Space-time formulations are computationally demanding when compared to time-stepping schemes. The talk will focus on diagonalization-based methods that allow to efficiently solve a isogeometrically discretized heat-equation on a single patch. Diagonalization based methods are not new, e.g. fast diagonalization can be used for elliptic problems, but their application to evolution problems can lead to numerical instabilities that ruin the accuracy of the solution. Few techniques to address those instabilities will be presented that ensure both computational efficiency and numerical stability.

The acoustic-gravity wave system of equations is used in geophysics for describing a linear compressible free-surface flow. It is used in particular for investigating the potential use of hydrophones in tsunami early-warning systems. In this talk, I will present simulations of acoustic gravity waves generated by a submarine landslide. In order to capture the relevant frequencies, a high-order spectral element method is used. Focusing on the spectrogram for the pressure field, we show how to reproduce a characteristic pattern observed in real data. This is a joint work with Anne Mangeney (IPGP) and Sébastien Imperiale (Inria Saclay).

We build upon recent works by some of the authors on high-order spline-based geometric methods for the three-dimensional initial boundary value problem of Maxwell's equations. Ina previous work of some of the authors, the proposed schemes were restricted to single-patch geometries. To address the computational challenges associated with inverting mass matrices in the multi-patch setting, we extend some results for 0-forms mass and stiffness matrices to 1-forms mass matrices associated with the Hodge star operator. Our approach leverages the tensor-product structure of spline spaces to develop a spectrally equivalent preconditioner for the 1-forms mass matrices. Specifically, we construct an efficient preconditioner for a single-patch domain that remains robust with respect to both mesh size and spline degree. In the multi-patch case, we extend this methodology by integrating the single-patch preconditioner with an Additive Schwarz method, ensuring robustness and scalability with respect to the number of patches. The proposed preconditioner achieves near-optimal computational complexity, requiring $\mathcal{O}(N_{\mathrm{dof}}^{4/3})$ floating-point operations, where $N_{\mathrm{dof}}$ denotes the number of degrees of freedom. Finally, we validate the robustness and computational efficiency of the proposed preconditioners through numerical experiments on realistic problems.

In this talk, we discuss space-time formulations for the scalar wave equation. Although the time variable is treated as an additional dimension, a straightforward discretization of the variational formulation using conforming finite element spaces results in only conditional stability, which is already known from time-stepping methods as the CFL--condition. To remedy this behavior, we consider the equivalent residual minimization problem, which leads to an unconditionally stable scheme. Furthermore, the minimization problem reveals an optimal transformation operator that, when applied to the test space, stabilizes the variational formulation. We discuss numerical realizations and approximations of the transformation operator, as well as the stability of the resulting schemes when these transformations are applied. The theoretical findings are complemented by numerical examples. This talk is based on joint work with Christian Köthe (TU Graz), Olaf Steinbach (TU Graz) and Marco Zank (Universität Wien).

In recent years, there has been growing interest in the study of open fluid systems, motivated by, among others, their numerical analysis. Open fluid systems are characterized by their ability to exchange matter with the external environment; that is, if $\partial\Omega$ denotes the domain boundary and $n$ the outward-pointing normal vector, the normal velocity component $u \cdot n$ is not sign-constrained. In this talk, we explore the existence of local-in-time strong solutions to the Navier--Stokes--Fourier system in the $L^p$–$L^q$ setting. We also examine criteria under which these solutions may be extended to exist globally in time.

Advection dominated problems represent still nowadays a great challenge for the Model Order Reduction (MOR) community, because of their intrinsic difficult nature: this idea is strictly related to a slow decay of the so called Kolmogorov n-width of the problem under consideration. This slow decay translates into a poor efficiency of more standard reduction techniques. In this talk, in particular, we will focus on hyperbolic problems with self-similar solutions. I will present a MOR approach for transport dominated problems, in the non-parametrized and in the parametrized setting, with a particular focus on the SOD problem in 1D, the Double Mach Reflection problem and the triple point problem in 2D. The approach is based on the definition of suitable deformation maps from the physical domain into itself: these maps are obtained by means of an optimization procedure. Once the map is found, a standard POD on the "modified" snapshots is performed. For the online phase, an Artificial Neural Network approach is used to compute the coefficients of the online solution. The whole procedure represents a first step towards an ALE approach, and is applied to problems where the solution presents multiple travelling discontinuities (shocks, rarefactions), whose location in the physical domain is unknown. Promising results are shown, to highlight the good performance of the whole methodology.

We present and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides an additional control of the time derivative of the discrete solution. We derive hp-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. We present numerical experiments that validate our theoretical results and assess some additional aspects of the proposed method.

We consider a PDE-constrained optimization problem of tracking type with parabolic state equation, where we assume to have only limited observation in time. We restrict ourselves to the heat equation as a model problem. The solution to this problem is characterized by the first order optimality system (Karush-Kuhn-Tucker conditions), which is a linear system with saddle point structure. In order to solve this system with a Krylov space solver efficiently, we need a preconditioner, preferably a preconditioner such that the condition number of the preconditioned system is robust in the relevant model parameters (like regularization parameters) and discretization parameters (like time step sizes, spacial grid sizes, and polynomial degrees). The preconditioner is based on a Schur complement formulation for the state variable. This means that we have to formulate the optimality system in a strong form, where the state has $H^2$-regularity in space and $H^1$-regularity in time. Consequently, the Lebesque space $L^2$ has to be chosen as function space for Lagrange multipliers and the control. A conforming discretization for the state variable would be required to have basis functions that are continuously differentiable in the spacial directions. This can be done with some effort with finite elements, in Isogeometric Analysis such basis functions can be set up with ease. Since the optimality system has a saddle-point structure, we have to choose the discretization such that the Ladyzhenskaya-Babuška-Brezzi conditions are satisfied. We will discuss possible choices, keeping in mind that we want to reduce the number of degrees of freedom as much as possible. Since our constructions preserve a tensor-product structure between space and time, the overall preconditioner can be simplified utilizing a fast diagonalization approach in time, which allows for a time-parallel solution of spacial sub-problems. Our theoretical findings are accompanied by numerical experiments.