arXiv subjects

Matteo Ferrari

Publications and source records attributed to Matteo Ferrari.

22 records · Page 2Linked to original sources

arXiv2022

CVEM-BEM coupling with decoupled orders for 2D exterior Poisson problems

For the solution of 2D exterior Dirichlet Poisson problems we propose the coupling of a Curved Virtual Element Method (CVEM) with a Boundary Element Method (BEM), by using decoupled approximation orders. We provide optimal convergence error estimates, in the energy and in the weaker $\textit{L}^\text{2}$-norm, in which the CVEM and BEM contributions to the error are separated. This allows taking advantage of the high order flexibility of the CVEM to retrieve an accurate discrete solution by using a low order BEM. The numerical results confirm the a priori estimates and show the effectiveness of the proposed approach.

math.NA↗

arXiv2021

On the minimal number of solutions of the equation $ ϕ(n+k)= M \, ϕ(n) $, $ M=1$, $2$

We fix a positive integer $k$ and look for solutions of the equations $ϕ(n+k) = ϕ(n)$ and $ϕ(n + k) = 2ϕ(n)$. We prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even and five for the second one when $k$ is odd. These results hold for $k \le 2 \cdot 10^{100}$. We also show that for the second equation with even $k$ there are at least three solutions for $k \le 4 \cdot 10^{58}$. Our work increases the previous minimal number of known solutions for both equations.

math.NT↗

arXiv2021

On the coupling of the Curved Virtual Element Method with the one-equation Boundary Element Method for 2D exterior Helmholtz problems

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the infinite region, in which the solution is defined, to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on this latter a non reflecting condition of boundary integral type. Then, we apply the curved virtual element method in the finite computational domain, combined with the one-equation boundary element method on the artificial boundary. We present the theoretical analysis of the proposed approach and we provide an optimal convergence error estimate in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.

math.NA↗

arXiv2020

On a basic mean value Theorem with explicit exponents

In this paper we follow a paper from A. Sedunova (2017) regarding R. C. Vaughan's basic mean value Theorem (Acta Arith. 1980) to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by A. C. Cojocaru and M. R. Murty (London Math. Soc. Stud. Texts 2006). As an application we derive a basic mean value Theorem for the von Mangoldt generalized functions.

math.NT↗