K. Gerdes

hp-discontinuous Galerkin time stepping for parabolic problems

Abstract We consider the hp-version of the discontinuous Galerkin (DG) time-stepping method for linear parabolic problems with non-symmetric elliptic spatial operators. We derive new analyticity estimates for the exact solutions by means of semigroup techniques. These estimates allow us to show that the hp-DG time-stepping method can resolve start-up singularities at exponential rates of convergence.

Fully discrete hp-finite elements : fast quadrature

A fully discrete hp-finite element method (FEM) is presented. It combines the features of the standard hp-FEM (conforming Galerkin formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spectral quadrature techniques). The speed-up (relative to standard hp elements) is analyzed in detail both theoretically and computationally.

On the pollution effect in FE solutions of the 3D-Helmholtz equation

We study convergence of the FEM for a 3D-problem of rigid scattering. Using separation of variables, analytical results are presented for stability and error estimation for an h-version Galerkin FEM with mesh-refinement in radial direction. We prove an upper bound that contains a pollution term similar to previous results in 1D and 2D. This estimate is evaluated computationally using a 3D FEM-IEM (infinite elements) implementation. The tabulated and plotted errors show the theoretically predicted pollution effect in the FE-error.

A summary of infinite element formulations for exterior Helmholtz problems

This work is devoted to a study and summary of different Infinite Element (IE) formulations for Helmholtz problems in arbitrary exterior domains. The theoretical setting for each of the different formulations is presented and related to the mathematical existence theory. The influence of a bilinear or a sesquilinear formulation is discussed as well as possible extensions to other elements. The implementation of the Infinite Element Method (IEM) incorporates the use of 2D and 3D hp Finite Elements and allows for hp-adaptive refinements. Numerical results show the computational efficiency of the coupled Finite-Infinite Element methodology.

Analysis of membrane locking in hp FEM for a cylindrical shell

In 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 boundary layers stemming from the edge effect are resolved. The numerical results illustrate the mesh independence of the exponential convergence of the hp-FEM.

hp -finite element simulations for Stokes flow—stable and stabilized

Abstract The stable Galerkin formulation and a stabilized Galerkin least squares formulation for the Stokes problem are analyzed in the context of the hp -version of the finite element method. Theoretical results for both formulations establish exponential rates of convergence under realistic assumptions on the input data. We confirm these results by a series of numerical experiments on an L -shaped domain where the solution exhibits corner singularities.

HIERARCHIC MODELS OF HELMHOLTZ PROBLEMS ON THIN DOMAINS

The Helmholtz equation in a three-dimensional plate is approximated by a hierarchy of two-dimensional models. Computable a posteriori error estimators of the modeling error in exponentially weighted norms are derived, and sharp, computable estimates for their effectivity indices are also obtained. The necessity of including, besides polynomials, a certain number of trigonometric director functions into the Ansatz, in order to prevent pollution effects at high wave numbers is demonstrated both theoretically and computationally.

Galerkin least squares hp-finite element method for the Stokes problem

Abstract A stabilized mixed hp -Finite Element Method (FEM) of Galerkin Least Squares type for the Stokes problem in polygonal domains is presented and analyzed. It is proved that for equal order velocity and pressure spaces this method leads to exponential rates of convergence provided that the data is piecewise analytic.

Stable and stabilized hp -finite element methods for the Stokes problem

Abstract Two hp–finite element methods for the Stokes problem in polygonal domains are presented: We discuss the S k × S k−2 elements which are stable on anisotropic and irregular meshes and introduce a stabilized Galerkin Least Squares approach featuring equal-order interpolation in the velocity and the pressure. Both methods lead to exponential rates of convergence provided that the data is piecewise analytic. Numerical studies on an L-shaped domain confirm these theoretical results.