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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 -
Boundary Element Methods for Maxwell Equations in Lipschitz Domains
(2001)SAM Research ReportWe consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderon projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain ...Report -
N-term Wiener Chaos Approximation Rates for elliptic PDEs with lognormal Gaussian random inputs
(2011)SAM Research ReportWe consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner-Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on $\mathbb{R}^\mathbb{N}$. It is shown that the weak solution can ...Report -
Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients
(2011)SAM Research ReportIn this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably innite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear ...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 -
Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond
(2012)SAM Research ReportThis paper is a contemporary review of QMC ("quasi-Monte Carlo") methods, i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube $[0; 1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original ...Report -
On the justification of plate models
(2009)SAM Research ReportIn this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff--Love and Reissner--Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern [Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Rational Mech. Anal. 4, 145--152 (1959)] and is based on the two-energies principle of Prager and Synge. This was half a centenium ...Report -
A multiscale hp-FEM for 2D photonic crystal band
(2010)SAM Research ReportA Multiscale generalized $hp$-Finite Element Method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, ...Report -
hp-dGFEM for Second-Order Elliptic Problems in Polyhedra I: Stability and Quasioptimality on Geometric Meshes
(2009)SAM Research ReportWe introduce and analyze $hp$-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary value problems in three dimensional polyhedral domains. In order to resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined towards the corresponding neighborhoods. Similarly, the local ...Report -
Static load balancing for multi-level Monte Carlo finite volume solvers
(2011)SAM Research ReportThe Multi-Level Monte Carlo finite volumes (MLMC-FVM) algorithm was shown to be a robust and fast solver for uncertainty quantification in the solutions of multi- dimensional systems of stochastic conservation laws. A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware. We describe this procedure together with other arising challenges in great detail. Finally, numerical experiments ...Report