11,270 research outputs found
Stability and Monotonicity for Some Discretizations of the Biot's Model
We consider finite element discretizations of the Biot's consolidation model
in poroelasticity with MINI and stabilized P1-P1 elements. We analyze the
convergence of the fully discrete model based on spatial discretization with
these types of finite elements and implicit Euler method in time. We also
address the issue related to the presence of non-physical oscillations in the
pressure approximation for low permeabilities and/or small time steps. We show
that even in 1D a Stokes-stable finite element pair fails to provide a monotone
discretization for the pressure in such regimes. We then introduce a
stabilization term which removes the oscillations. We present numerical results
confirming the monotone behavior of the stabilized schemes
Stabilized mixed finite element methods for linear elasticity on simplicial grids in
In this paper, we design two classes of stabilized mixed finite element
methods for linear elasticity on simplicial grids. In the first class of
elements, we use - and
- to approximate the stress
and displacement spaces, respectively, for , and employ a
stabilization technique in terms of the jump of the discrete displacement over
the faces of the triangulation under consideration; in the second class of
elements, we use - to
approximate the displacement space for , and adopt the
stabilization technique suggested by Brezzi, Fortin, and Marini. We establish
the discrete inf-sup conditions, and consequently present the a priori error
analysis for them. The main ingredient for the analysis is two special
interpolation operators, which can be constructed using a crucial
bubble function space of polynomials on each
element. The feature of these methods is the low number of global degrees of
freedom in the lowest order case. We present some numerical results to
demonstrate the theoretical estimates.Comment: 16 pages, 1 figur
Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity
A stress equilibration procedure for linear elasticity is proposed and
analyzed in this paper with emphasis on the behavior for (nearly)
incompressible materials. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs an -conforming, weakly symmetric stress reconstruction. Our focus is
on the Taylor-Hood combination of continuous finite element spaces of
polynomial degrees and for the displacement and the pressure,
respectively. Our construction leads then to reconstructed stresses by
Raviart-Thomas elements of degree which are weakly symmetric in the sense
that its anti-symmetric part is zero tested against continuous piecewise
polynomial functions of degree . The computation is performed locally on a
set of vertex patches covering the computational domain in the spirit of
equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local
problems need to satisfy consistency conditions associated with all rigid body
modes, in contrast to the case of Poisson's equation where only the constant
modes are involved. The resulting error estimator is shown to constitute a
guaranteed upper bound for the error with a constant that depends only on the
shape regularity of the triangulation. Local efficiency, uniformly in the
incompressible limit, is deduced from the upper bound by the residual error
estimator
A Review of Time Relaxation Methods
The time relaxation model has proven to be effective in regularization of Navier–Stokes Equations. This article reviews several published works discussing the development and implementations of time relaxation and time relaxation models (TRMs), and how such techniques are used to improve the accuracy and stability of fluid flow problems with higher Reynolds numbers. Several analyses and computational settings of TRMs are surveyed, along with parameter sensitivity studies and hybrid implementations of time relaxation operators with different regularization techniques
Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
In this brief note, we introduce a non-symmetric mixed finite element
formulation for Brinkman equations written in terms of velocity, vorticity and
pressure with non-constant viscosity. The analysis is performed by the
classical Babu\v{s}ka-Brezzi theory, and we state that any inf-sup stable
finite element pair for Stokes approximating velocity and pressure can be
coupled with a generic discrete space of arbitrary order for the vorticity. We
establish optimal a priori error estimates which are further confirmed through
computational example
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