Christoph Lehrenfeld

High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

Abstract In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H ( div ) -conforming finite elements to guarantee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.

The Nitsche XFEM-DG Space-Time Method and its Implementation in Three Space Dimensions

In the recent paper [C. Lehrenfeld, A. Reusken, SIAM J. Num. Anal., 51 (2013)] a new finite element discretization method for a class of two-phase mass transport problems is presented and analyzed. The transport problem describes mass transport in a domain with an evolving interface. Across the evolving interface a jump condition has to be satisfies. The discretization in that paper is a space-time approach which combines a discontinuous Galerkin (DG) technique (in time) with an extended finite element method (XFEM). Using the Nitsche method the jump condition is enforced in a weak sense. While the emphasis in that paper was on the analysis and one dimensional numerical experiments the main contribution of this paper is the discussion of implementation aspects for the spatially three dimensional case. As the space-time interface is typically given only implicitly as the zero-level of a level-set function, we construct a piecewise planar approximation of the space-time interface. This discrete interface is used to divide the space-time domain into its subdomains. An important component within this decomposition is a new method for dividing four-dimensional prisms intersected by a piecewise planar space-time interface into simplices. Such a subdivision algorithm is necessary for numerical integration on the subdomains as well as on the space-time interface. These numerical integrations are needed in the implementation of the Nitsche XFEM-DG method in three space dimensions. Corresponding numerical studies are presented and discussed.

Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes

Hybrid discontinuous Galerkin methods are popular discretization methods in applications from fluid dynamics and many others. Often large scale linear systems arising from elliptic operators have to be solved. We show that standard p-version domain decomposition techniques can be applied, but we have to develop new technical tools to prove poly-logarithmic condition number estimates, in particular on tetrahedral meshes.

Numerical and experimental analysis of local flow phenomena in laminar Taylor flow in a square mini-channel

The vertically upward Taylor flow in a small square channel (side length 2 mm) is one of the guiding measures within the priority program “Transport Processes at Fluidic Interfaces” (SPP 1506) of the German Research Foundation (DFG). This paper presents the results of coordinated experiments and three-dimensional numerical simulations (with three different academic computer codes) for typical local flow parameters (bubble shape, thickness of the liquid film, and velocity profiles) in different cutting planes (lateral and diagonal) for a specific co-current Taylor flow. For most quantities, the differences between the three simulation results and also between the numerical and experimental results are below a few percent. The experimental and computational results consistently show interesting three-dimensional flow effects in the rear part of the liquid film. There, a local back flow of liquid occurs in the fixed frame of reference which leads to a temporary reversal of the direction of the wall shear stress during the passage of a Taylor bubble. Notably, the axial positions of the region with local backflow and those of the minimum vertical velocity differ in the lateral and the diagonal liquid films. By a thorough analysis of the fully resolved simulation results, this previously unknown phenomenon is explained in detail and, moreover, approximate criteria for its occurrence in practical applications are given. It is the different magnitude of the velocity in the lateral film and in the corner region which leads to azimuthal pressure differences in the lateral and diagonal liquid films and causes a slight deviation of the bubble from the rotational symmetry. This deviation is opposite in the front and rear parts of the bubble and has the mentioned significant effects on the local flow field in the rear part of the liquid film.

Analysis of a Nitsche XFEM-DG Discretization for a Class of Two-Phase Mass Transport Problems

We consider a standard model for mass transport across an evolving interface. The solution has to satisfy a jump condition across an evolving interface. We present and analyze a finite element discretization method for this mass transport problem. This method is based on a space-time approach in which a discontinuous Galerkin (DG) technique is combined with an extended finite element method (XFEM). The jump condition is satisfied in a weak sense by using the Nitsche method. This Nitsche XFEM-DG method is new. An error analysis is presented. Results of numerical experiments are given which illustrate the accuracy of the method.

Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces

We present a new high-order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, Comp. Meth. Appl. Mech. Engrg., 300 (2016), pp. 716--733]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order

Optimal preconditioners for Nitsche-XFEM discretizations of interface problems

H^1(\Gamma)

Nitsche-XFEM with Streamline Diffusion Stabilization for a Two-Phase Mass Transport Problem

-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a stabilization method which is based on adding an anisotropic diffusion in the volume mesh is able to control the condition number of the stiffness matrix also for the case of higher-order discretizations. Results of numerical e...

Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell’s Equations

In the past decade, a combination of unfitted finite elements (or XFEM) with the Nitsche method has become a popular discretization method for elliptic interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper (Hansbo and Hansbo, Comput Methods Appl Mech Eng 191:5537–5552, 2002). In general, the resulting linear systems have very large condition numbers, which depend not only on the mesh size h, but also on how the interface intersects the mesh. This paper is concerned with the design and analysis of optimal preconditioners for such linear systems. We propose an additive subspace preconditioner which is optimal in the sense that the resulting condition number is independent of the mesh size h and the interface position. We further show that already the simple diagonal scaling of the stiffness matrix results in a condition number that is bounded by