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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 expli