538 research outputs found
A convergent nonconforming finite element method for compressible Stokes flow
We propose a nonconforming finite element method for isentropic viscous gas
flow in situations where convective effects may be neglected. We approximate
the continuity equation by a piecewise constant discontinuous Galerkin method.
The velocity (momentum) equation is approximated by a finite element method on
div-curl form using the nonconforming Crouzeix-Raviart space. Our main result
is that the finite element method converges to a weak solution. The main
challenge is to demonstrate the strong convergence of the density
approximations, which is mandatory in view of the nonlinear pressure function.
The analysis makes use of a higher integrability estimate on the density
approximations, an equation for the "effective viscous flux", and renormalized
versions of the discontinuous Galerkin method.Comment: 23 page
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Stable cheapest nonconforming finite elements for the Stokes equations
We introduce two pairs of stable cheapest nonconforming finite element space
pairs to approximate the Stokes equations. One pair has each component of its
velocity field to be approximated by the nonconforming quadrilateral
element while the pressure field is approximated by the piecewise constant
function with globally two-dimensional subspaces removed: one removed space is
due to the integral mean--zero property and the other space consists of global
checker--board patterns. The other pair consists of the velocity space as the
nonconforming quadrilateral element enriched by a globally
one--dimensional macro bubble function space based on
(Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure
field is approximated by the piecewise constant function with mean--zero space
eliminated. We show that two element pairs satisfy the discrete inf-sup
condition uniformly. And we investigate the relationship between them. Several
numerical examples are shown to confirm the efficiency and reliability of the
proposed methods
Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data
We give an introduction to discrete functional analysis techniques for
stationary and transient diffusion equations. We show how these techniques are
used to establish the convergence of various numerical schemes without assuming
non-physical regularity on the data. For simplicity of exposure, we mostly
consider linear elliptic equations, and we briefly explain how these techniques
can be adapted and extended to non-linear time-dependent meaningful models
(Navier--Stokes equations, flows in porous media, etc.). These convergence
techniques rely on discrete Sobolev norms and the translation to the discrete
setting of functional analysis results
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
Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results
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