9,447 research outputs found
Convergence of an adaptive mixed finite element method for general second order linear elliptic problems
The convergence of an adaptive mixed finite element method for general second
order linear elliptic problems defined on simply connected bounded polygonal
domains is analyzed in this paper. The main difficulties in the analysis are
posed by the non-symmetric and indefinite form of the problem along with the
lack of the orthogonality property in mixed finite element methods. The
important tools in the analysis are a posteriori error estimators,
quasi-orthogonality property and quasi-discrete reliability established using
representation formula for the lowest-order Raviart-Thomas solution in terms of
the Crouzeix-Raviart solution of the problem. An adaptive marking in each step
for the local refinement is based on the edge residual and volume residual
terms of the a posteriori estimator. Numerical experiments confirm the
theoretical analysis.Comment: 24 pages, 8 figure
Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
Recent works showed that pressure-robust modifications of mixed finite
element methods for the Stokes equations outperform their standard versions in
many cases. This is achieved by divergence-free reconstruction operators and
results in pressure independent velocity error estimates which are robust with
respect to small viscosities. In this paper we develop a posteriori error
control which reflects this robustness.
The main difficulty lies in the volume contribution of the standard
residual-based approach that includes the -norm of the right-hand side.
However, the velocity is only steered by the divergence-free part of this
source term. An efficient error estimator must approximate this divergence-free
part in a proper manner, otherwise it can be dominated by the pressure error.
To overcome this difficulty a novel approach is suggested that uses arguments
from the stream function and vorticity formulation of the Navier--Stokes
equations. The novel error estimators only take the of the
right-hand side into account and so lead to provably reliable, efficient and
pressure-independent upper bounds in case of a pressure-robust method in
particular in pressure-dominant situations. This is also confirmed by some
numerical examples with the novel pressure-robust modifications of the
Taylor--Hood and mini finite element methods
Residual-Based A Posteriori Error Estimates for Symmetric Conforming Mixed Finite Elements for Linear Elasticity Problems
A posteriori error estimators for the symmetric mixed finite element methods
for linear elasticity problems of Dirichlet and mixed boundary conditions are
proposed. Stability and efficiency of the estimators are proved. Finally, we
provide numerical examples to verify the theoretical results
Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
Recent works showed that pressure-robust modifications of mixed finite
element methods for the Stokes equations outperform their standard versions
in many cases. This is achieved by divergence-free reconstruction operators
and results in pressure-independent velocity error estimates which are robust
with respect to small viscosities. In this paper we develop a posteriori
error control which reflects this robustness. The main difficulty lies in the
volume contribution of the standard residual-based approach that includes the
L2-norm of the right-hand side. However, the velocity is only steered by the
divergence-free part of this source term. An efficient error estimator must
approximate this divergence-free part in a proper manner, otherwise it can be
dominated by the pressure error. To overcome this difficulty a novel approach
is suggested that uses arguments from the stream function and vorticity
formulation of the NavierStokes equations. The novel error estimators only
take the curl of the righthand side into account and so lead to provably
reliable, efficient and pressure-independent upper bounds in case of a
pressure-robust method in particular in pressure-dominant situations. This is
also confirmed by some numerical examples with the novel pressure-robust
modifications of the TaylorHood and mini finite element methods
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