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Approximation on Simplices with respect to Weighted Sobolev Norms
(1999)SAM Research ReportInequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we cannot use orthogonal polynomials, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but comparable estimates on rectangles show that we cannot expect rotational invariance of the terms with derivatives.Report -
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 -
Homogenization via p-FEM for Problems with Microstructure
(1999)SAM Research ReportA new class of $p$ version FEM for elliptic problems with microstructure is developed. Based on arguments from the theory of $n$-widths, the existence of subspaces with favourable approximation properties for solution sets of PDEs is deduced. The construction of such subspaces is addressed for problems with (patch-wise) periodic microstructure. Families of adapted spectral shape functions are exhibited which give exponential convergence ...Report -
Generalized p-FEM in Homogenization
(1999)SAM Research ReportA new finite element method for elliptic problems with locally periodic microstructure of length $\varepsilon >0$ is developed and analyzed. It is shown that the method converges, as $\varepsilon \rightarrow 0$, to the solution of the homogenized problem with optimal order in $\varepsilon$ and exponentially in the number of degrees of freedom independent of $\varepsilon > 0$. The computational work of the method is bounded independently ...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 -
Analysis of membrane locking in hp FEM for a cylindrical shell
(1997)SAM Research ReportIn this paper we analyze the performance of the hp-Finite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that appropriate boundary layer elements are used. The numerical results illustrate the robustness and exponential convergence properties of the hp-Finite Element Method.Report -
Mixed hp Finite Element Methods for Stokes and Non-Newtonian Flow
(1997)SAM Research ReportWe analyze the stability of hp finite elements for viscous incompressible flow. For the classical velocity-pressure formulation, we give new estimates for the discrete inf-sup constants on geometric meshes which are explicit in the polynomial degree k of the elements. In particular, we obtain new bounds for p-elements on triangles. For the three-field Stokes problem describing linearized non-Newtonian flow, we estimate discrete inf-sup ...Report -
Coupled Problems for Viscous Incompressible Flow in Exterior Domains
(1999)SAM Research ReportThe formulation of the {\it fluid flow in an unbounded exterior domain} $\Omega$ is not always convenient for computations and, therefore, the problem is often truncated to a bounded domain $\Omega^-\subset\Omega$ with an artificial exterior boundary $\Gamma$. Then the problem of the choice of suitable "transparent" boundary conditions on $\Gamma$ appears. Another possibility is to simulate the presence of the fluid in the domain $\Omega^+$ ...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 -
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