1,643 research outputs found
The Virtual Element Method with curved edges
In this paper we initiate the investigation of Virtual Elements with curved
faces. We consider the case of a fixed curved boundary in two dimensions, as it
happens in the approximation of problems posed on a curved domain or with a
curved interface. While an approximation of the domain with polygons leads, for
degree of accuracy , to a sub-optimal rate of convergence, we show
(both theoretically and numerically) that the proposed curved VEM lead to an
optimal rate of convergence
Basic principles of hp Virtual Elements on quasiuniform meshes
In the present paper we initiate the study of 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
and in the polynomial degree in the case of finite Sobolev regularity.
Exponential convergence is proved in the case of analytic solutions. The
theoretical convergence results are validated in numerical experiments.
Finally, an initial study on the possible choice of local basis functions is
included
Serendipity Face and Edge VEM Spaces
We extend the basic idea of Serendipity Virtual Elements from the previous
case (by the same authors) of nodal (-conforming) elements, to a more
general framework. Then we apply the general strategy to the case of
and conforming Virtual Element Methods, in two and three dimensions
Serendipity Nodal VEM spaces
We introduce a new variant of Nodal Virtual Element spaces that mimics the
"Serendipity Finite Element Methods" (whose most popular example is the 8-node
quadrilateral) and allows to reduce (often in a significant way) the number of
internal degrees of freedom. When applied to the faces of a three-dimensional
decomposition, this allows a reduction in the number of face degrees of
freedom: an improvement that cannot be achieved by a simple static
condensation. On triangular and tetrahedral decompositions the new elements
(contrary to the original VEMs) reduce exactly to the classical Lagrange FEM.
On quadrilaterals and hexahedra the new elements are quite similar (and have
the same amount of degrees of freedom) to the Serendipity Finite Elements, but
are much more robust with respect to element distortions. On more general
polytopes the Serendipity VEMs are the natural (and simple) generalization of
the simplicial case
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