Interested students are given several opportunities for a fruitful Ms and Ph.D. program. A common denominator for the research topics is that the numerical approach is based on rigorous mathematical analysis of the approximation schemes (well-posedness, stability, convergence, etc.) and on the numerical validation of the theoretical results.

**Research Topics**

**Approximation of eigenvalue problems arising from partial differential equations**

Eigenvalue problems arising from partial differential equations model several interesting phenomena. Surprisingly enough, the eigenvalue problem in mixed form requires much deeper mathematical analysis than the standard one in order to show the convergence of their finite element approximation. The abstract theory, in this case, is well understood and relies on suitable conditions ensuring a so-called discrete compactness property for the involved finite element spaces.

**People: **Linda Alzaben, Moataz Alghamdi, Abdul Halim

**Fluid-structure interaction problems**

The Immersed Boundary Method (IBM) is a method that has been introduced by Peskin in the 70's for the modeling and the approximation of fluid-structure interactions arising from biological models. The peculiarity of the IBM consists in the fact that the presence of the solid, immersed in the fluid, is modeled via an additional source term to the Navier-Stokes equations describing the fluid evolution. This allows for a fluid resolution on a fixed mesh; moreover, another important feature is that the source term referring to the immersed body is defined by making use of a description of the solid in its reference configuration. The original version of the IBM makes use of finite differences for the resolution of the Navier-Stokes equations. Professor Boffi when at the University of Pavia developed a finite element version of the IBM which present several interesting properties. The research in this field is still very active and offers opportunities, both from theoretical and practical points of view.

**People: **Fabio Credali, Najwa Alshehri

**Analysis and implementation of adaptive finite element schemes**

The analysis of adaptive finite element schemes has been the object of increasingly active research during the last decade. In particular, the convergence of the adaptive scheme and the optimal rate of convergence has been proved for several problems of interest. More recently, adaptive schemes for eigenvalue problems have been considered. In this case, it has been observed that it is more natural to consider the approximation of clusters of eigenvalues rather than of single eigenvalues. This is, in particular, necessary when multiple (or very close to each other) eigenvalues are present. Rigorous proof of the optimal rate of convergence for the adaptive scheme in case of mixed problems is a tough problem and has been solved only for very particular situations.

**People: **Najwa Alshehri

**Application of FEM to electromagnetism**

The finite element approximation of problems related to Maxwell's system requires special finite element spaces originally introduced by Nédélec. For these spaces, several properties are known: discrete compactness property, (lack of) approximation properties on distorted meshes, etc. The right setting for this study is the so-called *Finite element exterior calculus* where tools of differential topology related to De Rham complex are used for the construction of finite element spaces.

**Virtual Element Methods**

The virtual element methods are a generalization of the Finite Element Methods (FEM) where the usual FEM space is enriched by a non-polynomial function inside of each element. Such non-polynomial functions are usually obtained by solving an elliptic partial differential equation and therefore are impractical to work with. In particular, in order to deal with such non-polynomials functions, one works with easy to compute degrees of freedom without actually computing the virtual component. The Numerical Methods for PDEs research group is focused on constructing conforming virtual elements for problems in mixed formulation.

**People: **Linda Alzaben, Fabio Credali

**Least Square Finite Element Methods**

Least Square Finite Element Methods (LSQFEM) are an extension of the Finite Element Method (FEMI) that is based on finding a minimization principle associated with the PDE we aim to solve. Such an approach allows proving analytical error estimate in an easier manner rather than as usually done for FEM when dealing with a mixed formulation. In particular, the Numerical Methods for PDEs research group is focused on studying eigenvalue problems associated with the LSQFEM formulation of the PDE.

**People: **Linda Alzaben

**Surfactants** **Multiscale Modeling**

Multiscale model and numerical methods for the simulation of the diffusion of surfactants (anions and cations) in presence of an oscillating bubble. The ions satisfy a drift-diffusion equation subject to a potential describing the interaction with the bubble, while the underlying fluid satisfies an incompressible Navier-Stokes equation in a moving domain.

The problem is intrinsically multiscale in space, because both the range of action of the bubble and the amplitude of oscillation of the bubble surface are different orders of magnitude smaller than the domain. There is also a multiscale aspect in time because, in typical experiments, the oscillation period is different orders smaller than the diffusion time.

**People: **Clarissa Astuto