87 research outputs found
Virtual Element Method for fourth order problems: estimates
We analyse the family of -Virtual Elements introduced in
\cite{Brezzi:Marini:plates} for fourth-order problems and prove optimal
estimates in and in via classical duality arguments
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
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 accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
Virtual Element Methods for hyperbolic problems on polygonal meshes
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 estimates in H^1 semi-norm and L^2 norm. Moreover we develop a
theoretical analysis on the stability for the fully discrete problem by
comparing the Newmark method and the Bathe method. Finally we show the
practical behaviour of the proposed method through a large array of numerical
tests
Some Error Analysis on Virtual Element Methods
Some error analysis on virtual element methods including inverse
inequalities, norm equivalence, and interpolation error estimates are presented
for polygonal meshes which admits a virtual quasi-uniform triangulation
A virtual element method for the vibration problem of Kirchhoff plates
The aim of this paper is to develop a virtual element method (VEM) for the
vibration problem of thin plates on polygonal meshes. We consider a variational
formulation relying only on the transverse displacement of the plate and
propose an conforming discretization by means of the VEM which is
simple in terms of degrees of freedom and coding aspects. Under standard
assumptions on the computational domain, we establish that the resulting
schemeprovides a correct approximation of the spectrum and prove optimal order
error estimates for the eigenfunctions and a double order for the eigenvalues.
The analysis restricts to simply connected polygonal clamped plates, not
necessarily convex. Finally, we report several numerical experiments
illustrating the behaviour of the proposed scheme and confirming our
theoretical results on different families of meshes. Additional examples of
cases not covered by our theory are also presented
Lowest order Virtual Element approximation of magnetostatic problems
We give here a simplified presentation of the lowest order Serendipity
Virtual Element method, and show its use for the numerical solution of linear
magneto-static problems in three dimensions. The method can be applied to very
general decompositions of the computational domain (as is natural for Virtual
Element Methods) and uses as unknowns the (constant) tangential component of
the magnetic field on each edge, and the vertex values of the
Lagrange multiplier (used to enforce the solenoidality of the magnetic
induction ). In this respect the method can be seen
as the natural generalization of the lowest order Edge Finite Element Method
(the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost
arbitrary shape, and as we show on some numerical examples it exhibits very
good accuracy (for being a lowest order element) and excellent robustness with
respect to distortions
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