140 research outputs found
Aspects of guaranteed error control in CPDEs
Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretisation error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bounds (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects. First, the variational crimes from a nonconforming finite element discretisation and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB. Second, the reliable approximation of the discretisation error on curved boundaries and, finally, the reliable bounds of the error with respect to some goal-functional, namely, the error in the approximation of the directional derivative at a given point
Aspects of quaranteed error control in CPDEs
Whenever numerical algorithms are employed for a reliable computational
forecast, they need to allow for an error control in the final quantity of
interest. The discretisation error control is of some particular importance
in computational PDEs (CPDEs) where guaranteed upper error bounds (GUB) are
of vital relevance. After a quick overview over energy norm error control in
second-order elliptic PDEs, this paper focuses on three particular aspects.
First, the variational crimes from a nonconforming finite element
discretisation and guaranteed error bounds in the discrete norm with improved
postprocessing of the GUB. Second, the reliable approximation of the
discretisation error on curved boundaries and, finally, the reliable bounds
of the error with respect to some goal-functional, namely, the error in the
approximation of the directional derivative at a given poin
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
Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations
International audienceWe present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme
Guaranteed error control for the pseudostress approximation of the Stokes equations
The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in . Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy
Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method
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
Guaranteed error control for the pseudostress approximation of the Stokes equations
The pseudostress approximation of the Stokes equations rewrites the
stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet
boundary conditions as another (equivalent) mixed scheme based on a stress in
H (div) and the velocity in L2. Any standard mixed finite element function
space can be utilized for this mixed formulation, e.g. the Raviart-Thomas
discretization which is related to the Crouzeix-Raviart nonconforming finite
element scheme in the lowest-order case. The effective and guaranteed a
posteriori error control for this nonconforming velocity-oriented
discretization can be generalized to the error control of some piecewise
quadratic velocity approximation that is related to the discrete
pseudostress. The analysis allows for local inf-sup constants which can be
chosen in a global partition to improve the estimation. Numerical examples
provide strong evidence for an effective and guaranteed error control with
very small overestimation factors even for domains with large anisotropy
Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified
lowest-order nonconforming Crouzeix-Raviart finite element method for the
Stokes equations. The modification from [A. Linke 2014, On the role of the
Helmholtz-decomposition in mixed methods for incompressible flows and a new
variational crime] is based on the observation that only the divergence-free
part of the right-hand side should balance the vector Laplacian. The new
method has optimal energy error estimates and can lead to errors that are
smaller by several magnitudes, since the estimates are pressure-independent.
An efficient a posteriori velocity error estimator for the modified method
also should involve only the divergence-free part of the right-hand side.
Some designs to approximate the Helmholtz projector are compared and verified
by numerical benchmark examples. They show that guaranteed error control for
the modified method is possible and almost as sharp as for the unmodified
method
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