Domain decomposition methods for electronic structure calculations

• Applicant: Professor Dr. Benjamin Stamm
• Subject Area: Mathematics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 411724963

Electronic structure calculations are paramount in theoretical chemistry, physics, and material science. Indeed, among the top ten most cited scientific articles, two papers are related to this topic. Despite the fact that this field deals with the discretization of eigenvalue problems, which is a well-established subject in numerical analysis, the expertise of applied mathematics is little involved.Within the proposed project, the aim is to develop novel domain decomposition (DD) algorithms for eigenvalue problems which arise in electronic structure calculation like the Kohn-Sham DFT (Density Functional Theory) equations. The approach within this project is based on the idea that domain decomposition for eigenvalue problems is not fundamentally different than for source problems. Despite this fact, DD-methods for eigenvalue problems are less popular than for source problems. Additionally, recent work in the context of implicit solvation models and the analysis of those methods show that the DD-method is scalable for domains of an increasing number of fixed-size sub-domains, like e.g. for chain-like molecules or proteins, even without a so-called coarse-correction.The problems to be embraced within this project are manifold and contain: i) non-linear eigenvalue problems, ii) a large number of eigenvalues to be determined, iii) eigenvalue problems on unbounded domains and iv) dealing with potentials that contain Coulomb-like singularities.The project is a first step within a broader long-term plan to derive efficient local basis functions based on local reduced order modeling as an alternative of the widely-used but empirical contracted Gaussian basis functions. In fact, the domain decomposition strategy allows to localise the equations and opens the door to the application of reduced order modeling with certified a posteriori error estimates in a second step.

• DFG Programme: Research Grants
• International Connection: France, Italy
• Cooperation Partners: Professor Dr. Eric Cancès; Dr. Filippo Lipparini; Professor Dr. Yvon Maday