arXiv subjects

Nella Rotundo

Publications and source records attributed to Nella Rotundo.

6 recordsLinked to original sources

arXiv2026

Well-posedness and stability for a thermoelectromagnetic system

In this article we consider a system of coupled partial differential equations that mirrors the interconnection between the evolution of an electromagnetic field -- described by the Maxwell's system -- and that of the temperature distribution in a bounded region in three-dimension. A discussion of the mathematical model is provided. We establish the well-posedness of the initial-boundary value problems associated with the thermoelectromagnetic system, in an appropriate functional-analytic framework. Then, our investigation and main results pertain to the long-time behaviour of the solutions. It is shown that either exponential stability or convergence to stationary solutions hold true, according as the conductivity is positive definite or semidefinite, respectively; in the latter case, strong stability is attained in specific topological or analytical settings. This complements and expands earlier results obtained for the uncoupled Maxwell's system.

math.AP↗

arXiv2025

Regularity of solutions of the Navier-Stokes-{\alpha}{\beta} equations with wall-eddy boundary conditions

We establish global well-posedness and regularity for the Navier-Stokes-{\alpha}{\beta} system endowed with the wall-eddy boundary conditions proposed by Fried and Gurtin (2008). These conditions introduce a tangential vorticity traction proportional to wall vorticity and provide a continuum-mechanical model for near-wall turbulence. Our analysis begins with a variational formulation of the stationary fourth-order system, where we prove symmetry and a G{\aa}rding inequality for the associated bilinear form. We then verify Douglis-Nirenberg ellipticity and the Lopatinskii-Shapiro covering condition, establishing full Agmon-Douglis-Nirenberg regularity for the coupled system. Building on this framework, we derive a hierarchy of energy estimates for the nonlinear evolution equation, which yields global regularity, uniqueness, and stability. To our knowledge, this provides the first complete analytical treatment of the wall-eddy boundary model of Fried and Gurtin.

math.AP↗

arXiv2024

Forward lateral photovoltage scanning problem: Perturbation approach and existence-uniqueness analysis

In this paper, we present analytical results for the so-called forward lateral photovoltage scanning (LPS) problem. The (inverse) LPS model predicts doping variations in crystal by measuring the current leaving the crystal generated by a laser at various positions. The forward model consists of a set of nonlinear elliptic equations coupled with a measuring device modeled by a resistance. Standard methods to ensure the existence and uniqueness of the forward model cannot be used in a straightforward manner due to the presence of an additional generation term modeling the effect of the laser on the crystal. Hence, we scale the original forward LPS problem and employ a perturbation approach to derive the leading order system and the correction up to the second order in an appropriate small parameter. While these simplifications pose no issues from a physical standpoint, they enable us to demonstrate the analytic existence and uniqueness of solutions for the simplified system using standard arguments from elliptic theory adapted to the coupling with the measuring device.

math-ph↗

arXiv2022

A weighted Hybridizable Discontinuous Galerkin method for drift-diffusion problems

In this work we propose a weighted hybridizable discontinuous Galerkin method (W-HDG) for drift-diffusion problems. By using specific exponential weights when computing the $L^2$ product in each cell of the discretization, we are able to mimic the behavior of the Slotboom variables, and eliminate the drift term from the local matrix contributions, while still solving the problem for the primal variables. We show that the proposed numerical scheme is well-posed, and validate numerically that it has the same properties as classical HDG methods, including optimal convergence, and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter-Gummel finite volume scheme (i.e., it produces the same system matrix). The use of local exponential weights generalizes the Scharfetter-Gummel scheme (the state-of-the-art for finite volume discretization of transport dominated problems) to arbitrary high order approximations.

math.NA↗

arXiv2022

Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors

The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth, the recent redefinition of the 1kg to defect, and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system that describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three different data-driven approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by efficient physics-preserving finite volume solutions of the forward problem. While the linear least square method yields an average absolute $\ell^\infty$ error around $10\%$, the nonlinear networks roughly halve this error to $5\%$, respectively. Finally, we optimize the relevant hyperparameters and test the robustness of our approach with respect to noise.

math.NA↗

arXiv2019

Error estimates in weighted Sobolev norms for finite element immersed interface methods

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

math.NA↗