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Mixed hp-FEM on anisotropic meshes
(1997)SAM Research ReportMixed hp-FEM for incompressible fluid flow on anisotropic meshes are analyzed. A discrete inf-sup condition is proved with a constant independent of the meshwidth and the aspect ratio. For each polynomial degree $k\geq 2$, velocity-pressure subspace pairs are presented which are stable on quadrilateral mesh-patches, independently of the element aspect ratio implying in particular divergence stability on the so-called Shishkin-meshes. ...Report -
Mixed hp-FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes
(1997)SAM Research ReportDivergence stability of mixed hp-FEM for incompressible fluid flow for a general class of possibly highly irregular meshes is shown. The meshes may be refined anisotropically and contain hanging nodes on geometric patches. The inf-sup constant is independent of the aspect ratio of the elements and the dependence on the polynomial degree is given explicitly. Numerical estimates of inf-sup constants confirm our results.Report -
An hp a-priori error analysis of the DG time-stepping method for initial value problems
(1999)SAM Research ReportThe Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a-priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for problems with smooth and analytic time dependence, respectively. It is further shown that ...Report -
Coupling of an Interior Navier-Stokes Problem with an Exterior Oseen Problem
(1998)SAM Research ReportThe paper is concerned with the modelling of viscous incompressible flow in an unbounded exterior domain with the aid of the coupling of the nonlinear Navier--Stokes equations considered in a bounded domain with the linear Oseen system in an exterior domain. These systems are coupled on an artificial interface via suitable transmission conditions. The present paper is a continuation of the work [8], where the coupling of the Navier--Stokes ...Report -
Quadrature for hp-Galerkin BEM in R³
(1996)SAM Research ReportThe Galerkin discretization of a Fredholm integral equation of the second kind on a closed, piecewise analytic surface $\Gamma \subset \hbox {R}^3$ is analyzed. High order, $hp$-boundary elements on grids which are geometrically graded toward the edges and vertices of the surface give exponential convergence, similar to what is known in the $hp$ Finite Element Method. A quadrature strategy is developed which gives rise to a fully discrete ...Report -
Two notes on the implementation of wavelet Galerkin boundary element methods
(1997)SAM Research ReportWe report, in two notes, recent progress in the implementation of wavelet-based Galerkin BEM on polyhedra and study the performance.Report -
Fully Discrete Multiscale Galerkin BEM
(1995)SAM Research ReportWe analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in $R^3$. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with "compressed" stiffness matrices containing $O(N({\log N})^2)$ nonvanishing entries ...Report -
Wavelet Galerkin Algorithms for Boundary Integral Equations
(1997)SAM Research ReportThe implementation of a fast, wavelet-based Galerkin discretization of second kind integral equations on piecewise smooth surfaces $\Gamma\subset \R^3$ is described. It allows meshes consisting of triangles as well as quadrilaterals. The algorithm generates a sparse, approximate stiffness matrix with $N=O(N(log N)^2)$ nonvanishing entries in $O(N(\log N)^4)$ operations where N is the number of degrees of freedom on the boundary while ...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 -
HP90: a general & flexible fortran 90 hp-FE code
(1997)SAM Research ReportA general 2D-$hp$-adaptive Finite Element (FE) implementation inFortran 90 is described. The implementation is based on an abstractdata structure, which allows to incorporate the full $hp$-adaptivityof triangular and quadrilateral finite elements.The $h$-refinement strategies are based on $h2$-refinement ofquadrilaterals and $h4$-refinement of triangles. For $p$-refinementwe allow the approximation order to vary within any element. The ...Report