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Fully Discrete hp-Finite Elements: Fast Quadrature
(1999)SAM Research ReportA fully discrete hp finite element method is presented. It combines the features of the standard hp finite element method (conforming Galerkin Formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spectral quadrature techniques). The speed-up (relative to standard hp elements) is analyzed in detail both theoretically and computationally .Report -
The hp-Version of the Streamline Diffusion Finite Element Method in Two Space Dimensions
(1999)SAM Research ReportThe Streamline Diffusion Finite Element Method (SDFEM) for a two dimensional convection-diffusion problem is analyzed in the context of the hp-version of the Finite Element Method (FEM). It is proved that the appropriate choice of the SDFEM parameters leads to stable methods on the class of "boundary layer meshes" which may contain anisotropic needle elements of arbitrarily high aspect ratio. Consistency results show that the use of such ...Report -
The hp Streamline Diffusion Finite Element Method for Convection Dominated Problems in one Space Dimension
(1998)SAM Research ReportWe analyze the hp Streamline Diffusion Finite Element Method (SDFEM) and the standard Galerkin FEM for one dimensional stationary convection-diffusion problems. Under the assumption of analyticity of the input data, a mesh is exhibited on which approximation with continuous piecewise polynomials of degree p allows for resolution of the boundary layer. On such meshes, both the SDFEM and the Galerkin FEM lead to robust exponential convergence ...Report -
A spectral Garlekin method for hydrodynamic stability problems
(1998)SAM Research ReportA spectral Galerkin method for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of {\em scale resolution}, i.e., that $\re/p^2$ is small; here $\re$ is the Reynolds number and ...Report -
On n-widths for Elliptic Problems
(1998)SAM Research ReportThe n-width of solution sets of a class of elliptic partial differential equations is calculated. In particular singularly perturbed equations of elliptic-elliptic type are considered.Report -
Operator adapted spectral element methodsOperator Adapted Spectral Element Methods. I. Harmonic and Generalized Harmonic Polynomials
(1997)SAM Research ReportThe paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and sharp estimates for the approximation properties of harmonic polynomials ...Report -
hp FEM for Reaction-Diffusion Equations. II: Regularity Theory
(1997)SAM Research ReportA singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve, the boundary data are analytic, and the right hand side is analytic. We give asymptotic expansions of the solution and new error bounds that are uniform in the perturbation parameter as well as in the expansion order. Additionally, we provide growth estimates ...Report -
hp FEM for Reaction-Diffusion Equations. I: Robust Exponential Convergence
(1997)SAM Research ReportA singularly perturbed reaction-diffusion equation in two dimensions is considered. We assume analyticity of the input data, i.e., the boundary of the domain is an analytic curve and the right hand side is analytic. We show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width $O(p \varepsilon)$ is inserted near the domain boundary, that is, the rate ...Report -
An hp Finite Element Method for convection-diffusion problems
(1997)SAM Research ReportWe analyze an hp FEM for convection-diffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical $\alpha$-quadratically upwinded and the Hemker test-functions for piecewise linear trial spaces (see, e.g., [12] and the references there). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how ...Report -
On robust exponential convergence of hp finite element methods for problems with boundary layers
(1996)SAM Research ReportThe emhp/em version of the finite element method for a one dimensional, singularly perturbed elliptic model problem with analytic right hand side is considered. It is shown that the use of piecewise polynomials of degree emp/em on a mesh consisting of three suitably chosen elements leads to robust exponential convergence, i.e., the rate of convergence depends only on the right hand side and is independent of the perturbation parameter.Report