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Figure 1. Steps in the construction of a finite-element basis. 1) The domain (unit cell volume) is partitioned into subdomains called elements. 2) Polynomial basis functions are defined within each element. 3) Polynomials on neighboring elements are pieced together to form the strictly local piecewise-polynomial basis functions of the method. Because the basis functions are polynomials, the method is completely general and systematically improvable. Because they are strictly local, the method realizes the significant advantages of real-space grid approaches for the solution of large problems.

Finite-Element Method for Large-Scale Ab Initio Electronic-Structure Calculations


John Pask and Philip Sterne

The finite-element method is a powerful and general approach for solving partial differential and integral equations. The solutions of the Schrödinger and Poisson equations constitute the most time-consuming steps in density-functional based electronic-structure calculations. These calculations provide a means of determining materials properties completely from quantum-mechanical first principles (ab initio), with no adjustable parameters; and so provide a robust and system-independent means for understanding and predicting a wide range of properties in diverse materials systems. Many such electronic-structure methods provide accurate results for unit cells consisting of a small number of atoms (see Methods page for examples). Few methods, however, can provide accurate results for larger systems of several hundred atoms or more, a physically important regime where complex surfaces, interfaces, and materials defects can begin to be modeled. The goal of this research is to develop and apply a finite-element based method for large-scale ab initio electronic-structure calculations, extending the range of materials systems accessible by such rigorous, quantum mechanical means.

The finite-element method combines the significant advantages of both basis-oriented and real-space grid-based approaches. The finite-element method is an expansion method which uses a strictly local, piecewise polynomial basis (See Figure 1). Because the method is basis-oriented, it is variational, allows for arbitrarily accurate integrations, and allows for increased efficiency by choosing basis functions based upon physical insight. Because the basis functions are polynomials, the method is completely general and systematically improvable. Because they are strictly local, the method achieves a number of additional advantages with respect to large-scale calculations: the method produces sparse, structured matrices which are well suited to solution by efficient iterative methods and require only O(N) storage, where N is the number of atoms; the basis functions can be concentrated where needed in real-space (where the solution varies most rapidly) in order to increase the efficiency of the representation; all computations are performed directly in real-space, eliminating the need for computationally expensive transforms such as Fourier transforms which can incur large communications costs on large-scale computational platforms; and the method is well suited to parallel implementation. The method therefore combines the advantages of basis-oriented approaches such as the planewave method with the significant advantages with respect to large-scale calculations of real-space grid approaches such as the finite-difference method.

We have developed and implemented a finite-element based approach for the solution of the equations of density functional theory. We have used this approach routinely to calculate positron distributions and lifetimes in support of LLNL's experimental research program on positron annihilation in materials . A recent calculation of a nanoscale copper precipitate in iron involving over 5,000 atoms demonstrates the capacity of the finite-element based approach for such large ab initio calculations. We have incorporated finite-element Schrödinger and Poisson solvers into a fully self-consistent density-functional electronic-structure method employing nonlocal pseudopotentials. Current work focuses on the addition of quantum-mechanical forces, spin-polarization, and spin-orbit coupling; and optimization of iterative solver algorithms and implementations.

REFERENCES
  1. "Finite elements in ab initio electronic-structure calculations," J.E. Pask and P.A. Sterne, in Handbook of Materials Modeling, S. Yip (ed.), p. 423, Springer, Dordrecht, 2005. (Review)
  2. "Finite element methods in ab initio electronic structure calculations," J.E. Pask and P.A. Sterne, Modelling Simul. Mater. Sci. Eng. 13, R71 (2005). (Review)
  3. "Real-space formalism for the electrostatic potential and total energy of solids," J.E. Pask and P.A. Sterne, Phys. Rev. B, 71, 113101 (2005).
  4. "Finite-element methods in electronic-structure theory," J.E. Pask, B.M. Klein, P.A. Sterne, and C.Y. Fong, Comput. Phys. Commun. 135, 1 (2001).
  5. "Calculation of positron observables using a finite element-based approach," P.A. Sterne, J.E. Pask, and B.M. Klein, Appl. Surf. Sci. 149, 238 (1999).
  6. "Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach," J.E. Pask, B.M. Klein, C.Y. Fong, and P.A. Sterne, Phys. Rev. B 59, 12352 (1999).
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