Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem

For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic expansions of eigenvalues of the Stokes operator are derived by establishing two pseudostress interpolations, which admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results

Layer-adapted meshes for convection-diffusion problems

This is a book on numerical methods for singular perturbation problems - in particular stationary convection-dominated convection-diffusion problems. More precisely it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. An early important contribution towards the optimization of numerical methods by means of special meshes was made by N.S. Bakhvalov in 1969. His paper spawned a lively discussion in the literature with a number of further meshes being proposed and applied to various singular perturbation problems. However, in the mid 1980s this development stalled, but was enlivend again by G.I. Shishkin's proposal of piecewise- equidistant meshes in the early 1990s. Because of their very simple structure they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on competing meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue. With this contribution we try to counter this development and lay the emphasis on more general meshes that - apart from performing better than piecewise-uniform meshes - provide a much deeper insight in the course of their analysis. In this monograph a classification and a survey are given of layer-adapted meshes for convection-diffusion problems. It tries to give a comprehensive review of state-of-the art techniques used in the convergence analysis for various numerical methods: finite differences, finite elements and finite volumes. While for finite difference schemes applied to one-dimensional problems a rather complete convergence theory for arbitrary meshes is developed, the theory is more fragmentary for other methods and problems and still requires the restriction to certain classes of meshes

Adjoint recovery of superconvergent functionals from PDE approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

Non-axisymmetric instabilities in self-gravitating tori around black holes, and solving Einstein constraints with superconvergent finite element methods

This thesis contains results on two related projects. In the first project, we explore non-axisymmetric instabilities in general relativistic accretion disks around black holes. Such disks are created as transient structures in several astrophysical scenarios, including mergers of compact objects and core collapse of massive stars. These disks are suggested for the role of cenral engines of gamma-ray bursts. We address the stability of these objects against the runaway and non-axisymmetric instabilities in the three-dimensional hydrodynamical fully general relativistic treatment. We explore three slender and moderately slender disk models with varying disk-to-black hole mass ratio. None of the models that we consider develop the runaway instability during the time span of the simulations, despite large radial axisymmetric oscillations, induced in the disks by the initial data construction procedure. All models develop unstable non-axisymmetric modes on a dynamical timescale. In simulations with dynamical general relativistic treatment, we observe two distinct types of instabilities: the Papaloizou-Pringle instability and the so-called Intermediate instability. The development of the nonaxisymmetric mode with azimuthal number m=1 is enhanced by the outspiraling motion of the black hole. The overall picture of the unstable modes in our disk models is similar to the Newtonian case. In the second project, we experiment with solving the Einstein constraint equations using finite elements on semistructured triangulations of multiblock grids. We illustrate our approach with a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $\psi$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, we evolve the constructed initial data using a high order finite-differencing multi-block approach and extract gravitational waves from the numerical solution