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

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

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