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Exponential Convergence for hp-Version and Spectral Finite Element Methods for Elliptic Problems in Polyhedra
(2014)Research ReportWe establish exponential convergence of conforming hp-version and spectral finite element methods for second-order, elliptic boundary-value problems with constant coefficients and homogeneous Dirichlet boundary conditions in bounded, axiparallel polyhedra. The source terms are assumed to be piecewise analytic. The conforming hp-approximations are based on σ-geometric meshes of mapped, possibly anisotropic hexahedra and on the uniform and ...Report -
Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²
(2020)SAM Research ReportIn a polygon Ω ⊂ R2, we consider mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and hp spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in Ω, we prove exponential rates of convergence of the mixed ...Report -
An hp a-priori error analysis of the DG time-stepping method for initial value problems
(1999)SAM Research ReportThe Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a-priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for problems with smooth and analytic time dependence, respectively. It is further shown that ...Report -
Mixed hp-DGFEM for incompressible flows
(2002)SAM Research ReportWe consider several mixed discontinuous Galerkin approximations of the Stokes problem and propose an abstract framework for their analysis. Using this framework we derive a priori error estimates for hp-approximations on tensor product meshes. We also prove a new stability estimate for the discrete divergence bilinear form.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 -
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 -
hp Discontinuous Galerkin Time Stepping for Parabolic Problems
(2000)SAM Research ReportThe algorithmic pattern of the hp Discontinuous Galerkin Finite Element Method (DGFEM) for the time semidiscretization of abstract parabolic evolution equations is presented. In combination with a continuous $hp$ discretization in space we present a fully discrete hp-scheme for the numerical solution of parabolic problems. Numerical examples for the heat equation in a two dimensional domain confirm the exponential convergence rates which ...Report -
Local discontinuous Galerkin methods for the Stokes system
(2000)SAM Research ReportIn this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L^2-norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k-1 for the pressure, for ...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 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