arXiv subjects

Micol Pennacchio

Publications and source records attributed to Micol Pennacchio.

11 recordsLinked to original sources

arXiv2025

Artificial neural network evaluation of geometric constants for polygonal domains

We propose an approach based on Artificial Neural Networks (ANNs) to evaluate geometric constants relevant to the analysis and design of numerical schemes for partial differential equations. These constants play a central role, significantly influencing, for instance, a posteriori error estimates and the overall design of the computational strategy. Our technique leverages ANNs to learn the dependencies between these constants and a set of descriptive geometric features associated to polytopal mesh elements. The main computational costs are confined to data processing and training phases, which can be performed offline once and for all. This yields an effective tool for computing the constants, which we verify and show to be applicable across different scenarios, without substantial modifications - demonstrating its broader usability beyond the specific example considered.

math.NA↗

arXiv2024

The virtual element method on polygonal pixel-based tessellations

We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains. We focus on the case of approximating domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. This allows us to fully exploit the potential of higher order methods. Efficiency is ensured by a novel static condensation strategy acting on the edges of the decomposition.

math.NA↗

arXiv2023

Interior estimates for the Virtual Element Method

We analyze the local accuracy of the virtual element method. More precisely, we prove an error bound similar to the one holding for the finite element method, namely, that the local $H^1$ error in a interior subdomain is bounded by a term behaving like the best approximation allowed by the local smoothness of the solution in a larger interior subdomain plus the global error measured in a negative norm.

math.NA↗

arXiv2022

Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements

In the framework of virtual element discretizazions, we address the problem of imposing non homogeneous Dirichlet boundary conditions in a weak form, both on polygonal/polyhedral domains and on two/three dimensional domains with curved boundaries. We consider a Nitsche's type method [43,41], and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in [9]. We prove that also for the virtual element method (VEM), provided the stabilization parameter is suitably chosen (large enough for Nitsche's method and small enough for the Barbosa-Hughes Lagrange multiplier method), the resulting discrete problem is well posed, and yields convergence with optimal order on polygonal/polyhedral domains. On smooth two/three dimensional domains, we combine both methods with a projection approach similar to the one of [31]. We prove that, given a polygonal/polyhedral approximation $Ω_h$ of the domain $Ω$, an optimal convergence rate can be achieved by using a suitable correction depending on high order derivatives of the discrete solution along outward directions (not necessarily orthogonal) at the boundary facets of $Ω_h$. Numerical experiments validate the theory.

math.NA↗

arXiv2021

BDDC and FETI-DP for the Virtual Element Method

We build and analyze Balancing Domain Decomposition by Constraint (BDDC) and Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioners for elliptic problems discretized by the virtual element method (VEM). We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments confirm the theory.

math.NA↗

arXiv2021

High order VEM on curved domains

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $Ω$ with curved boundaries. Given a polygonal approximation $Ω_h$ of the domain $Ω$, the standard order $m$ VEM [6], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [16] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of $Ω_h$, which, to retain computability, is evaluated after applying the projector $Π^\nabla$ onto the space of polynomials. Numerical experiments confirm the theory.

math.NA↗

arXiv2021

FETI-DP for the three-dimensional Virtual Element Method

We deal with the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioner for elliptic problems discretized by the virtual element method (VEM). We extend the result of [22] to the three dimensional case. We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments validate the theory

math.NA↗

arXiv2021

Stabilization of the nonconforming virtual element method

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.

math.NA↗

arXiv2021

VEM and the Mesh

In this work we report some results, obtained within the framework of the ERC Project CHANGE, on the impact on the performance of the virtual element method of the shape of the polygonal elements of the underlying mesh. More in detail, after reviewing the state of the art, we present a) an experimental analysis of the convergence of the VEM under condition violating the standard shape regularity assumptions, b) an analysis of the correlation between some mesh quality metrics and a set of different performance indexes, and c) a suitably designed mesh quality indicator, aimed at predicting the quality of the performance of the VEM on a given mesh.

math.NA↗

arXiv2018

Algebraic Multigrid Methods For Virtual Element Discretizations: A Numerical Study

We investigate the performance of algebraic multigrid methods for the solution of the linear system of equations arising from a Virtual Element discretization. We provide numerical experiments on very general polygonal meshes for a model elliptic problem with and without highly heterogeneous diffusion coefficients and we draw conclusions regarding the efficacy of the method.

math.NA↗

arXiv2018

FETI-DP preconditioners for the Virtual Element Method on general 2D meshes

We analyze the performance of a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method, for the efficient solution of very large linear systems arising from elliptic problems discretized by the Virtual Element Method. We provide numerical experiments on a model linear elliptic problem with highly heterogeneous diffusion coefficients on arbitrary Voronoi meshes, which we modify by adding nodes and edges deriving from the intersection with an unrelated coarse decomposition. The experiments confirm also in this case that the FETI-DP method is numerically scalable with respect to both the problem size and number of subdomains, and its performance is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements.

math.NA↗