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In this paper we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to…

Numerical Analysis · Mathematics 2016-04-29 Nilsen Halvor , Nordbotten Jan , Raynaud Xavier

The Virtual Element Method for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence…

Numerical Analysis · Mathematics 2023-07-11 L. Beirao da Veiga , C. Lovadina , M. Trezzi

Hydrokinetic flapping foil turbines in swing-arm mode have gained considerable interest in recent years because of their enhanced capability to extract power, and improved efficiency compared to foils in simple mode. The performance of foil…

Fluid Dynamics · Physics 2023-02-28 Ahmed A. Hamada , Mirjam Fürth

In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence…

Numerical Analysis · Mathematics 2016-02-19 Giuseppe Vacca

This study presents a finite element and virtual element (FE-VE) coupled method for thermomechanical analysis in electronic packaging structures. The approach partitions computational domains strategically, employing FEM for regular…

Numerical Analysis · Mathematics 2026-01-14 Yanpeng Gong , Sishuai Li , Yue Mei , Bingbing Xu , Fei Qin , Xiaoying Zhuang , Timon Rabczuk

This work develops a polygonal finite element method (PFEM) for the analysis of steady-state and transient thermal stresses in two dimensional continua. The method employs Wachspress rational basis functions to construct conforming…

Numerical Analysis · Mathematics 2025-04-23 Yang Yang , Mingjiao Yan , Zongliang Zhang , Dengmiao Hao , Xuedong Chen , Weixiong Chen

The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses…

Numerical Analysis · Mathematics 2020-02-26 Ben Vadala-Roth , Shashank Acharya , Neelesh A Patankar , Simone Rossi , Boyce E Griffith

Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that…

Numerical Analysis · Mathematics 2020-02-06 Derk Frerichs , Christian Merdon

The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such…

Numerical Analysis · Mathematics 2021-12-28 Paola Francesca Antonietti , Gianmarco Manzini , Simone Scacchi , Marco Verani

We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g. water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g. air), it may come into contact with solid supports…

Analysis of PDEs · Mathematics 2021-04-27 Costante Bellettini

We extend the conforming virtual element method to the numerical resolution of eigenvalue problems with potential terms on a polytopal mesh. An important application is that of the Schrodinger equation with a pseudopotential term. This…

Numerical Analysis · Mathematics 2018-04-04 Ondrej Certik , Francesca Gardini , Gianmarco Manzini , Giuseppe Vacca

In this work we design a novel $C^1$-conforming virtual element method of arbitrary order $k \geq 2$, to solve the biharmonic problem on a domain with curved boundary and internal curved interfaces in two dimensions. By introducing a…

Numerical Analysis · Mathematics 2025-05-30 L. Beirão da Veiga , D. Mora , A. Silgado

This work focuses on model preparation for electrostatic simulations of CAD designs to realize a rapid virtual prototyping concept. We present a boundary element method (BEM) allowing discontinuous fields between surfaces. The corresponding…

Computational Engineering, Finance, and Science · Computer Science 2024-12-13 Benjamin Marussig , Thomas Rüberg , Jürgen Zechner , Lars Kielhorn , Thomas-Peter Fries

In this thesis, we investigate a novel local projection based stabilized conforming virtual element method for the generalized Oseen problem using equal-order element pairs on general polygonal meshes. To ensure the stability, particularly…

Numerical Analysis · Mathematics 2025-09-05 Sudheer Mishra , E Natarajan

We investigate through numerical simulations the hydrodynamic interactions between two rigid spherical particles suspended on the axis of a cylindrical tube filled with an elastoviscoplastic fluid subjected to pressure-driven flow. The…

Achieving accurate numerical results of hydrodynamic loads based on the potential-flow theory is very challenging for structures with sharp edges, due to the singular behavior of the local-flow velocities. In this paper, we introduce the…

Numerical Analysis · Mathematics 2021-09-23 Ying Wang , Yanlin Shao , Jikang Chen , Hui Liang

In this paper we propose and analyze a virtual element method to approximate the natural frequencies of the acoustic eigenvalue problem with polygonal meshes that allow the presence of small edges. With the aid of a suitable seminorm that…

Numerical Analysis · Mathematics 2023-10-13 Danilo Amigo , Felipe Lepe , Gonzalo Rivera

In this paper we introduce a finite element method for the Stokes equations with a massless immersed membrane. This membrane applies normal and tangential forces affecting the velocity and pressure of the fluid. Additionally, the points…

Numerical Analysis · Mathematics 2019-05-01 Kyle Dunn , Roger Lui , Marcus Sarkis

In the present paper we initiate the challenging task of building a mathematically sound theory for Adaptive Virtual Element Methods (AVEMs). Among the realm of polygonal meshes, we restrict our analysis to triangular meshes with hanging…

Numerical Analysis · Mathematics 2022-12-02 L. Beirao da Veiga , C. Canuto , R. H. Nochetto , G. Vacca , M. Verani

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation…

Numerical Analysis · Mathematics 2021-12-30 Gianmarco Manzini , Annamaria Mazzia