The Foundations of Numerical PDEs Workshop brings together researchers to discuss the latest advances in the numerical analysis of partial differential equations (PDEs). Focus topics of the workshop include
• Structure-preserving discretization of Hilbert complexes, including and generalizing FEEC (Finite Element Exterior Calculus) and offering a framework that unifies and extends traditional finite element methods, and forms the foundation of stable and robust numerical methods.
• Novel discretization techniques on polytopal meshes to improve flexibility, accuracy and efficiency of numerical methods for PDE-based multiphysics models in complex domains.
• Recent breakthroughs in the numerical modeling of wave propagation problems, including wave-number explicit analysis and wave-number robust numerical schemes.
The presentations at the workshop will address both theoretical developments, algorithmic challenges, and novel applications of numerical methods. This workshop will have natural connections the the other FoCM workshops “Geometric Integration and Computational Mechanics” and “Geometric Integration and Computational Mechanics”.
Organizers
Speakers
Semi-plenary speakers
Università di Milano Bicocca
Karlsruhe Institute of Technology
Invited speakers
Virginia Tech
Heriot-Watt University
University of Bergen
University Santiago de Compostela
UC Irvine
Université de Montpellier
TU Eindhoven
Santa Clara University
NTU Athens
Portland State University
University of Oxford
Ghent University
TU Dresden
Politecnico di Milano
University di Milano Bicocca
TU Wien
Radboud University
TU Wien
Thursday, 9 July
![[HS 14 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS14.png){kind=link}
14:00-14:30
Jay Gopalakrishnan (Portland State University)
We develop a notion of curvature for high-order Regge metrics on simplicial manifolds in arbitrary dimension. Classical Regge calculus models geometry using piecewise flat metrics, with curvature concentrated on lower-dimensional simplices through angle defects. More recently, arbitrary-order Regge finite elements have provided a framework for approximating smooth metrics with higher-order accuracy, allowing metrics that are smooth within simplices but generally discontinuous across interfaces, while remaining compatible under isometric gluing conditions. In this setting, linear derivatives of the metric can be interpreted in the sense of distributions even when the metric is not continuous across elements. But the nonlinear structure of the Riemann curvature tensor makes the definition of curvature substantially more delicate. Combining insights from prior works, we propose a generalized Riemann curvature tensor for such metrics that incorporates both classical angle defects and jumps in the second fundamental form across interfaces. The construction applies in dimensions two and higher and extends naturally to generalized notions of Ricci curvature, scalar curvature, and the Einstein tensor. Connections with finite element approximations of geometric partial differential equations will also be discussed.
14:30-15:00
Long Chen (UC Irvine)
In this talk, we present two frameworks for constructing finite elements: lattice decomposition and t–n decomposition. The first part focuses on the construction of finite elements via lattice decomposition. The simplicial lattice is the set of multi-indices with a fixed sum. Existing scalar finite elements can be systematically constructed through different lattice decompositions. In particular, we provide an explicit and computable framework for the smooth finite elements constructed by Hu, Lin, and Wu. The second part introduces a *t–n* decomposition for vector and tensor finite elements. At each subsimplex, the space is decomposed into tangential and normal components. Using the *t–n* frame, we construct decompositions for tensors and differential forms, leading to a new framework for building vector and tensor finite elements. Combined with lattice decomposition, we obtain a geometric decomposition of finite element spaces and, based on these decompositions, construct finite element complexes. This approach also allows us to use standard Lagrange elements for practical implementation. This is joint work with Xuehai Huang (Shanghai University of Finance and Economics).
15:00-15:30
Evan Gawlik (Santa Clara University)
If a simplicial triangulation is equipped with a piecewise smooth Riemannian metric that has single-valued tangential-tangential components on element interfaces, then there are various notions of curvature that one can define, even though the classical formulas for curvature involving derivatives of the metric no longer make sense. In this talk, I will explain the origins of these definitions from several perspectives. Some of these perspectives are extrinsic, meaning they rely on an embedding into Euclidean space, while others are intrinsic. I will then summarize what is known about the convergence of these discrete curvatures under refinement. I will conclude by mentioning some open problems, including the question of how to define and analyze discrete Lipschitz-Killing curvatures in the piecewise smooth (as opposed to piecewise flat) setting.
15:30-16:00
Martin Licht (TU Dresden)
Stability estimates in vector analysis are governed by the classical inequalities of Poincaré, Friedrichs, and Weber. Their finite element analogues play an equally central role in the stability and robustness of finite element methods. This talk presents practical estimates for the associated continuous and discrete constants, with particular emphasis on their dependence on polynomial degree, mesh geometry, and shape-regularity parameters.
The analysis combines Poincaré and Bogovskiĭ potential operators on convex sets, local estimates on finite element stars endowed with shellings, and global arguments on triangulated domains. The resulting bounds provide computable stability constants for both continuous and discrete formulations, with a combination of functional analysis, geometric combinatorics, and algebraic topology.
This is joint work with Théophile Chaumont-Frelet and Martin Vohralı́k.
16:00-16:30 Coffee Break
16:30-17:00
Lorenzo Mascotto (Università di Milano Bicocca)
Trace inequalities are crucial tools to derive the stability of partial differential equations with inhomogeneous, natural boundary conditions. In the analysis of corresponding Galerkin methods, they are also essential to show convergence of sequences of discrete solutions to the exact one for data with minimal regularity under mesh refinements and/or degree of accuracy increase. In nonconforming discretizations, such as Crouzeix-Raviart and discontinuous Galerkin, the trial and test spaces consists of functions that are only piecewise continuous: standard trace inequalities cannot be used in this case.
In this talk, we present several trace inequalities for piecewise $W^{1,p}$ functions. Compared to analogous results already available in the literature, our inequalities are established: on fairly general polytopic meshes (with arbitrary number of facets and arbitrarily small facets); without the need of finite dimensional arguments (e.g., inverse estimates, approximation properties
of averaging operators); for different ranges of maximal and nonmaximal Lebesgue indices.
17:00-17:30
Lehel Banjai (Heriot-Watt University)
As motivation we consider the attenuated Westervelt equation. In various standard models, the attenuation is controlled by a non-local in time term – often a fractional derivative. We consider a situtation where the fractional power is space-dependent. This has interesting consequences for both the analysis and fast and memory efficient implementation. We present the early results with focus on the fast methods. This is a joint work with Pascal Lehner (Klagenfurt) and Mariya Ptashnyk (Edinburgh).
17:30-18:30 semi-plenary talk
Kaibo Hu (University of Oxford)
Topology has long shaped the understanding of fluid motion, from Kelvin’s vortex theory and Helmholtz’s circulation laws to modern developments in magnetohydrodynamics and topological hydrodynamics. In fluids and plasmas, quantities such as helicity and enstrophy encode global constraints that strongly influence relaxation, magnetic-field generation, and turbulence.
Many of the central questions in this area are fundamentally asymptotic rather than local: whether magnetic fields grow or decay, how topological structures persist under advection and diffusion, and how relaxation selects large-scale structures. These problems motivate a form of global numerical analysis that goes beyond finite-time approximation and instead addresses topology preservation, long-time dynamics, and invariant structures.
In this talk I will discuss recent work developing such a perspective using finite element exterior calculus (FEEC) and finite element tensor calculus (FETC), Lie advection of differential forms, and generalized Hodge theory. The goal is to construct discretizations that faithfully preserve the topological and dynamical structure of the underlying continuum equations, including phenomena related to magnetic relaxation, dynamo and antidynamo, and cohomological constraints. More broadly, this suggests a computational perspective on topological hydrodynamics at the interface of geometry, PDEs, and scientific computing.
Friday, 10 July
14:00-15:00 semi-plenary talk
Lourenco Beirao da Veiga (Università di Milano Bicocca)
The field of magnetohydrodynamics (MHD) has garnered increased attention in the realm of computational mathematics in recent years. These equations, found in the study of plasmas and liquid metals, have diverse applications in geophysics, astrophysics, and engineering. The combination of equations from fluid dynamics and electromagnetism results in various models with different formulations and finite element (FEM) choices. In the present talk we focus on a three-field formulation (velocity, pressure, magnetic field) for the unsteady resistive MHD equations in three dimensions.
In the first part of the talk we will investigate the case of (polyhedral) convex domains, presenting two FEM schemes which are pressure robust and Reynolds quasi robust, i.e. enjoying error estimates which do not degenerate for large fluid and magnetic Reynold numbers. Such robustness is obtained by combining upwinding and an additional stabilization in the spirit of continuous interior penalty (CIP).
In the second part of the talk we will discuss the non-convex case, first by hinting at the simpler linearized equations, and afterwards presenting three different methods with different characteristics for the original nonlinear problem. In all cases, a major focus will be on pressure robustness and Reynolds quasi-robustness. A key aspect here is the use of Nedelec type elements for discretizing the magnetic field: such choice renders the scheme suitable (in a sense better defined in the talk) for problems on non-convex domains but introduces additional difficulties, especially in relation to its robustness in convective dominant cases.
During the talk a set of numerical tests which support and extend our theoretical findings will be presented.
15:00-15:30
Verani Marco (Politecnico di Milano)
In this talk, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian Stokes flows governed by the Carreau-Yasuda constitutive law, in the shear-thinning (r<2) and in the shear-thickening (r>2) regime. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. Numerical results demonstrate the practical performance of the proposed formulation. Joint work with P.F. Antonietti, L. Beirao da Veiga, M. Botti, A. Harnist and G. Vacca.
15:30-16:00
Daniele Di Pietro (Université de Montpellier)
The de Rham complex for three-dimensional domains is the sequence of spaces H1, Hcurl, Hdiv, and L2 connected by the vector calculus operators grad, curl, div. The well-posedness of important classes of partial differential equations (PDEs) is linked to the cohomology properties of this complex, which characterize the kernel of each operator depending on the topology of the domain. Emulating these properties at the discrete level is crucial in order to obtain stable discretizations of such PDEs. A classical way of doing so consists in using trimmed (Raviart—Thomas—Nédélec) finite elements [1,2]. These finite elements are, however, restricted to conforming meshes with elements of simple shape. In this talk I will illustrate the main principles to derive approximations of the de Rham complex that support much more general meshes, including general polyhedral elements and non-matching interfaces [3]. A crucial step consists in switching to a fully discrete formulation, where both spaces and operators in the de Rham complex are replaced by discrete counterparts obtained mimicking integration by parts formulas. The resulting Discrete de Rham framework thus provides a new approach to design inherently stable schemes for PDEs whose well-posedness relates to the de Rham complex. Originally designed in vector calculus notation, this framework has been recently extended to the de Rham complex of differential forms [4]. This extension has been used as a starting point to devise unified proofs covering arbitrary space dimensions and form degrees of key results such as discrete Poincaré inequalities [5] and adjoint consistency results [6].
[1] Raviart, P. A. and Thomas, J. M. (1977). A mixed finite element method for 2nd order elliptic problems. In Galligani, I. and Magenes, E., editors, Mathematical Aspects of the Finite Element Method. Springer, New York.
[2] Nédélec, J.-C. (1980). Mixed finite elements in R3. Numer. Math., 35(3):315–341.
[3] D. A. Di Pietro and J. Droniou. An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. Found. Comput. Math., 2023, 23:85–164. DOI: 10.1007/s10208-021-09542-8
[4] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods. J. Eur. Math. Soc., 2024. Accepted for publication
[5] D. A. Di Pietro, J. Droniou, M.-L. Hanot, and S. Pitassi, Uniform Poincaré inequalities for the discrete de Rham complex of differential forms, arXiv preprint 2501.16116 [math.NA], January 2025
[6] D. A. Di Pietro, J. Droniou, and S. Pitassi, Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms, arXiv preprint 2509.21449 [math.NA], September 2025
16:00-16:30 Coffee Break
16:30-17:00
Inga Berre (University of Bergen)
Engineered geothermal reservoirs are governed by strongly coupled thermo–hydro–mechanical–chemical (THMC) processes whose nonlinear interactions control reservoir evolution and operational response. For example, induced thermo-poroelastic forcings causing fault reactivation and deformation in geothermal reservoirs are a result of a complex multi-physics interactions. This will be illustrated with various cases, including how elevated seismicity can be associated with shut-in of production wells.
The contribution focus on the mathematical modeling and numerical simulation of coupled THM processes in geothermal reservoirs with a focus on fault reactivation caused by injection and production operations. Capturing these interactions in PDE-based multiphysics models is challenging due to complex nonlinear feedback mechanisms between processes.
Mathematical formulations and numerical solution approaches will be discussed, such as choices related to the complementarity formulation governing fault frictional contact mechanics and solvers for the numerical solution of the resulting nonlinear system of discretized equations.
17:00-17:30
Saray Busto Ulloa (University of Santiago de Compostela)
We present a semi-implicit hybrid methodology for the discretisation of the unified Godunov-Peshkov-Romenski (GPR) model for continuum mechanics. This system of partial differential equations is endowed with several structural properties, including involution constraints and asymptotic limits, allowing the modelling of media ranging from non-linear elasto-plastic solids to viscous fluids.
To handle the curl-free-type involutions arising in the solid limit of the model, the original system is augmented by adopting a thermodynamically compatible generalised Lagrangian multiplier (GLM) approach. Next, to ensure the preservation of the asymptotic limits, an operator splitting of the governing equations is performed. Accordingly, three subsystems are obtained: a transport subsystem, discretised using an explicit finite volume method; a source term subsystem, addressed implicitly; and a pressure subsystem, of the Poisson-type, solved employing continuous finite elements. This methodology leads to a mild CFL time-step restriction that neglects both the fast pressure waves and the effects of potentially stiff source terms. Consequently, the proposed approach is well suited for the simulation of all Mach number regimes.
Furthermore, to allow for small domain deformations, an arbitrary Lagrangian-Eulerian (ALE) framework is adopted. The transport subsystem is thus advanced by integration over closed space-time control volumes, ensuring verification of the geometric conservation law (GCL). A comprehensive set of benchmarks, in both the fluid and solid limits of the model, validates the accuracy and robustness of the resulting methodology.
17:30-18:00
Van Chien Le (Ghent University)
We present a boundary integral equation for the numerical modeling of electromagnetic scattering by composite dielectric objects. The formulation extends the classical Müller equation to composite structures through the global multi-trace method. The key ingredient enabling this extension is the use of the Stratton-Chu representation in complementary regions. The resulting block system is composed entirely of hypersingularity-free second-kind operators. Moreover, the multi-trace framework naturally supports a Petrov-Galerkin discretization based on standard and widely used spaces. Numerical results demonstrate that the discrete systems remain well-conditioned and stable under both dense-mesh refinement and low-frequency regimes, without requiring additional stabilization.
18:00-18:30
Michael Neunteufel (TU Wien)
Many applications, such as shell models and geometric flows, rely heavily on surface curvature. Discrete differential geometry (DDG) estimates curvature-related properties on discretized surfaces. Integrating DDG into a finite element framework provides rigorous tools for analyzing existing methods and systematically extending them. Distributional finite elements are crucial in this setup, allowing for weak regularity and requiring differential operators to be interpreted in the sense of distributions.
In this talk, we explore approximating the shape operator on surfaces described by Lagrange finite elements. Our method merges the DDG approach of dihedral angles with Hellan-Herrmann-Johnson finite elements. Using a novel integral error representation, we derive rigorous error bounds. Under mild assumptions, we show convergence in a negative Sobolev norm, thereby establishing the first proof that the dihedral-angle DDG approximation always converges. With stronger assumptions, we achieve optimal L2-convergence for the discrete L2-Riesz representative of the shape operator. We also discuss potential applications and demonstrate numerical results with the finite element library NGSolve (www.ngsolve.org). This work is in collaboration with Jay Gopalakrishnan (Portland State University).
Saturday, 11 July
14:00-15:00 semi-plenary talk
Marlis Hochbruck (Karlsruhe Institute of Technology)
In this talk, e address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh.
When using explicit time-integration schemes, the time stepsize is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local time-stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.
This is joint work with Malik Scheifinger, KIT Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173.
15:00-15:30
Victorita Dolean (TU Eindhoven)
Spectral coarse spaces based on local eigenproblems are a powerful tool for robust domain decomposition methods. Existing GenEO-type approaches, however, rely on positive semi-definite eigenproblems, which become ineffective for highly indefinite Helmholtz problems at large wavenumber. We introduce and analyse Hk-GenEO, a new coarse space construction based on local generalised eigenvalue problems involving the indefinite Helmholtz operator itself. These eigenproblems generate oscillatory coarse modes that are absent in SPD-based formulations and are essential for robustness at high frequency. A central contribution is an abstract spectral framework for indefinite eigenvalue problems, enabling the analysis of projections onto spaces spanned by negative and small positive modes. This leads to explicit GMRES convergence estimates for two-level Schwarz methods and conditions for robustness with respect to frequency and heterogeneity, without restricting the coarse subdomain size.
We also relate this approach to recent work on SPD coarse spaces, clarifying when indefinite spectral information becomes necessary and to other spectral approaches from the literature. Numerical results confirm stable iteration counts across a wide range of parameters.
15:30-16:00
Emmanuil Georgoulis (Heriot-Watt University & NTU Athens)
We present and analyse new space-time Galerkin discretisations nonlinear (semilinear and quasilinear) wave equations, based on a variational formulation derived from De Giorgi’s elliptic regularisation viewpoint. The method is shown to be well-posed through a minimisation approach and to be unconditionally stable for all choices of conforming discretisation spaces. Further, a priori error bounds are proven for sufficiently smooth solutions. Special attention is given to the conditioning of the method and its stable implementation. Both conforming and non-conforming variants of the method will be presented. Numerical experiments are provided to validate the theoretical findings.
16:00-16:30 Coffee Break
16:30-17:00
Daniel Appelo (Virginia Tech)
The WaveHoltz iteration has emerged as a robust framework for solving the Helmholtz equation and related frequency-domain wave problems by leveraging time-domain wave solvers. By filtering time solutions of the wave equation, WaveHoltz produces a symmetric positive definite (in the energy-conservative case) fixed-point iteration.
In this talk, I will survey recent developments in the theory, implementation, and application of WaveHoltz. On the analysis side, I will present sharper convergence results, and a clearer picture of how the iteration interacts with the spectrum of the underlying wave operator, including extensions to problems with impedance and absorbing boundary conditions. New algorithmic contributions to be discussed include a WaveHoltz approach to numerical homogenization, a low-rank WaveHoltz variant that exploits compressibility of the solution manifold to reduce memory and compute costs, and domain decomposition strategies that enable scalable parallel solves across complex geometries while delivering multigrid complexity.
17:00-17:30
Théophile Chaumont-Frelet (INRIA)
This talk considers the finite element discretization of scattering problems involving acoustic waves in the time-harmonic regime. More specifically, we are interested in providing guaranteed and fully computable error bounds that can be use to certify the accuracy of the numerical solution. Standard a posteriori error estimation procedures can not be directly applied, because the inf-sup constant of the weak formulation is usually unknown for these problems. This is further complicated by the fact that scattering problems are naturally posed in unbounded domains. In this talk, I will present a new a posteriori error estimation strategy that solves both issues. The resulting error estimate is efficient, and in particular, suited for high-order finite element discretizations which are known to be particularly efficient in the high-frequency regime.
17:30-18:00
Vanja Nikolic (Radboud University)
We present a space–time finite element approach for Westervelt’s nonlinear acoustic wave equation. The method combines a conforming finite element spatial discretization with a discontinuous-continuous Galerkin time stepping. A key difficulty in the analysis is that standard Galerkin testing for wave problems does not yield control of the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach’s fixed-point argument, we show the well-posedness of the scheme and establish a priori
error estimates. Moreover, we show that the scheme preserves the asymptotic behavior of the continuous problem in the singular vanishing dissipation limit. The talk is based on joint work with Sergio Gómez (University of Milano-Bicocca).
18:00-18:30
Joachim Schöberl (TU Wien)
Vector-valued function spaces, their finite element sub-spaces, and relations between these spaces are well understood within the de Rham complex. The framework of differential forms and Hilbert complexes provides a unified framework for any space dimension. Various matrix-valued finite element spaces have been introduced and analyzed more or less independently. In this presentation we put these spaces into a so called 2-complex. We present applications in fluid dynamics, solid mechanics and relativity.
Posters
![[PZ14 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Alessandra Cancrini
(Politecnico Di Milano)
This poster focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by applications in non-Newtonian fluids and interface problems, where the stress field plays a central role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method in time, leading at each step to a symmetric and positive definite linear system whose conditioning deteriorates as the time-step size $\Delta t$ decreases. Numerical results indicate that standard iterative solvers, including common Krylov methods and basic block preconditioners, show a significant deterioration in convergence as $\Delta t$ decreases, a behaviour directly linked to the conditioning of the system matrix, which scales as $1 / \Delta t$. To address this issue, we investigate two strategies tailored to the structure of the fully discrete system: (i) deflated Conjugate Gradient, which mitigates the influence of the most problematic eigenmodes, and (ii) a collective Block-Jacobi preconditioner which exploits the block structure induced by the pseudo-stress formulation. Numerical tests show that these approaches achieve iteration counts independent of $\Delta t$, demonstrating robustness with respect to the time step. Overall, the poster highlights the proposed solvers as a robust framework for PolyDG discretisations on general polytopal meshes, while outlining ongoing research on multigrid strategies and theoretical analysis.
Elisa Coenen
(Radboud University)
With this poster session, I will present some results on the Westervelt equation with space-dependent time-fractional attenuation, which models nonlinear acoustic waves in biological media. I will start with a well-posedness result and give a brief summary of the proof. The proof is done via a Galerkin method in space, where we have to pay extra attention to the space-dependency of the convolution kernel for the attenuation.
Furthermore, I will present a full discretization in space and time to numerically solve the equation. To deal with the fractional diffusion, the numerical scheme will involve a convolution quadrature method in time. In space, I will apply a finite element method.
I will demonstrate some example simulations from the numerical scheme to show the effects of the space-dependent attenuation and do a numerical convergence analysis.
Femke de Wit
(KU Leuven)
The poster introduces a joint research project by KU Leuven, TU Graz and TU Munich. Its aim is to extend the Wave Based Method (WBM), a frequency domain approach, to transient time-signals in the context of vibro-acoustics. The WBM is a Trefftz method that was shown to be well-suited for the mid-frequency regime in a range of applications. The reason it performs well in this frequency range is because the method consists of only a few large subdomains, whose size is independent of the wave number. Because of this, the resulting discretization matrix is typically considerably smaller than that of classical FEM techniques. On the other hand, the matrices involved are fully populated, complex-valued, ill-conditioned and frequency-dependent.
Until now, the WBM has been used for time-harmonic systems, where these issues have been overcome. Yet, the assumption of time-harmonicity is not always valid. Transient problems are not easily characterized in the frequency domain and the temporal coordinate may require high resolution. In this project we consider two general approaches to discretize in time: (1) time-stepping schemes and (2) transformation-based methods, in particular using the Laplace transform. Both of these approaches transform an initial boundary value problem (e.g. of the wave equation) to a sequence of boundary value problems (e.g. of the Helmholtz equation). Each approach comes with its own benefits and challenges and leads to possible modifications of the WBM. To this end, also a mathematical analysis of the WBM in its current form is conducted.
Julian Dörner
(Karlsruhe Institute of Technology)
The following is joint work with Théophile Chaumont-Frelet.
We investigate the scalar Helmholtz equation and its high-order FEM discretization in a cavity setting with Wentzell boundary condition
involving a Laplace-Beltrami operator. This problem class is well understood in the literature for parameter regimes that yield variational formulations of the “coercive + compact” type. In contrast, our analysis covers the full parameter range whereby we allow the parameter in front of the Laplace–Beltrami operator in the Wentzell boundary condition to be real and negative. In this case, the boundary condition contributes a negative definite, non-compact boundary term to the variational formulation. These nonstandard regimes are motivated by applications in physics, for instance in the modeling of thin-coated scatterers or non-dissipative metamaterials, where the resulting boundary conditions naturally lead to non-coercive formulations or to operators with only weak damping.
From a mathematical perspective, our results are also relevant for problems with parameters that do admit a coercive formulation but with a
very small coercivity constant. Our estimates do not rely on the size of this constant and remain robust in such nearly non-coercive regimes.
We establish stability and wavenumber-explicit quasi-optimality of the method, with estimates that show the benefit of the high-order discretization.
The resulting resolution condition matches the expected k-h scaling for high-order FEM discretizations of Helmholtz problems. Our results extend the range of rigorous numerical analysis available for Helmholtz problems with second-order boundary operators beyond the “coercive+compact” type.
Vivienne Ehlert
(University of Augsburg)
We consider here the dynamics of self-gravitating astrophysical flows, where the governing equations can be split into the hyperbolic hydrodynamic equations describing the flow and an elliptic Poisson equation describing the gravitational potential. For this problem we show that we can use already existing solvers for elliptic and hyperbolic equations, connect them through a joint hierarchical Cartesian mesh and couple them via their source terms depending on each other’s state variables. By performing the coupling of these two systems in this way we obtain a multiphysics solver directly combining efficient solvers for their respective coupled governing equations. This coupling also allows indicators for adaptive mesh refinement to take all state variables of the coupled equations into account, which is also true for time step control. Here we couple Trixi.jl, a high-order discontinuous Galerkin framework for solving hyperbolic conservation laws featuring shock capturing and structure preserving methods written in Julia, and deal.II, a finite-element library written in C++, on a shared adaptive mesh. We apply the resulting solver to some example problems, including a self-gravitating Sedov blast. Our approach is also extendable to other multiphysics systems following a common Eulerian formulation. In the future we will use this approach to evaluate different structure preserving numerical methods, including a broken FEEC ansatz.
Sônia Gomes
(Universidade Estadual de Campinas)
The importance of exact finite element (FE) de Rham sequences is well recognized for stable and conservative mixed formulations of multiphysic systems. The principles rely on appropriate choice of the adopted approximation spaces for each variable, defining the way in which the differential equations are approximated. In the construction of these exact FE complexes it is necessary to keep a careful track of traces over element interfaces and of bubble components (having vanishing traces) of the functions in the sequence. For the most popular H1, H(curl), H(div}, and L2 conforming case, we provide guidelines for the construction of high order frameworks combined with advanced numerical strategies [1]. For instance, they support general polytopal meshes and/or allow trace interface constraints. We describe how these attributes can be explored, both with regard to the generality of mesh geometry and of the adopted local polynomial approximations. For instance, one idea is to apply local bubble enrichment while keeping the traces at coarser resolution. This trace-constraint technique is crucial for hp-adaptivity or for a better resolution of refined details in multiscale problems. The computational implementation and the definition of associated projection-based interpolants, commuting the de Rham diagram, can be performed with minimal modifications of a methodology and a coding capability already available to treat hp-adaptive frameworks. Illustrations shall be presented for a multiscale Darcy flow combined with a reduced elasticity model [2].
[1] Devloo, Durán, Gomes, Exact sequences of conforming finite element spaces with interface constraints for macro polytopal meshes, CMAME 134: 124-139, 2023
[2] Durán, Devloo, Gomes, A reduced surrogate model for poroelasticity. Preprint.
Mieszko Grodzicki
(University of Warsaw)
We develop and evaluate a GPU implementation of the massively parallel domain decomposition preconditioner proposed by Dryja and Krzyzanowski, investigating both its additive and hybrid Schwarz variants. We fine-tune the preconditioner’s parameters, including the subdomain and coarse problem sizes, and leverage mixed-precision computation to further increase throughput. The resulting solver demonstrates promising efficiency while maintaining a relatively small memory footprint. It outperforms NVIDIA’s AmgX solver in all tested scenarios, achieving a maximum speedup of 4.92× in 3D for higher-order elements.
Pascal Lehner
(University of Klagenfurt)
In this poster, we introduce and analyze a space-time p-adaptive discontinuous Galerkin method
for nonlinear acoustics.
We first present the underlying mathematical model, which is based on a recently derived
formulation involving, in particular, only first order in time derivatives. We then propose a spacetime discontinuous Galerkin discretization of this model, combining a symmetric Friedrichs systems discretization for symmetric hyperbolic systems with an interior penalty discretization for damping terms. The resulting nonlinear system is solved using Newton’s method.
Next, we present a well-posedness analysis of the discrete problem. The analysis begins with
a linearized system, for which stability is shown. Using a fixed point argument, these results
are extended to the fully discrete nonlinear system, yielding a priori error estimates in a natural
discontinuous Galerkin norm. Finally, we present numerical experiments demonstrating the parallel solvability of the spacetime formulation and the effectiveness of p-adaptivity. The results confirm the theoretical convergence rates and show that adaptive refinement can reduce the number of degrees of freedom required to accurately approximate selected goal functionals. Moreover, the experiments demonstrate that the model reproduces characteristic phenomena of nonlinear acoustics, such as harmonic generation, thereby validating the proposed model.
Marialetizia Mosconi
(Università Di Milano-Bicocca)
We construct new Crouzeix-Raviart (CR) spaces of even degree p that are spanned by basis functions mimicking those of the standard odd-degree case. Compared to the standard even-order CR case, the present construction allows for the use of nested bases of increasing degree and is particularly suited for the design of variable-order CR methods. We analyze a nonconforming discretization of a two-dimensional Poisson problem, which requires a DG-type stabilization. The design of variable-degree CR global spaces and a corresponding method are discussed, together with an assessment of its numerical performance for the approximation of corner singularities using hp-refinements.
Andreas Myklebust
(Norwegian University of Life Sciences)
Connections on principal bundles are central objects in differential geometry and appear naturally in gauge field theories in physics. Yang–Mills connections, the critical points of the Yang–Mills action functional, are of particular interest. Computing these on non-trivial bundles requires a discretization that reflects the topological structure of the bundle.
We present a finite element method for computing Yang–Mills connections on principal bundles over compact Riemannian manifolds. Given a triangulation of the base manifold and local sections of the bundle over each simplex, the connection form is represented piecewise. The transfer functions between overlapping sections prescribe tangential and normal jump conditions across interfaces. These jumps encode the topology of the bundle in a way that is amenable to finite element discretization.
In the case of an abelian structure group, the jump conditions become linear and can be enforced weakly through Lagrange multipliers, leading to a saddle-point problem over broken Sobolev spaces of differential forms. We characterize the associated jump and dual spaces, prove well-posedness via inf-sup conditions, and derive error estimates with optimal convergence rate using the trimmed polynomial spaces from finite element exterior calculus.
The poster will also present a numerical experiment on the Hopf bundle over the two-sphere, where the Yang–Mills connections have known analytical representations. The implementation uses the Firedrake library, and the results confirm the predicted convergence rate.
Valentina Pederzoli
(Politecnico di Milano)
This work aims to introduce, analyze, and validate a novel mathematical framework that describes biological tissue atrophy induced by the diffusion of a biological agent, with particular relevance to neurodegenerative diseases. We propose a new mathematical and computational model consisting of a Fisher–Kolmogorov equation, which is the simplest model governing species diffusion and widely used to model prion-like behavior, coupled with an elasticity equation that accounts for tissue mass loss. The novelty of this work is that the coupling between these equations is achieved through a logistic law that regulates the progressive reduction of the medium’s mass. This model is applied to the study of the onset and progression of Alzheimer’s disease, where it captures the spread of misfolded τ -proteins and the resulting brain atrophy that characterizes the pathology. To numerically address the complexity of the coupled system, we employ a Discontinuous Galerkin method for spatial discretization and a Crank–Nicolson scheme for time integration. We present the mathematical formulation of the model, examine its key properties, and detail the proposed numerical discretization. Finally, we provide convergence studies to validate the numerical implementation, along with simulation results that demonstrate the model’s capability to reproduce key features of Alzheimer’s disease onset and progression.
Halima Usman
(Usmanu Danfodiyo University)
This study presents a numerical and analytical investigation of Magnetohydrodynamics (MHD) Casson nanofluid over a semi-infinite flat plate embedded in a porous medium, taking into account pressure gradient effects, Soret diffusion, and thermal radiation. The governing nonlinear partial differential equations describing momentum, energy, and mass transport are derived and nondimensionalized using similarity transformations. A regular perturbation scheme is developed to obtain approximate analytical solutions, which are then validated against finite difference simulations to examine accuracy, stability, and convergence. This study analyzes how key physical parameters such as the Casson number, magnetic field parameter, radiation coefficient, and Soret number affect the velocity, temperature, and concentration fields. The computed Nusselt and Sherwood numbers quantify the rates of heat and mass transfer, respectively.
The results show that increasing the Casson parameter suppresses velocity and momentum boundary layer thickness due to yield stress resistance. At the same time, copper (Cu) nanoparticles enhance heat transfer compared to titanium dioxide (TiO₂), owing to their higher thermal conductivity. From a computational perspective, the proposed perturbation–numerical hybrid framework offers a structure-preserving and computationally efficient means for studying non-Newtonian nanofluid transport governed by coupled PDEs. The methodology can be extended to polytopal and irregular domains, providing a basis for robust discretization in complex multiphysics models.
Pratibha Verma
(Poznań University Of Technology)
We present a review of recent developments and results on the scalar auxiliary variable (SAV) approach for the numerical simulation of gradient flow systems. The SAV methods provide an efficient and accurate methodology for constructing time- and space-discretization schemes for a broad class of gradient flows. They are motivated by the invariant energy quadratization (IEQ) approach and retain its main advantages while addressing several of its limitations. In particular, the SAV method yields numerical schemes that are unconditionally energy stable and computationally efficient, since only decoupled linear systems with constant coefficients are solved at each time step. A key strength of the SAV technique is that it does not rely on restrictive assumptions about the nonlinear part of the free energy, thereby enabling its application to a wide range of gradient-flow systems. Numerical simulations are presented to demonstrate the accuracy, stability, and efficiency of SAV-based schemes compared to existing methods. Furthermore, we discuss
several extensions of the standard SAV formulation that have been proposed to improve flexibility and applicability for more complex models. These include exponential SAV (E-SAV), improved SAV (I-SAV), and multiple scalar auxiliary variable (MSAV) approaches. Such variants aim to enhance stability properties and accommodate more general energy structures. Finally, we briefly illustrate the E-SAV method for gradient systems with different types of energy functionals and outline possible extensions of the SAV methodology to develop energy-stable numerical schemes for fractional differential equations.
David Wörgötter
(TU Wien)
When numerically solving wave-propagation problems at a large wavenumber k using low-order
Galerkin schemes one usually encounters the pollution effect: Even if the number of degrees of freedom per wavenumber is held constant, the gap between the Galerkin error and the best-approximation in the discrete space widens. We consider the time-harmonic Maxwell equations with piecewise analytic coefficients for which we present recently developed wavenumber-explicit regularity estimates. These estimates are the key to a regularity splitting result for Maxwell’s equations, according to which the solution can, under certain conditions, be decomposed into a piecewise analytic part with explicit wavenumber-dependency and a less regular part that is uniformly bounded in the wavenumber. From this regularity splitting we conclude that hp-FEM for Maxwell’s equations does not suffer from the pollution effect, provided that there holds the scale-resolution condition a) that kh/p is sufficiently small and b) that p/ log k is bounded from below. At last, we present numerical experiments that confirm our findings.