5,320 research outputs found
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
Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we investigate the application of pseudo-transient-continuation
(PTC) schemes for the numerical solution of semilinear elliptic partial
differential equations, with possible singular perturbations. We will outline a
residual reduction analysis within the framework of general Hilbert spaces,
and, subsequently, employ the PTC-methodology in the context of finite element
discretizations of semilinear boundary value problems. Our approach combines
both a prediction-type PTC-method (for infinite dimensional problems) and an
adaptive finite element discretization (based on a robust a posteriori residual
analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
We derive a posteriori error estimates for the hybridizable discontinuous
Galerkin (HDG) methods, including both the primal and mixed formulations, for
the approximation of a linear second-order elliptic problem on conforming
simplicial meshes in two and three dimensions.
We obtain fully computable, constant free, a posteriori error bounds on the
broken energy seminorm and the HDG energy (semi)norm of the error. The
estimators are also shown to provide local lower bounds for the HDG energy
(semi)norm of the error up to a constant and a higher-order data oscillation
term. For the primal HDG methods and mixed HDG methods with an appropriate
choice of stabilization parameter, the estimators are also shown to provide a
lower bound for the broken energy seminorm of the error up to a constant and a
higher-order data oscillation term. Numerical examples are given illustrating
the theoretical results
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