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Lucia Gastaldi

Publications and source records attributed to Lucia Gastaldi.

At least 19 recordsLinked to original sources
arXiv2026

Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.

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arXiv2025

On the stability and conditioning of a fictitious domain formulation for fluid-structure interaction problems

We consider a distributed Lagrange multiplier formulation for fluid-structure interaction problems in the spirit of the fictitious domain approach. This is an unfitted method, which does not require the construction of meshes conforming to the interface. We focus on the stationary problem arising from the time discretization and we analyze the behavior of the condition number with respect to mesh refinement. At the numerical level, the computation of the term coupling the fluid and the solid mesh requires the knowledge of the intersection between fluid and mapped solid elements and it might happen that a portion of the intersected elements is very small. We show that our formulation is stable independently of such intersections and that the conditioning is not affected by the interface position.

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arXiv2025

Advances on the finite element discretization of fluid-structure interaction problems

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.

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arXiv2024

A posteriori error estimator for elliptic interface problems in the fictitious formulation

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.

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arXiv2024

Quadrature error estimates on non-matching grids in a fictitious domain framework for fluid-structure interaction problems

We consider a fictitious domain formulation for fluid-structure interaction problems based on a distributed Lagrange multiplier to couple the fluid and solid behaviors. How to deal with the coupling term is crucial since the construction of the associated finite element matrix requires the integration of functions defined over non-matching grids: the exact computation can be performed by intersecting the involved meshes, whereas an approximate coupling matrix can be evaluated on the original meshes by introducing a quadrature error. The purpose of this paper is twofold: we prove that the discrete problem is well-posed also when the coupling term is constructed in approximate way and we discuss quadrature error estimates over non-matching grids.

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arXiv2024

On the stabilization of a virtual element method for an acoustic vibration problem

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.

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arXiv2023

Approximation of the Maxwell eigenvalue problem in a Least-Squares setting

We discuss the approximation of the eigensolutions associated with the Maxwell eigenvalues problem in the framework of least-squares finite elements. We write the Maxwell curl curl equation as a system of two first order equation and design a novel least-squares formulation whose minimum is attained at the solution of the system. The eigensolution are then approximated by considering the eigenmodes of the underlying solution operator. We study the convergence of the finite element approximation and we show several numerical tests confirming the good behavior of the method. It turns out that nodal elements can be successfully employed for the approximation of our problem also in presence of singular solutions.

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arXiv2023

A parallel solver for FSI problems with fictitious domain approach

We present and analyze a parallel solver for the solution of fluid structure interaction problems described by a fictitious domain approach. In particular, the fluid is modeled by the non-stationary incompressible Navier-Stokes equations, while the solid evolution is represented by the elasticity equations. The parallel implementation is based on the PETSc library and the solver has been tested in terms of robustness with respect to mesh refinement and weak scalability by running simulations on a Linux cluster.

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arXiv2022

A parallel solver for fluid structure interaction problems with Lagrange multiplier

The aim of this work is to present a parallel solver for a formulation of fluid-structure interaction (FSI) problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The fluid subproblem, consisting of the non-stationary Stokes equations, is discretized in space by $\mathcal{Q}_2$-$\mathcal{P}_1$ finite elements, whereas the structure subproblem, consisting of the linear or finite incompressible elasticity equations, is discretized in space by $\mathcal{Q}_1$ finite elements. A first order semi-implicit finite difference scheme is employed for time discretization. The resulting linear system at each time step is solved by a parallel GMRES solver, accelerated by block diagonal or triangular preconditioners. The parallel implementation is based on the PETSc library. Several numerical tests have been performed on Linux clusters to investigate the effectiveness of the proposed FSI solver.

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arXiv2022

Unfitted mixed finite element methods for elliptic interface problems

In this paper, new unfitted mixed finite elements are presented for elliptic interface problems with jump coefficients. Our model is based on a fictitious domain formulation with distributed Lagrange multiplier. The relevance of our investigations is better seen when applied to the framework of fluid structure interaction problems. Two finite elements schemes with piecewise constant Lagrange multiplier are proposed and their stability is proved theoretically. Numerical results compare the performance of those elements, confirming the theoretical proofs and verifying that the schemes converge with optimal rate.

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arXiv2022

On the interface matrix for fluid-structure interaction problems with fictitious domain approach

We study a recent formulation for fluid-structure interaction problems based on the use of a distributed Lagrange multiplier in the spirit of the fictitious domain approach. In this paper, we focus our attention on a crucial computational aspect regarding the interface matrix for the finite element discretization: it involves integration of functions supported on two different meshes. Several numerical tests show that accurate computation of the interface matrix has to be performed in order to ensure the optimal convergence of the method.

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arXiv2021

Existence, uniqueness, and approximation of a fictitious domain formulation for fluid-structure interactions

In this paper we describe a computational model for the simulation of fluid-structure interaction problems based on a fictitious domain approach. We summarize the results presented over the last years when our research evolved from the Finite Element Immersed Boundary Method (FE-IBM) to the actual Finite Element Distributed Lagrange Multiplier method (FE-DLM). We recall the well-posedness of our formulation at the continuous level in a simplified setting. We describe various time semi-discretizations that provide unconditionally stable schemes. Finally we report the stability analysis for the finite element space discretization where some improvements and generalizations of the previous results are obtained.

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arXiv2020

Virtual element approximation of eigenvalue problems

We discuss the approximation of eigenvalue problems associated with elliptic partial differential equations using the virtual element method. After recalling the abstract theory, we present a model problem, describing in detail the features of the scheme, and highligting the effects of the stabilizing parameters. We conlcude the discussion with a survey of several application examples.

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arXiv2020

Approximation of PDE eigenvalue problems involving parameter dependent matrices

We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=λ\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples we show that the presence of one (or both) parameters can produce unexpected results.

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arXiv2020

Splitting schemes for a Lagrange multiplier formulation of FSI with immersed thin-walled structure: stability and convergence analysis

The numerical approximation of incompressible fluid-structure interaction problems with Lagrange multiplier is generally based on strongly coupled schemes. This delivers unconditional stability but at the expense of solving a computationally demanding coupled system at each time-step. For the case of the coupling with immersed thin-walled solids, we introduce a class of semi-implicit coupling schemes which avoids strongly coupling without compromising stability and accuracy. A priori energy and error estimates are derived. The theoretical results are illustrated through numerical experiments in an academic benchmark.

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arXiv2019

Analysis of finite element approximations of Stokes equations with non-smooth data

In this paper we analyze the finite element approximation of the Stokes equations with non-smooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces, and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid-driven cavity problem improving the error estimates obtained in [Z. Cai and Y. Wang, Math. Comp., 78(266):771-787, 2009]. Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and efficiency, and show some numerical examples which suggest that optimal order of convergence is obtained by an adaptive procedure based on our estimator.

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