GeoFEM
The ERC Starting Project GeoFEM (Geometric Finite Element Methods) explores the discretization of geometric objects using finite element methods, connections between finite elements and discrete differential geometry, and their applications.
GeoFEM hosts lectures and short courses delivered by visitors, collaborators, and occasionally group members. These sessions focus on emerging topics and recent advances in finite element methods (broadly defined) and Finite Element Exterior Calculus, particularly those lacking comprehensive documentation. Some lectures offer online participation. To attend remotely, please contact.
Lectures and Short Courses
- Mathematical Theory for Deep Neural Networks
Jinchao Xu (KAUST)
Location: TBA
Time: 15–19 June 2026
Abstract: Show AbstractTBA - TBA
Gunnar Hornig (University of Dundee)
Location: TBA
Time: 14–16 April 2026
Abstract: Show AbstractTBA - TBA
Young-Ju Lee (Texas State University)
Location: TBA
Time: 23–24 March 2026
Abstract: Show AbstractTBA - Auxiliary Space Theory: Simple Construction of Sophisticated Iterative Methods for Linear Systems
Jongho Park (KAUST)
Location: Andrew Wiles Building, Oxford
Time: 17 February 2026, 14:00–15:00; 20 February 2026, 14:00–16:00
Videos: Lecture 1, Lecture 2, Lecture 3
Lecture Notes: Lecture Notes
Abstract: Show AbstractWe present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems. By interpreting a given iterative method for the original system as an equivalent, yet more elementary, iterative method for an auxiliary system defined on a larger space, we derive sharp convergence estimates using elementary linear algebra. In particular, we establish identities for the error propagation operator and the condition number associated with iterative methods, which generalize and refine existing results. The proposed auxiliary space theory is applicable to the analysis of numerous advanced numerical methods in scientific computing. To demonstrate its utility, we present a variety of applications, including subspace correction methods, Hiptmair–Xu preconditioners, saddle point problems, and iterative substructuring methods, and show how the proposed framework yields refined analyses in each case.References:
[1] Jongho Park. Unified analysis of saddle point problems via auxiliary space theory. arXiv preprint 2509.11434 (2025).
[2] Jongho Park and Jinchao Xu. Auxiliary space theory for the analysis of iterative methods for semidefinite linear systems. arXiv preprint 2509.07179 (2025).
[3] Jinchao Xu and Ludmil Zikatanov. Algebraic multigrid method. Acta Numer., 26 (2017), pp. 591–721. (Section 4 only) - Riemannian Geometry with Finite Elements
Evan Gawlik (Santa Clara University), October 11–17, 2025
Location: VC1, Andrew Wiles Building, Oxford
Time: October 13, 14:00–16:00; October 14, 10:00–12:00; October 15, 13:00–15:00
Abstract: Show AbstractThis short course will cover techniques for discretizing the building blocks of Riemannian geometry---metrics, connections, and curvature---with finite elements. It will start with a brief introduction to the geometry of smooth surfaces and classical notions of discrete curvature on piecewise flat triangulated surfaces. We will explain how to generalize these notions of discrete curvature to triangulated manifolds equipped with piecewise polynomial metrics. Then we will explain how one can prove that these curvature discretizations converge to their smooth counterparts under refinement of the triangulation. Along the way, key concepts from Riemannian geometry will be reviewed, and links between discrete differential geometry and finite element theory will be highlighted and used extensively. - Finite elements in categorical language
Snorre Christiansen (University of Oslo), October 6–14, 2025
Location: VC1, Andrew Wiles Building, Oxford
Time: October 8, 14:00–16:00; October 10, 14:00–16:00; October 13, 11:00–12:00
Abstract: Show AbstractI will show how finite elements can be described in terms of constructs from category theory. In particular they may be interpreted as presheaves on the poset category of a mesh. I will assume no prior knowledge of categories or sheaves. Ciarlet's definition of a finite element imposes a unisolvency condition which can be related to softness of sheaves. De Rham theorems relating different cohomologies can be proved in the general setting. Examples will be discussed. - A Posteriori Error Estimation by Preconditioning
Yuwen Li (Zhejiang University), August 4–8, 2025
Location: James Clerk Maxwell Building (JCMB), Edinburgh. August 7, 13:30–15:30: JCMB 6206; August 8, 13:30–15:30: JCMB 6201; August 11, 13:30–15:30: JCMB 6206
Videos: Lecture 1, Lecture 2, Lecture 3
Lecture Notes: Lecture Notes
Abstract: Show AbstractThis short course explores preconditioning and a posteriori error estimation in finite element methods, focusing on H(curl) and H(div) spaces. We begin by reviewing fundamental concepts in a posteriori error estimates for adaptive methods and iterative solvers for linear algebraic systems. The course emphasizes the interplay between iterative solvers and a posteriori error estimates, demonstrating the derivation of novel parameter-robust and p-robust error estimators in H(curl) and H(div) using nodal auxiliary space preconditioning at the continuous level. For comparison, we also examine classical a posteriori error analysis in H(grad), H(curl), and H(div). - Spectral Theory and Spectral Practice
Umberto Zerbinati (University of Oxford), May 13–16, 2025
Course Website: https://www.uzerbinati.eu/teaching/spectral_theory/
Location: James Clerk Maxwell Building (JCMB) 5328 (May 13–14), JCMB 5326 (May 16), Edinburgh
Time: May 13, 2025, 10:00–12:00; May 14, 2025, 14:00–16:00; May 16, 2025, 15:00–17:00
Show AbstractThis short course explores finite element discretizations of eigenvalue problems involving non-normal operators, with a focus on the advection-diffusion equation as a guiding example. We begin by revisiting fundamental spectral notions—self-adjointness, normality, spectra, and pseudospectra—with particular emphasis on how an operator’s spectrum informs the physical behavior of time-dependent PDEs. The core of the course is devoted to the classical analysis of finite element approximations: we present the Bramble-Osborn results for non-self-adjoint eigenvalue problems, including full proofs, and discuss their implications for convergence and approximation quality. For comparison, we also review the celebrated Babuška-Osborn theory for self-adjoint cases. If time permits, we will conclude with a discussion on iterative solvers and preconditioning strategies tailored to non-normal eigenvalue problems. The course requires a basic background in functional analysis and finite element methods.