Talks

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Michele Aldé  

TU Wien 

Title: Weak and strong convergence of a BDF2-type integrator for the Landau–Lifshitz–Gilbert equation in micromagnetics 

Abstract: We consider the Landau–Lifshitz–Gilbert (LLG) equation, a nonlinear partial differential equation that models time-dependent micromagnetic phenomena. We present a fully discrete scheme combining first-order finite elements in space with a projection-free BDF2-type method of second order in time. Despite the nonlinearities of LLG, our scheme requires the solution of only one linear system at each time step. The unit-length constraint |m| = 1 is not enforced by normalization at each step, but is recovered in the limit. For a reduced model with only exchange energy and an applied external field, this is, to the best of our knowledge, the first linear higher-order-in-time integrator for LLG for which unconditional weak convergence to weak solutions [1] and optimal a-priori error estimates of order O(h +k2) for sufficiently smooth solutions [2] have both been established. Here, h and k denote the spatial mesh size and the time-step size, respectively. We then extend the scheme through an implicit-explicit (IMEX) strategy to incorporate additional energy contributions, including anisotropy, the nonlocal stray field, and the Dzyaloshinskii–Moriya interaction. The extension retains one linear solve per time step, unconditional weak convergence [3], and optimal a-priori error estimates [4]. Numerical experiments illustrate the theoretical results. 

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Bernhard Endtmayer 

Leibniz University Hannover 

Title: Goal-Oriented Error Control via Anisotropic hp DG Finite Element Spaces 

Abstract: In many applications, one is not interested in the solution itself, but in some specific quantity of interest (goal), evaluated at the solution. Here, we present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-dominated problems. The estimator is elementwise separated with respect to the single directions in space and time. This leads to directional error indicators, whichs quantify the anisotropy of the solution with respect to the goal . Finally, we present some numerical examples. 

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Dario Ferloni 

University of Twente 

Title: Optimal Complexity of Adaptive FEM for Second-Order Linear Elliptic PDEs Driven by Non-Residual Estimators

Abstract: We consider adaptive finite element methods for second-order lin ear elliptic PDEs, where the adaptive algorithm steers the local mesh refinement as well as an iterative algebraic solver. Under abstract as sumptions on the underlying a-posteriori error estimator and the solver, we prove that the usual adaptive algorithm leads to unconditional full R-linear convergence, independently of the user-chosen adaptivity pa rameters. For sufficiently small parameters, this guarantees optimal complexity in the sense that the decay rate of an appropriate error quantity is optimal with respect to the overall computation cost (and hence time) measured in terms of the usual nonlinear approximation classes. Unlike available results in the literature, the main focus is on the analytical treatment and understanding of non-residual estimators like averaging-based estimators as proposed by Zienkiewicz and Zhu or estimators based on equilibrated fluxes. In particular, our results ex tend [1, 2], where R-linear convergence at optimal rate with respect to the number of mesh elements is shown for adaptive algorithms steered by non-residual estimators, under the assumption that the discrete lin ear systems are solved exactly. 

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Samantha Glory Guiate Simo 

University of Göttingen 

Title: Embedded Trefftz Discontinuous Galerkin Methods for Parabolic PDEs.

Abstract: Trefftz discontinuous Galerkin (DG) methods are finite element methods built on standard DG methods, in which the local basis functions satisfy the differential equation on each element. This greatly reduces the number of degrees of freedom, especially for high polynomial orders. The recent embedded Trefftz variant [1] builds this space implicitly inside a standard DG space, extending the method to a broad class of mostly elliptic equations. In this talk, we extend the embedded Trefftz approach to transient problems along two routes: time stepping within a class of time discretization schemes, and a space-time formulation. We introduce and discuss both methods and demonstrate their performance numerically. 

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Fatima Hasanova 

RICAM, Austrian Academy of Sciences 

Title: Robust approximation error estimates for AS-G1 multi-patch isogeometric spaces.  

Abstract: Conforming discretizations of fourth-order problems, such as the biharmonic equation or Kirchhoff–Love shells, need 𝐻2-regularity. In isogeometric analysis (IGA), this means 𝐶1-smooth splines. On multi-patch domains, such spaces can be constructed with analysis-suitable 𝐺1 (AS-𝐺1) parametrizations [1]. These spaces work well in practice, but so far the approximation error estimates have been missing. In this talk, we close this gap. We construct a local, patchwise projector that keeps the continuity across patch interfaces, and we use it to derive error estimates for AS-𝐺1 multi-patch spaces. The estimates give optimal rates in the mesh size ℎ and are robust in the spline degree 𝑝. The theory is based on [2, 3]. We use these results to develop multigrid solvers for biharmonic problems and analyze their robustness with respect to mesh size, degree and geometry.

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Chiara Hergl 

Leibniz University Hannover 

Title: Nonlinear Proper Orthogonal Decomposition Using POD-Enriched Snapshots 

Abstract: In this work, we present an extension of the proper orthogonal decomposition (POD) combined with sampling methods,  such as the discrete empirical interpolation method or S-optimal.  The error in nonlinear reduced order simulations have various causes.   An error arises from spanning the nonlinear basis using full-order solutions.  The error induced by the POD can cause the nonlinear component calculated in the reduced formulation to lie outside  the spanned space, causing the solution to deviate further from the actual solution over time.  We enrich the snapshots of the nonlinear portion using POD-error-induced calculations, which can reduce this error. The enrichment is tested on the well-known “Flow around a cylinder” benchmark, as well as on a fluid-structure-concentration interaction example. 

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Pablo Munoz 

University of Klagenfurt, Austria 

Title: Operator Splitting for the Non-Autonomous Bloch–Torrey Equation: Parameter Reconstruction and Transmission Problems 

Abstract: Abstract. We present a computational framework for model-based quantitative imaging based on the Bloch–Torrey equation [5], a non-autonomous PDE governing magnetiza tion dynamics in magnetic resonance imaging. The equation couples time-dependent RF-induced precession, longitudinal and transverse relaxation, and diffusion–advection transport, the interplay of rotational, dissipative, and elliptic operators on different time scales poses challenges for discretization, analysis, and adjoint derivation. For the forward problem, we prove first-order convergence of Lie–Trotter splitting for non-autonomous evolution equations comprising three parts [1], combined with a finite element discretization in space [4]. On this basis, we derive a discrete adjoint consistent with the splitting scheme [3] and use it within an L-BFGS method [2] to identify the spatially varying parameters R1, R2, and Meq. We present numerical experiments that validate the theoretical convergence orders of the forward problem and illustrate the reconstruction based on the splitting scheme and finite elements. Finally, we present an extension of the splitting scheme to non-autonomous time-dependent PDEs with discontinuous parameters, which leads to transmission problems. Distinguishing between self-adjoint operators and non-self-adjoint perturbations, we generalize the error and convergence estimates to coefficients that jump across interfaces inside the domain. 

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Göksu Oruc 

Ulm University 

Title: Non-existence of travelling waves in fractional Camassa-Holm equation 

Abstract: In this study, we consider the fractional Camassa-Holm equation (fCH) which models the propagation of small-amplitude long unidirectional waves in a homogenous medium made of nonlocally and nonlinearly elastic material.  We address the non-existence of solitary wave solutions to the fCH equation and investigate numerically different aspects of these solutions which arise due to delicate balance between nonlinearity and dispersion.  The Petviashvili method is proposed for the numerical generation of positive and negative solitary wave solutions. 

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Dirk Praetorius 

TU Wien 

Title: Rate-optimal adaptive FEM with dynamic Dörfler marking 

Abstract: Meanwhile convergence and optimal convergence rates of adaptive FEM (AFEM) are fairly well understood,at least, for linear elliptic PDEs. The current state of the art on the standard AFEM loop  SOLVE — ESTIMATE — MARK — REFINE   can be paraphrased as follows: First, any Dörfler marking parameter 0 < \theta <= 1 guarantees R-linear convergence of the estimator sequence. Second, if \theta is sufficiently small, then the estimator sequence decays at rate s > 0 with respect to the number of degrees of freedom, where s is optimal, i.e., as large as possible in terms of certain nonlinear approximation classes.   In practice, one observes that also „moderate“ \theta lead to optimal convergence rates, but the thorough mathematical argument for this observation is still open.  In our talk, we propose a slight modification of the adaptive loop, where the Dörfler marking is chosen dynamically (and decays logarithmically to 0). If a convergence rate s > 0 is possible, then the extended algorithm guarantees convergence at any rate 0 < r < s. Eventually, this proves that the extended AFEM loop is almost optimal. 

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Mira Schedensack 

Universität Leipzig 

Title: A posteriori error estimates for a modified Morley FEM 

Abstract: Residual-based a posteriori error estimators are derived for the modified Morley FEM, proposed by Wang, Xu, Hu for the singularly perturbed biharmonic equation and the nonlinear von Karman equations. The error estimators are reliable and efficient. Moreover, an adaptive algorithm driven by these error estimators is investigated in numerical experiments. 

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Mechthild Thalhammer 

Universität Innsbruck 

Title: On the design and analysis of modified operator splitting methods for nonlinear evolution equations 

Abstract: Exponential operator splitting methods constitute a favourable class of time integration methods for various kinds of linear and nonlinear evolution equations. They rely on the presumption that the defining right-hand side comprises two (or more) operators and that the numerical approximation of the associated subproblems is significantly simpler compared to the numerical approximation of the original problem. Under these premises, their excellent behaviour with respect to stability, accuracy, and the preservation of conserved quantities has been confirmed by a remarkable amount of contributions.  In my talk, I will review well-known achievements and recent advances on the design and analysis of exponential operator splitting methods. I will contrast standard splitting schemes involving real coefficients with two alternative approaches that are based on the incorporation of complex coefficients or double commutators, respectively. As fundamental test problems, I will study Gross-Pitaevskii equations modelling Bose-Einstein condensates, their parabolic counterparts arising in ground and excited state computations, complex Ginzburg-Landau equations having a similar structure, and high-order semilinear parabolic equations describing quasicrystalline patterns. 

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Ngoc Tien Tran 

University of Augsburg 

Title: On numerical approximation of obstacle problems with Hamilton-Jacobi-Bellman differential operators 

Abstract: Hamilton-Jacobi-Bellman (HJB) equations are a class of fully nonlinear partial differential equations that arises from applications in stochastic control.  When a stopping criterion is involved, we obtain an obstacle problem associated with an HJB differential operator. Even in the popular setting with the Cordes condition in convex domains, the solution is not characterized by a variational formulation. This talk presents an early design and analysis of numerical methods for this class of problems. 

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Martin Vetter 

TU Wien 

Title: „Inertial Manifold Neural Operators for the Kuramoto–Sivashinsky equation“ 

Abstract: The Kuramoto–Sivashinsky equation (KSE) exhibits chaotic dynamics, making accurate long-time approximation particularly challenging. In this talk, we present a neural operator framework for approximating its solutions. We show that the network size grows at most polynomially with the KSE scale parameter $L$, thereby avoiding the curse of dimensionality.  Our analysis is based on an approximate inertial manifold (AIM), which provides a finite-dimensional approximation of the PDE dynamics by reducing to a system of ODEs. We establish holomorphic regularity of both the AIM map and the solution map of the reduced ODE system, which provides the foundation for our error analysis.  For the numerical realization, we approximate the AIM map by a finite number of Banach fixed-point iterations, which have been shown to be learned effectively. Combined with a continuous Galerkin hp-discretization of the reduced ODEs, this yields a constructive approach to neural operator approximation. 

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Enrico Zampa 

University of Vienna 

Title: „Second- vs. First-Order-in-Time Space–Time Methods for the Wave Equation: Stability, Convergence, and Geometric Properties“ 

Abstract: Second-order-in-time space–time Galerkin methods for wave equations offer an attractive alternative to first-order formulations, as they require only the wave field as an unknown and can be implemented efficiently. However, standard discretizations may be only conditionally stable, and the theoretical properties of stabilized variants have remained unclear.  In this talk, we establish a precise equivalence between a stabilized second-order-in-time space–time formulation, based on continuous piecewise-polynomial trial functions and a projection of the test functions onto discontinuous polynomials of one degree lower in time, and the established first-order-in-time discontinuous Galerkin–continuous Galerkin (DG–CG) method. For the linear wave equation, this equivalence yields unconditional stability, optimal-order error estimates, and energy conservation at time nodes, while reducing the number of unknowns by half. We also discuss an efficient implementation based on under-integration in time. The framework is then extended to semilinear wave equations, including the sine-Gordon equation.  This talk is based on joint work with Matteo Ferrari and Ilaria Perugia.  

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Posters

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Oskar Broukal

TU Wien

Title: Wavelet compression for sampling Deep Gaussian Processes

Abstract: Deep Gaussian processes (DGPs) use iterative sampling to create non-stationary random fields. This is notoriously expensive because the covariance at each non-initial layer depends on the realised field from the layer below, so it changes with every sample. Moreover, standard fast sampling techniques like circulant embedding or matrix compression fail since the underlying kernel usually lacks the necessary regularity and structure. We develop and analyse a wavelet compression method for convolution Matérn DGPs. At each layer the covariance matrix is compressed using wavelets and a Lanczos algorithm computes the random sample. For specific parameter ranges involving the smoothness of the Matérn kernel as well as the smoothness of the wavelets, we show that the accumulative compression and sampling errors through all layers decay exponentially with increasing truncation parameter. We obtain a practical DGP sampling algorithm by choosing a truncation that gives log-linear covariance storage and matrix-vector costs, enabling DGP sampling at nearly linear computational cost.

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Felix Engertsberger 

JKU 

Title: Energy-based modeling and simulation of magnetic hysteresis 

Abstract: Accurate simulations of electrical machines and transformers require models that capture the nonlinear response of ferromagnetic materials. At low frequencies, magnetic hysteresis is particularly important and can be described by rate-independent, nonsmooth constitutive laws. This poster presents a systematic approach to incorporating magnetic hysteresis into magnetic field simulations within an energy-based modeling framework. Using tools from convex analysis and optimization, we establish well-posedness of the resulting field problem and global convergence of iterative solvers under suitable assumptions. Numerical simulations of an electrical machine then illustrate the theoretical results. 

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Eva Maria Haslhofer 

JKU 

Title: Model Order Reduction for Thermal and Electromagnetic Problems 

Abstract: Model Order Reduction is an essential tool for speeding up simulations of electric machines and devices and reducing needed memory storage while guaranteeing the required level of accuracy. We present the Reduced Basis (RB) Method and demonstrate how it can be efficiently split into a costly offline phase and a very fast online phase for a linear heat transfer problem in a squirrel cage induction machine. For nonlinear problems, it is more difficult to evaluate the nonlinearity in a way that also results in an efficient online phase. We employ the Empirical Interpolation Method (EIM) to approximate the nonlinear terms as linear combinations of parameter-independent basis functions, with coefficients determined by evaluating the nonlinearities at selected interpolation points. We demonstrate its performance for a nonlinear magnetostatic problem involving an inductor and investigate its limitations in highly nonlinear regimes. 

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Chiara Hergl 

Leibniz University Hannover 

Title: Model Order Reduction: POD-DEIM for Laser Heating

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Joseph Holten

Karlsruhe Institute of Technology

Title: Efficient Simulation of Dynamic Elastic Beam Networks

Abstract: Fiber-based materials such as paper can be modeled as spatial networks ofTimoshenko beams, coupled at the nodes by continuity of displacements androtations and by balance of forces and moments. We discretize the resultingelastic wave propagation problem in space by a hybridizable discontinuousGalerkin (HDG) method. Since the coupling points are zero-dimensional, theglobal problem in each time step reduces to a symmetric positive definitesystem on the network nodes whose size is independent of the polynomial degree.As the edge lengths of realistic fiber networks vary over several orders ofmagnitude, explicit time stepping is ruled out by the CFL condition. Wetherefore integrate the semi-discrete system with energy-conservative implicitschemes and prove a priori error estimates of optimal order in space and time.The condensed systems are solved by preconditioned conjugate gradients, withpreconditioners that exploit the network structure and whose convergence isuniform in the network size and the time step. Numerical experiments onnetworks with millions of nodes confirm the theory.

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Tobias Kaltenbacher

Institute of Applied Mathematics, Graz University of Technology

Title: Space-time finite element methods for quasi-static elliptic problems

Abstract: This poster presents space-time finite element methods for quasi-static Poisson and Stokes problems with time-dependent data. Using suitable Bochner spaces, we formulate both problems in a variational setting that allows us to discretize space and time within a unified finite element framework.

We first consider tensor-product space-time meshes, obtained by combining a spatial mesh with a partition of the time interval. We then extend the approach to fully unstructured simplicial space-time meshes, which provide more flexibility in mesh construction. For the Stokes problem, we use a mixed formulation in which velocity and pressure are approximated simultaneously.

Finally, numerical examples demonstrate the methods for both problems on tensor-product and fully unstructured space-time meshes.

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Dorian Zeilinger

JKU

Title: Structured Neural Network Models for Anisotropic Magnetic Constitutive Laws

Abstract: High-performance electrical steel used as core material in power transformers and generators exhibits significant nonlinearity and anisotropy. While the modeling and simulation of isotropic nonlinear behavior is well-understood, the incorporation of magnetic anisotropy remains challenging. This work proposes to utilize a specific structure neural network architecture to systematically describe anisotropic nonlinear anhysteretic magnetic material response of electrical steel sheets. By constructing a convex and smooth co-energy density with dedicated behavior at infinity, we can ensure thermodynamic consistency and correct physical behavior and further guarantee basic mathematical properties which provide well-posedness of corresponding magnetic field problems and efficient numerical solvers. Numerical tests demonstrate the capability of the proposed method to describe strongly anisotropic behavior as well as its robust convergence in finite-element simulations.