Related papers: Some Error Analysis on Virtual Element Methods
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…
Virtual element methods is a new promising finite element methods using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover…
We present the design of a mesh quality indicator that can predict the behavior of the Virtual Element Method (VEM) on a given mesh family or finite sequence of polyhedral meshes (dataset). The mesh quality indicator is designed to measure…
We design the conforming virtual element method for the numerical approximation of the two dimensional elastodynamics problem. We prove stability and convergence of the semi-discrete approximation and derive optimal error estimates under…
In the present paper we introduce a Virtual Element Method (VEM) for the approximate solution of general linear second order elliptic problems in mixed form, allowing for variable coefficients. We derive a theoretical convergence analysis…
Since its introduction, the Virtual Element Method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the…
The purpose of the present paper is to develop $C^1$ Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We…
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…
This article presents novel proof methods for estimating interpolation errors, predicated on the understanding that one has already studied foundational error analysis using the finite element method.
A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background…
We introduce a novel residual-based a posteriori error estimator for the conforming $C^1$ Virtual Element Method (VEM) applied to the buckling eigenvalue problem, incorporating nonlinear plane stress effects in both two and three…
The lowest-order Neural Approximated Virtual Element Method on polygonal elements is proposed here. This method employs a neural network to locally approximate the Virtual Element basis functions, thereby eliminating issues concerning…
In the context of adaptive remeshing, the virtual element method provides significant advantages over the finite element method. The attractive features of the virtual element method, such as the permission of arbitrary element geometries,…
We rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials on elemental…
We present a class of nonconforming virtual element methods for general fourth order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element…
In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical…
This work concerns adaptive refinement procedures for meshes of polygonal virtual elements. Specifically, refinement procedures previously proposed by the authors for structured meshes are generalized for the challenging case of arbitrary…
The mesh flexibility offered by the virtual element method through the permission of arbitrary element geometries, and the seamless incorporation of `hanging' nodes, has made the method increasingly attractive in the context of adaptive…
This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations…
In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the…