We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

In 2019, diabetes caused 1.5 million global deaths, with 48% occurring before age 70. While Type 1 diabetes strongly depends on genetic components and is usually diagnosed in childhood, Type 2 diabetes is primarily caused by long term consumption of high calories foods. Lifestyle choices significantly influence the risk of Type 2 diabetes and obesity, including energy intake, diet composition, physical activity, and smoking.

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Analogies between financial markets and critical phenomena have long been observed empirically. So far, no convincing theory has emerged that can explain these empirical observations. Here, we take a step towards such a theory by modeling financial markets as a lattice gas.

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

In this short course, we will introduce some elements in deriving the hp a posteriori error estimate for a high-order unfitted finite element method for elliptic interface problems. The key ingredient is an hp domain inverse estimate, which allows us to prove a sharp lower bound of the hp a posteriori error estimator.

The studies in numerical approximation of partial differential equations are characterized by the necessity of managing complex geometries and their discretization. We focus our attention on two different fields where complex geometries are very common: the mathematical modeling of fluid-structure interaction problems and the family of virtual element methods.

Tensor network techniques are known for their ability to approximate low-rank structures and beat the curse of dimensionality. They are also increasingly acknowledged as fundamental mathematical tools for efficiently solving high-dimensional Partial Differential Equations (PDEs). In this talk, we present a novel method that incorporates the Tensor Train (TT) and Quantized Tensor Train (QTT) formats for the computational resolution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian coordinates.

A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties.

In this talk we propose and validate a Space Multiscale model for the description of particle diffusion in the presence of trapping boundaries. We start from a drift diffusion equation in which the drift term describes the effect of bubble traps, and it is simulated by the Lennard–Jones potential.

Abstract

Multifluid has attracted a lot of attention in recent years.

Abstract

In this talk, we will discuss the m

After a brief introduction to the field of Computational Cardiology and cardiac reentry, we introduce and study some scalable domain decomposition preconditioners for cardiac reaction-diffusion models, discretized with splitting semi-implicit techniques in time and isoparametric finite elements in space.

In this seminar, we will present our work on Virtual Element Method (VEM) approximations. The Virtual Element Method is a recent numerical technique for solving partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. This work aims to develop effective linear solvers for general order VEM approximations of three-dimensional scalar elliptic equations in mixed form and Stokes equations. To this end, we consider block algebraic multigrid preconditioners and balancing domain decomposition by constraints (BDDC) preconditioners. The latter allows us to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the differential problems are indefinite, ill-conditioned, and of saddle point nature.

This talk discusses the non-conforming approximation of Biot's consolidation model. In the first part of the talk, we discuss posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. In the second part of the talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young’s modulus and hydraulic conductivity field.

Eigenvalue problems arising from partial differential equations are used to model several applications in science and engineering, ranging from vibrations of structures, industrial microwaves, photonic crystals, and waveguides, to particle accelerators.

An efficient method is proposed for the numerical solution of the Stokes equations in a domain with a moving bubble and two techniques for the treatment of the boundary conditions are adopted and then compared. The treatment of diffusion of surfactants (anions and cations) in presence of an oscillating bubble is an interesting interdisciplinary problem, with applications to chemistry and biology.

Real life multi-phase and multi-physics problems coupled across different scales present outstanding challenges, whose practical resolution often require unconventional numerical methods.

Finite element approximation of Stokes equations with non-smooth data

Monolithic matrix-free solver for fluid-structure interaction problems.

In the classical theory of the finite element approximation of elliptic partial differential equations, based on standard Galerkin schemes, the energy norm of the error decays with the same rate of convergence as the best finite element approximation, without any additional requirements on the involved spaces.

Semi-implicit schemes for evolutionary partial differential equations. Topic 3 - construction of more general schemes for evolutionary partial differential equations, in which the stiffness may be of a different type than the one previously considered. Several examples will be given illustrating the general procedure.

Implicit-Explicit schemes for hyperbolic systems with stiff relaxation. Topic 2 - hyperbolic relaxation models and to the methods for their numerical solution. After introduction of hyperbolic-hyperbolic and hyperbolic-parabolic type relaxation problem, conservative finite difference space discretization will be introduced.

Construction of high order finite volume and finite difference shock-capturing schemes for conservation laws. Topic 1 - illustrating how to construct shock capturing schemes for conservation laws. We focus on semi-discrete schemes based on the method of lines.