Friedhelm Schieweck

A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems

Abstract We consider a nonconforming streamline-diffusion finite element method for solving convection-diffusion problems. The theoretical and numerical investigation for triangular and tetrahedral meshes recently given by John, Maubach and Tobiska has shown that the usual application of the SDFEM gives not a sufficient stabilization. Additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case. The error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space. Thus, the analysis can be applied for example to the Crouzeix/Raviart element but not to the nonconforming quadrilateral elements proposed by Rannacher and Turek. In this paper, parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case. Numerical experiments support the theoretical predictions.

A-stable discontinuous Galerkin–Petrov time discretization of higher order

Abstract We construct and analyze a discontinuous Galerkin–Petrov time discretization of a general evolution equation in a Hilbert space. The method is A-stable and exhibits an energy decreasing property. The approach consists in a continuous solution space and a discontinuous test space such that the time derivative of the discrete solution is contained in the test space. This is the key to get stability. We prove A-stability and optimal error estimates. Numerical results confirm the theoretical results.

Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

Abstract We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the “length” of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles. These analytical investigations are complemented by presenting a numerical algorithm, based on C 1-elements, and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.

Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation

Abstract We discuss numerical properties of continuous Galerkin–Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.

A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

In this note, we extend our recent work for the heat equation in [1] and describe and compare by means of numerical experiments the continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization applied to the nonstationary Stokes equations in the two-dimensional case. For the space discretization, we use the well-known LBB-stable quadrilateral finite element which consists of conforming biquadratic elements for the velocity and discontinuous linear elements for the pressure. We discuss implementation aspects as well as methods for solving the resulting block systems using monolithic multigrid solvers based on Vanka-type smoothers. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs. We show that the convergence behavior of the multigrid method is almost independent of the mesh size in space and the time step size which means (at least for these examples) that we have created an efficient solution process. 2000 Mathematics Subject Classification (MSC): 65M12, 65M55, 65M60.

Parallel Finite Element Methods for the Incompressible Navier-Stokes Equations

We consider parallel and adaptive algorithms for the incompressible Navier-Stokes equations discretized by an upwind type finite element method. Two parallelization concepts are used, a first one based on a static domain decomposition into macroelements and a second one based on a dynamic load balancing strategy. We investigate questions of the scalability up to the massive parallel case and the use of a posteriori error estimators. The arising discrete systems are solved by parallelized multigrid methods which are applied either directly to the coupled system or within a projection method.

H 1 -interpolation on quadrilateral and hexahedral meshes with hanging nodes

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H1-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H1. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.

A parameter-free smoothness indicator for high-resolution finite element schemes

This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.

Higher Order Galerkin Time Discretization for Nonstationary Incompressible Flow

In this paper, we extend our work for the heat equation in (Hussain et al., J Numer Math 19(1):41–61, 2011) and for the Stokes equations in (Hussain et al., Open Numer Methods J 4:35–45, 2012) to the nonstationary Navier-Stokes equations in two dimensions. We examine continuous Galerkin-Petrov (cGP) time discretization schemes for nonstationary incompressible flow. In particular, we implement and analyze numerically the higher order cGP(2)-method. For the space discretization, we use the LBB-stable finite element pair \(Q_{2}/P_{1}^{\mathit{disc}}\). The discretized systems of nonlinear equations are treated by using the fixed-point as well as the Newton method and the associated linear subproblems are solved by using a monolithic multigrid solver with GMRES method as smoother. We perform nonstationary simulations for a benchmarking configuration to analyze the temporal accuracy and efficiency of the presented time discretization scheme.

Local CIP Stabilization for Composite Finite Elements

We propose a continuous interior penalty (CIP) method for the pure transport problem and for the viscosity dependent “Stokes--Brinkman” problem where the gradient jump penalty is localized to faces in the interior of subdomains. Special focus is given to the case where the subdomains are so-called composite finite elements, e.g., quadrilateral, hexahedral or prismatic elements which are composed by simplices such that the arising global simplicial mesh is regular. The advantage of this local CIP is that it allows for static condensation in contrast to the classical CIP method. If the degrees of freedom in the interior of the composite finite elements are eliminated using static condensation then the resulting couplings of the skeleton degrees of freedom are comparable to those for classical conforming finite element methods which leads to a substantially smaller matrix stencil than for the standard global CIP method. Optimal stability and error estimates are proved and numerical tests are presented. For t...