Project description
Crucial framework for complex science and geometric models
Advancements in physics, particularly in general relativity and curvature, have introduced significant complexity through sophisticated experiments. While these developments deepen understanding and benefit multiple disciplines, they often depend on cutting-edge computational simulations, which can face limitations or precision issues over time. The ERC-funded GeoFEM project aims to address these challenges by developing an algebraic framework to systematically construct tensorial finite elements with symmetries. This framework will enable the creation of more complex and reliable models, integrating concepts such as partial differential equations and discrete physics. GeoFEM highlights the role of numerical relativity and the need for robust codes that can provide consistent solutions for physics and geometry calculations.
Objective
Partial differential equations (PDEs) describe important models in science and engineering. Many of these PDE-based models encode fundamental geometric and topological principles. For general relativity, gravity is described as the curvature of spacetime governed by the Einstein equations. For materials, defects and microstructures can be modelled as geometric quantities such as curvature. Since controlled experiments and analytical solutions are only available in very special cases, it is essential to simulate these equations on computers. Despite significant progress in the past decades, cutting-edge applications still call for reliable numerical methods. In numerical relativity, codes may break down or significantly lose precision in long term simulation of black holes due to the violation of geometric constraints. For continuum with microstructures, convergence may degenerate as multiple length scales are present. The common challenge behind these examples is to find an intrinsic way to discretise high-order tensors in geometry with certain symmetries.
My research will address the fundamental problem of discretising high-order tensors by bringing together geometry, algebra, PDEs and numerical analysis. I will develop an algebraic framework and a systematic construction of tensorial finite elements with symmetries. By clarifying mathematical structures at both continuous and discrete levels, I will investigate reliable methods for discretising the Einstein equations and continuum models with microstructures. The new framework will also inspire the development of
fundamental concepts and models, and establish novel connections between numerical schemes, discrete geometry, measure-valued solutions of PDEs, and discrete physics, e.g. quantum gravity and lattice gauge theory.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
- natural sciencesmathematicspure mathematicsalgebra
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
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Keywords
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
EH8 9YL Edinburgh
United Kingdom