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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 -
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 -
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 -
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 -
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