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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