7,621 research outputs found
Nonconforming Virtual Element Method for -th Order Partial Differential Equations in
A unified construction of the -nonconforming virtual elements of any
order is developed on any shape of polytope in with
constraints and . As a vital tool in the construction, a
generalized Green's identity for inner product is derived. The
-nonconforming virtual element methods are then used to approximate
solutions of the -harmonic equation. After establishing a bound on the jump
related to the weak continuity, the optimal error estimate of the canonical
interpolation, and the norm equivalence of the stabilization term, the optimal
error estimates are derived for the -nonconforming virtual element
methods.Comment: 33page
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
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