55 research outputs found
Pre- and postprocessing techniques for determining goodness of computational meshes
Research in error estimation, mesh conditioning, and solution enhancement for finite element, finite difference, and finite volume methods has been incorporated into AUDITOR, a modern, user-friendly code, which operates on 2D and 3D unstructured neutral files to improve the accuracy and reliability of computational results. Residual error estimation capabilities provide local and global estimates of solution error in the energy norm. Higher order results for derived quantities may be extracted from initial solutions. Within the X-MOTIF graphical user interface, extensive visualization capabilities support critical evaluation of results in linear elasticity, steady state heat transfer, and both compressible and incompressible fluid dynamics
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
Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems
We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved. Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases
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 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
Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
Equilibration error estimators have been shown to commonly lead to very
accurate guaranteed error bounds in the a posteriori error control of finite
element methods for second order elliptic equations. Here, we extend previous
results by the design of equilibrated fluxes for higher-order finite element
methods with nonconstant coefficients and illustrate the favourable
performance of different variants of the error estimator within two
deterministic benchmark settings. After the introduction of the respective
parametric problem with stochastic coefficients and the stochastic Galerkin
FEM discretisation, a novel a posteriori error estimator for the stochastic
error in the energy norm is devised. The error estimation is based on the
stochastic residual and its decomposition into approximation residuals and a
truncation error of the stochastic discretisation. Importantly, by using the
derived deterministic equilibration techniques for the approximation
residuals, the computable error bound is guaranteed for the considered class
of problems. An adaptive algorithm allows the simultaneous refinement of the
deterministic mesh and the stochastic discretisation in anisotropic Legendre
polynomial chaos. Several stochastic benchmark problems illustrate the
efficiency of the adaptive process
Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process
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