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
- Traces, Bubbles, and Supersmoothness: Constructing Conforming Finite Elements for Tensor Complexes
Bowen Shi (Oden Institute, UT Austin)
Location: Andrew Wiles Building, Oxford
Time: 20-24 July, 2026
Abstract: Show AbstractFinite element exterior calculus (FEEC) provides a systematic framework for discretizing differential complexes while preserving their underlying cohomological structure. Starting from the de Rham complex and the classical $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$-conforming finite elements, this perspective has been extended through BGG constructions to tensor-valued complexes such as the Hessian, elasticity, and divdiv complexes. These developments reveal a common construction principle: conforming finite element spaces should be designed together with the traces, bubbles, and cohomological properties of the complex.
In these lectures, we discuss this philosophy and its extension to a third-order tensor complex: the three-dimensional conformal deformation complex. Its middle spaces consist of symmetric traceless tensor fields, connected by the conformal Killing operator, the linearized Cotton–York operator, and divergence. We explain how trace complexes determine interelement continuity, how bubble complexes reveal necessary vertex supersmoothness, and how these ingredients lead to conforming finite element subcomplexes that are exact on contractible domains. The construction also yields an inf–sup stable $H(\mathrm{div})$-conforming pair for symmetric traceless tensors, providing a structure-preserving discretization of transverse-traceless fields. - Mathematical Theory for Deep Neural Networks
Jinchao Xu (KAUST)
Location: TBA
Time: 15–19 June 2026
Abstract: Show AbstractTBA - Solving PDEs with FEM using the open source finite element package NGSolve
Joachim Schöberl (TU Wien)
Details: GeoFEM will fund this course as part of the Research School at SciCADE2026.
Application: Open until 1 March 2026
Website: https://scicade.org/research-school
Abstract: Show AbstractThis course introduces solving partial differential equations (PDEs) using finite element methods with the open-source software package NGSolve. It covers practical aspects of FEM implementation, mesh generation, assembly, solvers, and visualization in NGSolve, aimed at researchers and students working on scientific computing and PDE problems. - Topological perspectives of MHD
Gunnar Hornig (University of Dundee)
Location: VC1, Andrew Wiles Building, Oxford
Time: 14–16 April 2026
Abstract: Show AbstractTALK 1: Magnetic Helicity and Related Quantities (Tuesday 14th April, 2:30pm)
Magnetic helicity is a widely used concept in astrophysics and plasma physics that measures the linking of magnetic flux within a domain. We will explain its relation to the Gauss linking number, and explore various generalisations. First, we extend the definition to multiply connected domains and to magnetically open domains. We then introduce the concept of field-line helicity, its relation to the Calabi invariant, and, ultimately, Arnold's inequality, which relates helicity to magnetic energy.TALK 2: Magnetic Relaxation and the Parker Problem (Wednesday 15th April, 2:30pm)
The equations of Magnetohydrodynamics (MHD) describe the evolution of a plasma. For most astrophysical and many technical plasmas, the dissipative term in the equations is very small. If one neglects this term (ideal MHD), the equations have an infinite number of conserved quantities related to the conservation of the topology of the magnetic field. On the other hand, even with very small resistivity, strong current sheets can develop in localised regions, leading to magnetic reconnection. This raises the question of which ideal invariants are still approximate invariants in the limit of small, but non-vanishing, resistivity. We will discuss several numerical relaxation experiments we have performed on plasmas with complex, braided magnetic fields to determine which invariants govern the relaxation process toward a minimum-energy state. This includes a critical review of Taylor's theory, a discussion of the role of turbulence, and the Parker problem, i.e., whether equilibria are accessible from arbitrary initial configurations.TALK 3: Topological Invariants of Magnetic Fields Beyond Helicity (Thursday 16th April, 2:30pm)
We explain how higher-order linking, e.g. of the type of Borromean rings, can be measured in magnetic fields using invariants derived from Massey products. We discuss the limitations of these integrals, in particular the ordering problem that has hindered their broader application. We then discuss extensions to higher-dimensional spaces, e.g., link invariants such as the Novikov invariant, which can be applied to the electromagnetic field in 4-dimensional spacetime. - Iterative solvers and subspace correction
Young-Ju Lee (Texas State University)
Location: TBA
Time: 23–24 March 2026
Abstract: Show AbstractSession 1: Parallel Subspace Correction Methods for Semicoercive and Nearly Semicoercive Convex Optimization with Applications to Nonlinear PDEs
We develop new convergence results for parallel subspace correction methods applied to unconstrained semicoercive and nearly semicoercive convex optimization problems. This extends the classical theory for singular and nearly singular linear systems to a broader class of nonlinear problems. In particular, we show that the theoretical framework established for linear singular and nearly singular problems also applies in the convex optimization setting. For semicoercive problems, the convergence rate is characterized through a seminorm-stable decomposition involving the subspaces and the kernel of the operator, in close analogy with the singular linear case. For nearly semicoercive problems, we prove a parameter-independent convergence rate under the assumption that the kernel of the semicoercive component admits a decomposition into local kernels, paralleling the theory for nearly singular problems. As an application, we analyze two-level additive Schwarz methods for a class of nonlinear partial differential equations with Neumann boundary conditions within this abstract framework.
Session 2: A High-Order Augmented Lagrangian Method with Arbitrarily Fast Convergence
We introduce a high-order augmented Lagrangian method for convex optimization problems with linear constraints that attains arbitrarily fast, and even superlinear, convergence rates. Our analysis begins with the high-order proximal point method, for which we establish convergence rates under suitable uniform convexity assumptions on the objective functional. Building on these results, we formulate a high-order augmented Lagrangian scheme and derive its convergence properties by relating it to the high-order proximal point framework. We conclude by demonstrating the method’s applicability to several scientific problems, including data fitting, porous media flow, and scientific machine learning.
- 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.