Search
Results
-
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 -
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 -
hp-dGFEM for second-order elliptic problems in polyhedra. II: Exponential convergence
(2009)SAM Research ReportThe goal of this paper is to establish exponential convergence of $hp$ -version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [33] which are based on $\sigma$-geometric ...Report -
hp-dGFEM for Second-Order Elliptic Problems in Polyhedra II
(2009)Research reportThe goal of this paper is to establish exponential convergence of hp-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with piecewise analytic data in threedimensional polyhedral domains. More precisely, we shall analyze the convergence of the hp-IP dG methods considered in [30] which are based on !-geometric anisotropic ...Report -
Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs
(2009)SAM Research ReportDeterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D_Rd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(!)=(yi(!)). This yields an equivalent parametric deterministic PDE whose ...Report -
Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels
(2009)SAM Research ReportGalerkin discretizations of integral equations in $\mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $ \int_{S^{{(2)}}}$ $g(x,y)dydx$ where $S^{{(1)}}$, $S^{{(2)}}$ are $d$-simplices and $g$ has a singularity at $x=y$. We assume that g is Gevrey smooth for $x \neq y$ and satisfies bounds for the derivatives which allow algebraic singularities at $x = y$. This holds for kernel functions commonly occuring in integral ...Report -
Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels
(2009)SAM Research ReportGalerkin discretizations of integral equations in $\mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $ \int_{S^{{(2)}}}$ $g(x,y)dydx$ where $S^{{(1)}}$, $S^{{(2)}}$ are $d$-simplices and $g$ has a singularity at $x=y$. We assume that g is Gevrey smooth for $x \neq y$ and satisfies bounds for the derivatives which allow algebraic singularities at $x = y$. This holds for kernel functions commonly occuring in integral ...Report -
On Kolmogorov equations for anisotropic multivariate Lévy processes
(2008)Research reportsFor d-dimensional exponential L´evy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate L´evy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Representations of ...Report -
Sparse p-version BEM for first kind boundary integral equations with random loading
(2008)Research ReportReport -
Sparse high order FEM for elliptic sPDEs
(2008)Research ReportWe describe the analysis and the implementation of two Finite Element (FE) algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients. They are based on separation of deterministic and stochastic parts of the input data by a Karhunen-Lo`eve expansion, truncated after M terms. With a change of measure we convert the problem to a sequence of M-dimensional, parametric ...Report