206 research outputs found
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
Nonconforming finite element Stokes complexes in three dimensions
Two nonconforming finite element Stokes complexes ended with the
nonconforming - element for the Stokes equation in three dimensions
are constructed. And commutative diagrams are also shown by combining
nonconforming finite element Stokes complexes and interpolation operators. The
lower order -nonconforming finite
element only has degrees of freedom, whose basis functions are explicitly
given in terms of the barycentric coordinates. The -nonconforming elements are applied to solve the
quad-curl problem, and optimal convergence is derived. By the nonconforming
finite element Stokes complexes, the mixed finite element methods of the
quad-curl problem is decoupled into two mixed methods of the Maxwell equation
and the nonconforming - element method for the Stokes equation, based
on which a fast solver is developed.Comment: 20 page
p- and hp- virtual elements for the Stokes problem
We analyse the p- and hp-versions of the virtual element method (VEM) for the
the Stokes problem on a polygonal domain. The key tool in the analysis is the
existence of a bijection between Poisson-like and Stokes-like VE spaces for the
velocities. This allows us to re-interpret the standard VEM for Stokes as a
VEM, where the test and trial discrete velocities are sought in Poisson-like VE
spaces. The upside of this fact is that we inherit from [7] an explicit
analysis of best interpolation results in VE spaces, as well as stabilization
estimates that are explicit in terms of the degree of accuracy of the method.
We prove exponential convergence of the hp-VEM for Stokes problems with regular
right-hand sides. We corroborate the theoretical estimates with numerical tests
for both the p- and hp-versions of the method
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