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Andrea Cangiani

Publications and source records attributed to Andrea Cangiani.

At least 19 recordsLinked to original sources
arXiv2026

Virtual element methods for a class of fully nonlinear elliptic PDEs

We study virtual element discretizations of a well-known variational formulation in $H^2$ of Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients. We show that the use of $H^2$ conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda-Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.

math.NA↗
arXiv2026

Conforming Virtual Element Method for Biharmonic Poisson-Nernst-Planck Navier-Stokes systems

We develop and analyze a conforming Virtual Element Method (VEM) for the fourth-order Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The proposed scheme is based on compatible discretizations of each component: an \(H^2\)-conforming VEM for a fourth-order electrostatic potential equation, an \(H^1\)-conforming VEM for the Nernst--Planck equations, and a \textit{divergence-free} and \textit{pressure-robust} VEM for the Navier--Stokes equations. Time integration is performed using a backward Euler scheme to ensure stability. We establish the well-posedness of the continuous problem up to three-dimensions and also establish existence and uniqueness of the fully discrete solution via a fixed-point argument. Further, we derive a priori error estimates showing that the electrostatic potential, concentration and velocity converge optimally in Bochner norms \(L^\infty\bigl(0,T; H^2(\Omega)\bigr)\), \(L^2\bigl(0,T; H^1(\Omega)\bigr)\), and \(L^2\bigl(0,T; \bm{H}^1(\Omega)\bigr)\), respectively. The analysis requires a sophisticated argument to avoid any restrictive assumptions on the coercivity and continuity constants and to handle the trilinear form involving three different variables. The pressure-robust design permits the use of lowest-order pressure approximations without compromising convergence rates of the other variables, reducing computational cost. Numerical experiments confirm the theoretical convergence rates for various polynomial orders and demonstrate the scheme's robustness, including in low-viscosity regimes.

math.NA↗
arXiv2025

An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations

We present new aposteriori error estimates for the interior penalty discontinuous Galerkin method applied to non-stationary convection-diffusion equations. The focus is on strongly convection-dominated problems without zeroth-order reaction terms, which leads to the absence of positive L^2-like components. An important specific example is the energy/temperature equation of the Boussinesq system arising from the modelling of mantle convection of the Earth. The key mathematical challenge of mitigating the effects of exponential factors with respect to the final time, arising from the use of Gronwall-type arguments, is addressed by an exponential fitting technique. The latter results to a new class of aposteriori error estimates for the stationary problem, which are valid in cases of convection and reaction coefficient combinations not covered by the existing literature. This new class of estimators is combined with an elliptic reconstruction technique to derive new respective estimates for the non-stationary problem, exhibiting reduced dependence on Gronwall-type exponents and, thus, offer more accurate estimation for longer time intervals. We showcase the superior performance of the new class of aposteriori error estimators in driving mesh adaptivity in Earth's mantle convection simulations, in a setting where the energy/temperature equation is discretised by the discontinuous Galerkin method, coupled with the Taylor-Hood finite element for the momentum and mass conservation equations. We exploit the community code ASPECT, to present numerical examples showing the effectivity of the proposed approach.

math.NA↗
arXiv2024

R3MG: R-tree based agglomeration of polytopal grids with applications to multilevel methods

We present a novel approach to perform agglomeration of polygonal and polyhedral grids based on spatial indices. Agglomeration strategies are a key ingredient in polytopal methods for PDEs as they are used to generate (hierarchies of) computational grids from an initial grid. Spatial indices are specialized data structures that significantly accelerate queries involving spatial relationships in arbitrary space dimensions. We show how the construction of the R-tree spatial database of an arbitrary fine grid offers a natural and efficient agglomeration strategy with the following characteristics: i) the process is fully automated, robust, and dimension-independent, ii) it automatically produces a balanced and nested hierarchy of agglomerates, and iii) the shape of the agglomerates is tightly close to the respective axis aligned bounding boxes. Moreover, the R-tree approach provides a full hierarchy of nested agglomerates which permits fast query and allows for efficient geometric multigrid methods to be applied also to those cases where a hierarchy of grids is not present at construction time. We present several examples based on polygonal discontinuous Galerkin methods, confirming the effectiveness of our approach in the context of challenging three-dimensional geometries and the design of geometric multigrid preconditioners.

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arXiv2024

Isoparametric Virtual Element Methods

We present two approaches to constructing isoparametric Virtual Element Methods of arbitrary order for linear elliptic partial differential equations on general two-dimensional domains. The first method approximates the variational problem transformed onto a computational reference domain. The second method computes a virtual domain and uses bespoke polynomial approximation operators to construct a computable method. Both methods are shown to converge optimally, a behaviour confirmed in practice for the solution of problems posed on curved domains.

math.NA↗
arXiv2024

Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \ge 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $\Omega$ and is rather elementary. We prove stability and error estimates in $H^2(\Omega)$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(\Omega)$.

math.NA↗
arXiv2023

A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of simplicial and/or box-type elements is included in the analysis as a special case. In particular, for the upper bounds, an arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. As a corollary, the present analysis generalizes known a posteriori error bounds for dG methods, allowing in particular for meshes with an arbitrary number of irregular hanging nodes per element. The proof hinges on a new conforming recovery strategy in conjunction with a Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds are also proven for a number of practical cases. Numerical experiments are also presented, highlighting the practical value of the derived a posteriori error bounds as error estimators.

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arXiv2021

$hp$-Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements

We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for \emph{curved} element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new $H^1-L_2$-type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

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arXiv2020

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces used on consecutive time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the $L^2(H^1)$ and $L^{\infty}(L^2)$ norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

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arXiv2019

A posteriori error estimates for mixed virtual element methods

We present an a posteriori error analysis for the mixed virtual element method (mixed VEM) applied to second order elliptic equations in divergence form with mixed boundary conditions. The resulting error estimator is of residual-type. It only depends on quantities directly available from the VEM solution and applies on very general polygonal meshes. The proof of the upper bound relies on a global inf-sup condition, a suitable Helmholtz decomposition, and the local approximation properties of a Cl\'ement-type interpolant. In turn, standard inverse inequalities and localization techniques based on bubble functions are the main tools yielding the lower bound. Via the inclusion of a fully local postprocessing of the mixed VEM solution, we also show that the estimator provides a reliable and efficient control on the broken $\mathrm{H}(\mathrm{div})$-norm error between the exact and the postprocessed flux. Numerical examples confirm the theoretical properties of our estimator, and show that it can be effectively used to drive an adaptive mesh refinement algorithm.

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arXiv2017

Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework

Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models such as Lotka-Volterra competition models with spatial dependence. In this work, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterised by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proven to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns both in two and three spatial dimensions.

q-bio.PE↗
arXiv2017

Virtual Element Method for Quasilinear Elliptic Problems

We present a Virtual Element Method (VEM) for the solution of Dirichlet problems for the quasilinear equation $-\text{div} (k(u)\text{grad} u)=f$ with essential boundary conditions. Within the VEM the nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal order a priori error estimates in the $H^1$ and $L^2$ norms are proven. In addition, the convergence of fixed point iterations for the solution of the resulting nonlinear system is established. Numerical examples confirm the convergence analysis.

math.NA↗
arXiv2016

The non-conforming virtual element method for the Stokes equations

We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

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arXiv2016

$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes

We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of \emph{total} degree, say $p$, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete $hp$-dG scheme using less degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with \emph{arbitrary} number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments.

math.NA↗
arXiv2016

A posteriori error estimates for the virtual element method

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.

math.NA↗
arXiv2015

Conforming and nonconforming virtual element methods for elliptic problems

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal $H^1$- and $L^2$-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.

math.NA↗
arXiv2015

Adaptivity and blow-up detection for nonlinear evolution problems

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

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