The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.

Schwarz Alternating Method

Numerical models of surface tension play an increasingly important role in our capacity to understand and predict a wide range of multiphase flow problems The accuracy and robustness of these models have improved markedly in the past 20 years, so that they are now applicable to complex, three-dimensional configurations of great theoretical and practical interest In this review, I attempt to summarize the most significant recent developments in Eulerian surface tension models, with an emphasis on well-balanced estimation, curvature estimation, stability, and implicit time stepping, as well as test cases and applications The advantages and limitations of various models are discussed, with a focus on common features rather than differences Several avenues for further progress are suggested

/pdf/numerical-models-of-surface-tension-10umm7lm0b.pdf

Numerical Models of Surface Tension

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

This paper introduces a quasi-interpolation operator for scalar-and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator is stable in L^1 , is a projection, whether homogeneous boundary conditions are imposed or not, and, assuming regularity in the fractional Sobolev spaces W^{s,p} where p ∈ [1, ∞] and s can be arbitrarily close to zero, gives optimal local approximation estimates in any L^p-norm. The theory is illustrated on H^1-, H(curl)-and H(div)-conforming spaces.

/pdf/finite-element-quasi-interpolation-and-best-approximation-38ibadsfdp.pdf

Finite element quasi-interpolation and best approximation

High order unfitted finite element methods on level set domains using isoparametric mappings

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 $H^1(\Gamma)$-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...

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

Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $$\nabla u \in L^1(0,T;L^\infty (\varOmega ))$$
 which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free $$H^1$$
 -conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

In a recent paper [C. Lehrenfeld and A. Reusken, SIAM J. Numer. Anal., 51 (2013), pp. 958--983] 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 satisfied. 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 disc...

/pdf/the-nitsche-xfem-dg-space-time-method-and-its-implementation-2071qwrfc2.pdf

Christoph Lehrenfeld

Papers

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

High order unfitted finite element methods on level set domains using isoparametric mappings

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

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

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