### Advanced Finite Element Methods for the numerical solution of Partial Differential Equations

This research area focuses on the development and application of advanced finite element methods (FEM) to tackle complex problems in computational mechanics and physics. Key topics include the use of discontinuous and discontinuous Galerkin approximations, non-matching techniques for finite element approximation of interface problems, and adaptive FEM for elliptic problems using regularized forcing data. The motivation behind these studies is to improve the accuracy, efficiency, and applicability of FEM in scenarios where traditional methods face limitations due to geometric complexities, material discontinuities, or the need for adaptive mesh refinement. A particular emphasis is given to open-source software development, and to high-performance computing applications based on the deal.II library.

*Members*: L. Heltai*Collaborators*: Andrea Cangiani (SISSA), Wenyu Lei (University of Electronic Science and Technology of China), Wolfgang Bangerth (Colorado State University), Martin Kronbichler (Ruhr University Bochum), Nella Rotundo (University of Florence)

### Computational Techniques for Fluid-Structure Interaction (FSI)

This area delves into the numerical simulation of fluid-structure interaction problems, employing cutting-edge computational techniques. The research includes the immersed finite element method, arbitrary Lagrangian-Eulerian discretizations, and the development of efficient preconditioners for complex FSI solvers. These works aim to address the challenges in accurately modelling and predicting the behaviour of complex FSI problems, which are critical in many engineering and biomedical applications.

*Members*: L. Heltai*Collaborators*: D. Boffi (KAUST), L. Gastaldi (University of Brescia), M. Wichrowski (Heidelberg University)

### Non-matching, Interface, and Coupling Methods for mixed-dimensional PDEs

Research in this area explores innovative numerical strategies for the effective coupling of mixed-dimensional domains and the finite element approximation of problems involving interfaces. Topics include immersed finite element methods, cut-fem, and the reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains. The objective is to develop robust and efficient methods for the simulation of multi-physical phenomena that span across different physical dimensions or require the coupling of disparate mathematical models, enhancing the simulation accuracy and computational performance in applications ranging from engineering to applied sciences.

*Members*: L. Heltai*Collaborators*: P. Zunino (Politecnico di Milano), L. O. Mueller (Università di Trento), A. Caiazzo (WIAS)

### Structure-preserving numerical methods

In recent years it has been realized that formulating numerical methods compatible with the physical properties of a model problem – and not just approximating them – has paved the way for numerical discretization with superior accuracy and stability properties. To this aim, structure-preserving or geometric numerical methods seek to achieve physically consistent numerical solutions by preserving the geometric and topological structure of the governing mathematical model. We work at the development of numerical methods and model order reduction for the structure-preserving approximation of differential equations arising in e.g. Hamiltonian mechanics, dissipative dynamics and fluid models.

*Members*: C. Pagliantini*Collaborators*: J. S. Hesthaven (EPFL), R. Hiptmair (ETH Zurich), B. A. de Dios (Milano Bicocca), F. Vismara (TU Eindhoven), D. Lombardi (INRIA Paris), A. Benvenuti (UniPi)

### Computational Plasma Physics

Plasma phenomena are ubiquitous in many physical applications ranging from astrophysics to controlled thermonuclear fusion. Efficient and accurate simulations of plasma models are however challenged by the complex, highly nonlinear and inherent multiscale nature of the physical phenomena involved. Research activities of the group are devoted to the development of numerical methods for fluid and kinetic plasma models based on discontinuous Galerkin, spectral methods and discrete differential forms. Explicit and implicit time integration schemes are investigated. The emphasis is on the study of computationally efficient techniques able to preserve the conservation properties of the problem while accurately coupling the microscopic physics with the macroscopic system-scale dynamics.

*Members*: C. Pagliantini*Collaborators*: G.L. Delzanno (Los Alamos National Laboratory), O. Koshkarov (Los Alamos National Laboratory), V. Roytershteyn (Space Science Institute)

### Model order reduction

Numerical simulation of parametrized differential equations is of crucial importance

in the study of real-world phenomena in applied science and engineering. The parameter here has to be understood in a broad sense as it can represent time, boundary condition, a physical parameter, etc. Computational methods for the simulation of such problems often

require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. In this line of research we are interested in theoretical aspects and application of model order reduction techniques.

*Members*: L. Heltai, C. Pagliantini*Collaborators*: J. S. Hesthaven (EPFL), D. Lombardi (INRIA Paris), F. Vismara (TU Eindhoven)

### Filtering and data assimilation

In this line of research we are concerned with the inverse problem of reconstructing an unknown function u or an unknown parameter y from a finite set of measurements of u(y). To counteract ill-posedness additional a priori information is given by assuming that the studied phenomena can be well described by certain physical models. These models can come in the form of ordinary or partial differential equations or they can be given by a prior probability distribution. The first hypothesis is associated with deterministic reconstruction algorithms (e.g. PBDW), while the second hypothesis is the starting point of Bayesian inversion where the data are used to compute a posterior distribution that represents the uncertainty in the solution. In both research directions, we are interested in the development of efficient filtering algorithms and in the study of related aspects, such as accuracy quantification, computational efficiency of the sampling strategies, dynamical approximation of the state, adaptation of the measurements via sensors’ update, observation noise, modeling errors, etc.

- Members: C. Pagliantini
*Collaborators*: O. Mula (TU/e), K. Veroy (TU/e), M. Grepl (RTWH), F. Silva (TU/e), F. Vismara (TU/e)

### Fractional Differential Equations

Research on numerical methods for Fractional Partial Differential Equations (FPDEs) is an emerging and rapidly growing field at the intersection of fractional calculus and computational mathematics. It focuses on developing efficient and accurate numerical techniques for solving FPDEs, which generalize classical partial differential equations by incorporating fractional derivatives. We focus on designing efficient solvers for the resulting discrete systems, considering the potentially large dense and structured matrices that arise from discretizing FPDEs. This includes developing preconditioning techniques and leveraging parallel computing architectures for improved performance. Our research is also involved in exploring applications of FPDEs in various scientific and engineering fields, including anomalous diffusion, viscoelastic materials, optimal control, and application to the field of complex networks.

*Members*: F. Durastante, S. Massei, L. Robol*Collaborators*: L. Aceto (UNIUPO), M. Mazza (UNITOV)