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We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element…

Numerical Analysis · Mathematics 2021-06-01 Dibyendu Adak , Gianmarco Manzini , Sundararajan Natarajan

The virtual element method (VEM) is a stabilized Galerkin method that is robust and accurate on general polygonal meshes. This feature makes it an appealing candidate for simulations involving meshes with embedded interfaces and evolving…

Numerical Analysis · Mathematics 2025-10-03 Ramsharan Rangarajan , N. Sukumar

A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in [R. Oyarz\'ua and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity, SIAM J. Numer.…

Numerical Analysis · Mathematics 2019-12-13 Raimund Bürger , Sarvesh Kumar , David Mora , Ricardo Ruiz-Baier , Nitesh Verma

In this paper, we design and analyze a Virtual Element discretization for the steady motion of non-Newtonian, incompressible fluids. A specific stabilization, tailored to mimic the monotonicity and boundedness properties of the continuous…

Numerical Analysis · Mathematics 2024-03-07 P. F. Antonietti , L. Beirao da Veiga , M. Botti , G. Vacca , M. Verani

Finite element methods are well-known to admit robust optimal convergence on simplicial meshes satisfying the maximum angle conditions. But how to generalize this condition to polyhedra is unknown in the literature. In this work, we argue…

Numerical Analysis · Mathematics 2022-12-15 Ruchi Guo

This paper proposes a methodology to estimate stress in the subsurface by a hybrid method combining finite element modeling and neural networks. This methodology exploits the idea of obtaining a multi-frequency solution in the numerical…

Machine Learning · Computer Science 2020-08-27 Xavier Garcia , Adrian Rodriguez-Herrera

An optimization-based strategy is proposed for coupling three-dimensional and one-dimensional problems (3D-1D coupling) in the context of soil-root interaction simulations. This strategy, originally designed to tackle generic 3D-1D coupled…

Numerical Analysis · Mathematics 2024-12-18 Stefano Berrone , Stefano Ferraris , Denise Grappein , Gioana Teora , Fabio Vicini

In this paper we propose and analyze a virtual element method for the two dimensional non-symmetric diffusion-convection eigenvalue problem in order to derive a priori and a posteriori error estimates. Under the classic assumptions of the…

Numerical Analysis · Mathematics 2023-09-29 Danilo Amigo , Felipe Lepe , Gonzalo Rivera

A virtual element discretisation of an Arbitrary Lagrangian-Eulerian method for two-dimensional convection-diffusion equations is proposed employing an isoparametric Virtual Element Method to achieve higher-order convergence rates on curved…

Numerical Analysis · Mathematics 2024-04-30 H. Wells

In the present paper we describe the computational implementation of some integral terms that arise from mixed virtual element methods (mixed-VEM) in two-dimensional pseudostress-velocity formulations. The implementation presented here…

Numerical Analysis · Mathematics 2020-12-15 Filánder A. Sequeira , Helen Guillén-Oviedo

This paper proposes a novel first-order and a novel second-order fully discrete virtual element schemes based on the scalar auxiliary variable method for the three dimensional inductionless magnetohydrodynamics problem. The backward Eular…

Analysis of PDEs · Mathematics 2024-12-13 Xianghai Zhou , Haiyan Su

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 $d \ge 2$.…

Numerical Analysis · Mathematics 2024-10-16 Guillaume Bonnet , Andrea Cangiani , Ricardo H. Nochetto

In this work, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the Carreau-Yasuda constitutive law, in the shear-thickening regime (r > 2) including both…

We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element…

Numerical Analysis · Mathematics 2014-10-08 Sundararajan Natarajan , Stéphane P. A. Bordas , Ean Tat Ooi

We develop a cut finite element method (CutFEM) for convection-diffusion problems posed on mixed-dimensional domains, i.e., unions of manifolds of different dimensions arranged in a hierarchical structure where lower-dimensional components…

Numerical Analysis · Mathematics 2026-04-09 Erik Burman , Peter Hansbo , Mats G. Larson , Karl Larsson , Shantiram Mahata

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent…

Numerical Analysis · Mathematics 2023-02-22 Nicolás A. Barnafi , Franco Dassi , Simone Scacchi

Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable…

Numerical Analysis · Mathematics 2018-02-22 Konrad Simon , Jörn Behrens

We study the $h$- and $p$-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet-Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use…

Numerical Analysis · Mathematics 2018-07-30 Lorenzo Mascotto , Ilaria Perugia , Alexander Pichler

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve…

Numerical Analysis · Mathematics 2023-11-01 Zhibao Zheng , Udo Nackenhorst

It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the…

Numerical Analysis · Mathematics 2016-12-28 Paola F. Antonietti , Matteo Bruggi , Simone Scacchi , Marco Verani