163 research outputs found
A Return to Innocence: Hopeful Words on the Rise and Fall of Man
A thesis presented to the faculty of the College of Business and Public Affairs at Morehead State University in partial fulfillment of the requirements for the Degree of Master of Public Administration by Philip Lederer in March of 2012
Asymptotically exact a posteriori error estimates for the BDM finite element approximation of mixed Laplace eigenvalue problems
We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the eigenfunction of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence without compromising the approximation properties. Consequently, the new flux can be used to derive optimal and asymptotically exact upper bounds for the eigenfunction, and optimal upper bounds for the corresponding eigenvalue. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.</p
Recommended from our members
Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations
This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1
A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators
The embedded discontinuous Galerkin (EDG) finite element method for the
Stokes problem results in a point-wise divergence-free approximate velocity on
cells. However, the approximate velocity is not H(div)-conforming and it can be
shown that this is the reason that the EDG method is not pressure-robust, i.e.,
the error in the velocity depends on the continuous pressure. In this paper we
present a local reconstruction operator that maps discretely divergence-free
test functions to exactly divergence-free test functions. This local
reconstruction operator restores pressure-robustness by only changing the right
hand side of the discretization, similar to the reconstruction operator
recently introduced for the Taylor--Hood and mini elements by Lederer et al.
(SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error
analysis of the discretization showing optimal convergence rates and
pressure-robustness of the velocity error. These results are verified by
numerical examples. The motivation for this research is that the resulting EDG
method combines the versatility of discontinuous Galerkin methods with the
computational efficiency of continuous Galerkin methods and accuracy of
pressure-robust finite element methods
Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations
This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1
Gradient-robust hybrid DG discretizations for the compressible Stokes equations
This paper studies two hybrid discontinuous Galerkin (HDG) discretizations
for the velocity-density formulation of the compressible Stokes equations with
respect to several desired structural properties, namely provable convergence,
the preservation of non-negativity and mass constraints for the density, and
gradient-robustness. The later property dramatically enhances the accuracy in
well-balanced situations, such as the hydrostatic balance where the pressure
gradient balances the gravity force. One of the studied schemes employs an
H(div)-conforming velocity ansatz space which ensures all mentioned properties,
while a fully discontinuous method is shown to satisfy all properties but the
gradient-robustness. Also higher-order schemes for both variants are presented
and compared in three numerical benchmark problems. The final example shows the
importance also for non-hydrostatic well-balanced states for the compressible
Navier-Stokes equations
- …