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Higher order Quasi Monte Carlo integration for holomorphic, parametric operator equations
(2014)Research ReportWe analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space X. Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of X render the random inputs and the solutions of the forward problem countably parametric. ...Report -
Computational Higher Order Quasi-Monte Carlo Integration
(2014)Research ReportThe efficient construction of higher-order interlaced polynomial lattice rules introduced recently in [4] is considered. After briefly reviewing the principles of their construction by the “fast component-by-component” (CBC) algorithm due to [1, 10] as well as recent theoretical results on their convergence rates, we indicate algorithmic details of their construction. Instances of such rules are applied to highdimensional test integrands ...Report -
Covariance regularity and H-matrix approximation for rough random fields
(2014)Research ReportReport -
hp-dGFEM for Second-Order Mixed Elliptic Problems in Polyhedra
(2013)SAM Research ReportWe prove exponential rates of convergence of $hp$-dG interior penalty (IP) methods for second-order elliptic problems with mixed boundary conditions in polyhedra which are based on axiparallel, $\sigma$ -geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of $ \mu$-bounded variation. Compared to homogeneous Dirichlet boundary conditions in [10,11], or problems with mixed Dirichlet-Neumann ...Report -
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Exponential convergence of hp-DGFEM for elliptic problems in polyhedral domains
(2012)SAM Research ReportWe review the recent results of [21, 22], and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and constant coefficients in threedimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.Report -
Boundary element methods for Maxwell's equations on non-smooth domains
(2001)SAM Research ReportVariational boundary integral equations for Maxwell's equations on Lipschitz surfaces in R3 are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary ...Report -
The hp-Version of the Streamline Diffusion Finite Element Method in Two Space Dimensions
(1999)SAM Research ReportThe Streamline Diffusion Finite Element Method (SDFEM) for a two dimensional convection-diffusion problem is analyzed in the context of the hp-version of the Finite Element Method (FEM). It is proved that the appropriate choice of the SDFEM parameters leads to stable methods on the class of "boundary layer meshes" which may contain anisotropic needle elements of arbitrarily high aspect ratio. Consistency results show that the use of such ...Report -
High order Galerkin appoximations for parametric second order elliptic partial differential equations
(2012)SAM Research ReportLet $D \subset \mathbb{R}^d, d=2,3$, be a bounded domain with piecewise smooth boundary $\partial D$ and let $U$ be an open subset of Banach space $Y$. We consider a parametric family $P_y$ of uniformly strongly elliptic, parametric second order partial differential operators $P_y$ on $D$ in divergence form, where the parameter $y$ ranges in the parameter domain $U$ so that, for a given set of data $f_y$, the solution $u$ and the ...Report -
Analysis of membrane locking in hp FEM for a cylindrical shell
(1997)SAM Research ReportIn this paper we analyze the performance of the hp-Finite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that appropriate boundary layer elements are used. The numerical results illustrate the robustness and exponential convergence properties of the hp-Finite Element Method.Report