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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 -
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 -
Exponential Convergence in a Galerkin Least Squares hp-FEM for Stokes Flow
(1999)SAM Research ReportA stabilized hp-Finite Element Method (FEM) of Galerkin Least Squares (GLS) type is analyzed for the Stokes equations in polygonal domains. Contrary to the standard Galerkin FEM, this method admits equal-order interpolation in the velocity and the pressure, which is very attractive from an implementational point of view. In conjunction with geometrically refined meshes and linearly increasing approximation orders it is shown that thehp-GLSFEM ...Report -
Fourier mode analysis of layers in shallow shell deformations
(1999)SAM Research ReportWe investigate here the length scales of the boundary or interior layer effects in shell deformation. Quantitative information on the layers is obtained by considering two (simplified) `shallow' shell models corresponding to the `classical' three-field (Love-Koiter-Novozhilov), resp. five-field (Reissner-Naghdi) shell models. We start by analysing the layers as functions of the thickness of the shell, while keeping the other geometric ...Report -
Time Discretization of Parabolic Problems by the hp-Version of the Discontinuous Galerkin Finite Element Method
(1999)SAM Research ReportThe Discontinuous Galerkin Finite Element Method (DGFEM) for the time discretization of parabolic problems is analyzed in a hp-version context. Error bounds which are explicit in the time step as well as the approximation order are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential ...Report -
hp-FEM for Hyperbolic Problems
(1999)SAM Research ReportThis paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial differential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equations and second-order ...Report -
Advanced boundary element algorithms
(1999)SAM Research ReportWe review recent algorithmic developments in the boundary element method (BEM) for large scale engineering calculations. Two classes of algorithms, the clustering and the wavelet-based schemes are compared. Both have $O(N(\log N)^a)$ complexity with some small $a \ge 0$ and allow in-core simulations with up to $N = O(10^6)$ DOF on the boundary on serial workstations. Clustering appears more robust for complex surfaces.Report -
Fast numerical solution of the linearized Molodensky problem
(1999)SAM Research ReportWhen standard boundary element methods (BEM) are used to solve the linearized vector Molodensky problem we are confronted with two problems: (i) the absence of $O(|x|^{-2})$ terms in the decay condition is not taken into account, since the single layer ansatz, which is commonly used as representation of the perturbation potential, is of the order $O(|x|^{-1})$ as $x \to \infty$. This implies that the standard theory of Galerkin BEM is not ...Report -
hp-DGFEM for Partial Differential Equations with Nonnegative Characteristic Form
(1999)SAM Research ReportWe develop the error analysis for the hp-version of a discontinuous finite element approximation to second-order partial differential equations with nonnegative characteristic form. This class of equations includes classical examples of second-order elliptic and parabolic equations, first-order hyperbolic equations, as well as equations of mixed type. We establish an a priori error bound for the method which is of optimal order in the ...Report -
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