Partition-Of-Unity Finite Elements For Quantum Mechanical Calculations in Condensed Matter

Lecturer: Dr. John E. Pask, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
Date: September 24, 2009 (Thursday), 1:30pm
Location: Institute of Physics AS CR, Na Slovance 2, Praha 8, Lecture Hall on the ground floor

The Institute of Physics AS CR - Dept. of Theory of Condensed Matter, the Institute of Thermomechanics AS CR and the Czech Society for Mechanics – the expert group for Computational mechanics, jointly invite you to a lecture by

Dr. John E. Pask
Lawrence Livermore National Laboratory
P.O. Box 808, L-045
Livermore, CA 94551 USA

Institute of Physics AS CR, Na Slovance 2, Praha 8 (entrance from st. Pod vodárenskou věží 1), Lecture Hall on the ground floor

1:30pm Part 1
2:15pm Break
2:35pm Part 2, Discussion


Over the course of the past few decades, quantum mechanical calculations have emerged as a key component of modern materials research. Such calculations allow the understanding and prediction of materials properties from first principles (ab initio), with no empirical inputs or adjustable parameters. The planewave (PW) pseudopotential method, as implemented in widely used VASP and ABINIT codes, among many others, has established itself as the dominant method for large, accurate, density-functional calculations in condensed matter. However, due to the underlying Fourier representation of the required quantum mechanical wavefunctions, the PW method suffers from substantial inefficiencies in parallelization and applications involving highly localized states, such as those with 1st-row, transition-metal, or other atoms at extreme conditions. Modern real-space approaches, such as finite-difference (FD) and finite-element (FE) methods, can address these deficiencies without sacrificing rigorous, systematic improvability but have until now required much larger bases/grids to attain the required accuracy.

In this talk, I will discuss the application of finite-element (FE) techniques to solve the required quantum mechanical equations with the goal of pushing back the current limits on such calculations, while retaining both locality and systematic improvability, and thus accuracy and parallelizability. In particular, I will highlight our recent work using modern partition-of-unity finite-element (PUFE) techniques to substantially reduce the number of basis functions required in the representation of the required wavefunctions and to overcome the main disadvantage that has plagued all such "real space" solution approaches until now: excessive degrees of freedom needed to achieve the required accuracies. We will discuss the required coupled, 3D Schroedinger-Poisson quantum mechanical problem and current state-of-the-art approaches to its solution, such as planewaves (Fourier), finite-differences, finite-elements, and wavelets. Particular focus will be given to the issues which arise in solving the required equations in a strictly local, C0 FE/PUFE basis: boundary conditions, nonlocal operators, infinite lattice sums, and enriched (PUFE) bases, in particular. It will be shown that PUFE techniques can be employed to substantially reduce the number of basis functions required in the representation of the required solutions (wavefunctions) by building known atomic physics into the basis: while retaining both locality and systematic improvability of the basis as a whole, and thus accuracy and parallelizability. We show direct comparisons of PW, classical FE, adaptive-mesh FE, and new PUFE methods for model and physical problems and discuss pseudopotential as well as all-electron (singular Coulomb) applications. Initial results show order-of-magnitude improvements relative to current state-of-the-art PW and adaptive-mesh FE methods for systems involving localized states such as d- and f-electron metals and/or other atoms at extreme conditions.

John Pask is a theoretical and computational physicist in the EOS & Materials Theory group at the Lawrence Livermore National Laboratory (LLNL). He received his undergraduate degree in physics from the University of California, Davis in 1988. He taught mathematics, physics, and reactor dynamics to naval officers and civilian engineers at the Naval Nuclear Power School in Orlando, Florida from 1988-1994, where he served as Director of the Mathematics and Physics division from 1993-94. He received his Ph.D. in physics from the University of California, Davis in 1999. His thesis work focused on the development and implementation of a new finite-element based approach to large-scale ab initio electronic-structure calculations. During the latter part of his graduate studies, he worked at the Materials Research Institute of the Lawrence Livermore National Laboratory on the extension and application of the finite-element based electronic-structure method to large-scale ab initio positron calculations. He was the recipient of the Nicholas Metropolis Award for Outstanding Doctoral Thesis Work in Computational Physics from the American Physical Society in 2001. Dr. Pask was the recipient of a National Research Council Associateship in 1999 to continue work on electronic-structure method development and applications at the Naval Research Laboratory in Washington, DC. While there, he studied transition-metal compounds, using the full-potential linearized augmented planewave method; and continued work on the finite-element electronic-structure method and associated large-scale positron applications until moving to LLNL in 2001. At LLNL, Dr. Pask continues work on new real-space ab initio electronic-structure methods, associated large-scale iterative linear- and eigen-solvers, and applications to semiconductor, transition-metal, and actinide materials at ambient and extreme conditions.

More information: Dr. Jiří Plešek

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