 for parallel implementation. The method thus combines significant advantages of both real-space-grid and basis-oriented approaches, and so is well suited for large, accurate ab initio calculations. We review the construction and properties of the required FE bases and their use in the self-consistent solution of the Kohn-Sham equations of density functional theory.