arXiv subjects

Ilaria Perugia

Publications and source records attributed to Ilaria Perugia.

At least 37 records · Page 2Linked to original sources

arXiv2023

On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation

We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d+1)$-dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by Gómez, Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same $h$-convergence rates as for polynomials of degree $p$ in $(d + 1)$ variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree $2p$ in $d$ variables, and bases are easily constructed.

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arXiv2023

Design and performance of a space-time virtual element method for the heat equation on prismatic meshes

We present a space-time virtual element method for the discretization of the heat equation, which is defined on general prismatic meshes and variable degrees of accuracy. Strategies to handle efficiently the space-time mesh structure are discussed. We perform convergence tests for the $h$- and $hp$-versions of the method in case of smooth and singular solutions, and test space-time adaptive mesh refinements driven by a residual-type error indicator.

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arXiv2022

On the exponential time-decay for the one-dimensional wave equation with variable coefficients

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.

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arXiv2022

An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time approach to the proof of existence of bounded weak solutions of cross-diffusion systems, making use of the system entropy to examine long-term behavior and to show that the solution is nonnegative, even when a maximum principle is not available. This approach naturally gives rise to a novel space-time Galerkin method for the numerical approximation of cross-diffusion systems that conserves their entropy structure. We prove existence and convergence of the discrete solutions, and present numerical results for the porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.

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arXiv2022

Exponential Convergence of $hp$-Time-Stepping in Space-Time Discretizations of Parabolic PDEs

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "$h$-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of $D$, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in $D$. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

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arXiv2022

An adaptive finite element method for high-frequency scattering problems with smoothly varying coefficients

We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency $ω$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $ω$. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes. Numerical examples demonstrate the advantages of the proposed method and the fact that the method can also be applied with external source terms such as point sources and sound-soft scatterers. The gained efficiency, however, is limited in the presence of trapping modes.

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arXiv2021

Tent pitching and Trefftz-DG method for the acoustic wave equation

We present a space-time Trefftz discontinuous Galerkin method for approximating the acoustic wave equation semi-explicitly on tent pitched meshes. DG Trefftz methods use discontinuous test and trial functions, which solve the wave equation locally. Tent pitched meshes allow to solve the equation elementwise, allowing locally optimal advances in time. The method is implemented in NGSolve, solving the space-time elements in parallel, whenever possible. Insights into the implementational details are given, including the case of propagation in heterogeneous media.

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arXiv2021

A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems

We present an equilibration-based a posteriori error estimator for Nédélec element discretizations of the magnetostatic problem. The estimator is obtained by adding a gradient correction to the estimator for Nédélec elements of arbitrary degree presented in [J. Gedicke, S. Geevers, and I. Perugia. An equilibrated a posteriori error estimator for arbitrary-order Nédélec elements for magnetostatic problems. Journal of Scientific Computing, 83:1-23, 2020]. This new estimator is proven to be reliable, with reliability constant 1, and efficient, with an efficiency constant that is independent of the polynomial degree of the approximation. These properties are demonstrated in a series of numerical experiments on three-dimensional test problems.

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arXiv2021

Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.

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arXiv2021

A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

We study numerically the dispersion and dissipation properties of the plane wave virtual element method and the nonconforming Trefftz virtual element method for the Helmholtz problem. Whereas the former method is based on a conforming virtual partition of unity approach in the sense that the local (implicitly defined) basis functions are given as modulations of lowest order harmonic virtual element functions with plane waves, the latter one represents a pure Trefftz method with local edge-related basis functions that are eventually glued together in a nonconforming fashion. We will see that the qualitative and quantitative behavior of dissipation and dispersion of the method hinges upon the level of conformity and the use of Trefftz basis functions. To this purpose, we also compare the results to those obtained in [15] for the plane wave discontinuous Galerkin method, and to those for the standard polynomial based finite element method.

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arXiv2020

A p-robust polygonal discontinuous Galerkin method with minus one stabilization

We introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree $p$. In the setting of [S. Bertoluzza and D. Prada, A polygonal discontinuous Galerkin method with minus one stabilization, ESAIM Math. Mod. Numer. Anal. (DOI: 10.1051/m2an/2020059)], the stabilization is obtained by penalizing, in each mesh element $K$, a residual in the norm of the dual of $H^1(K)$. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a $p$-explicit stability and error analysis, proving $p$-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

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arXiv2020

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $Ω\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $Ω$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.

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arXiv2020

FEM-BEM mortar coupling for the Helmholtz problem in three dimensions

We present a FEM-BEM coupling strategy for time-harmonic acoustic scattering in media with variable sound speed. The coupling is realized with the aid of a mortar variable that is an impedance trace on the coupling boundary. The resulting sesquilinear form is shown to satisfy a Garding inequality. Quasi-optimal convergence is shown for sufficiently fine meshes. Numerical examples confirm the theoretical convergence results.

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arXiv2020

An equilibrated a posteriori error estimator for arbitrary-order Nédélec elements for magnetostatic problems

We present a novel \textit{a posteriori} error estimator for Nédélec elements for magnetostatic problems that is constant-free, i.e. it provides an upper bound on the error that does not involve a generic constant. The estimator is based on equilibration of the magnetic field and only involves small local problems that can be solved in parallel. Such an error estimator is already available for the lowest-degree Nédélec element [D. Braess, J. Schöberl, \textit{Equilibrated residual error estimator for edge elements}, Math. Comp. 77 (2008)] and requires solving local problems on vertex patches. The novelty of our estimator is that it can be applied to Nédélec elements of arbitrary degree. Furthermore, our estimator does not require solving problems on vertex patches, but instead requires solving problems on only single elements, single faces, and very small sets of nodes. We prove reliability and efficiency of the estimator and present several numerical examples that confirm this.

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arXiv2019

Fast Least-Squares Padé approximation of problems with normal operators and meromorphic structure

In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g. the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Padé approximation technique introduced in [6] following [11]. In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the "fast" in the name). Moreover, we prove several theoretical results that improve and extend those in [6], including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Padé approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in [6] for differential operators with normal and compact resolvent.

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arXiv2019

A structure-preserving discontinuous Galerkin scheme for the Fischer-KPP equation

An implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L2 norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.

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arXiv2019

Robust adaptive hp discontinuous Galerkin finite element methods for the Helmholtz equation

This paper presents an $hp$ a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree $p$ and the wave number $k$. For the discretization, we consider a discontinuous Galerkin formulation that is unconditionally well posed. The a posteriori error analysis is based on the technique of equilibrated fluxes applied to a shifted Poisson problem, with the error due to the nonconformity of the discretization controlled by a potential reconstruction. We prove that the error estimator is both reliable and efficient, under the condition that the initial mesh size and polynomial degree is chosen such that the discontinuous Galerkin formulation converges, i.e., it is out of the regime of pollution. We confirm the efficiency of an $hp$-adaptive refinement strategy based on the presented robust a posteriori error estimator via several numerical examples.

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arXiv2018

Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.

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