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HP90: a general & flexible fortran 90 hp-FE code
(1997)SAM Research ReportA general 2D-$hp$-adaptive Finite Element (FE) implementation inFortran 90 is described. The implementation is based on an abstractdata structure, which allows to incorporate the full $hp$-adaptivityof triangular and quadrilateral finite elements.The $h$-refinement strategies are based on $h2$-refinement ofquadrilaterals and $h4$-refinement of triangles. For $p$-refinementwe allow the approximation order to vary within any element. The ...Report -
Hierarchic Models for Laminated Plates and Shells
(1997)SAM Research ReportThe definition, essential properties and formulation of hierarchic models for laminated plates and shells are presented. The hierarchic models satisfy three essential requirements: approximability; asymptotic consistency, and optimality of convergence rate. Aspects of implementation are discussed and the performance characteristics are illustrated by examples.Report -
The hp Streamline Diffusion Finite Element Method for Convection Dominated Problems in one Space Dimension
(1998)SAM Research ReportWe analyze the hp Streamline Diffusion Finite Element Method (SDFEM) and the standard Galerkin FEM for one dimensional stationary convection-diffusion problems. Under the assumption of analyticity of the input data, a mesh is exhibited on which approximation with continuous piecewise polynomials of degree p allows for resolution of the boundary layer. On such meshes, both the SDFEM and the Galerkin FEM lead to robust exponential convergence ...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 -
An hp Finite Element Method for convection-diffusion problems
(1997)SAM Research ReportWe analyze an hp FEM for convection-diffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical $\alpha$-quadratically upwinded and the Hemker test-functions for piecewise linear trial spaces (see, e.g., [12] and the references there). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how ...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 -
Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs
(2011)SAM Research ReportWe investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a-priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor-, Legendre- and ...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 -
Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions
(2011)SAM Research ReportSpace-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space $H$ are developed. The well-posedness of these equations in the Hilbert space $L^2(H,μ)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $μ$ on $H$, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin ...Report