# Finite Element Discretizations for Problems Arising in CFD

Numerical Analysis for Time-Dependent Equations

 PSPG for transient Stokes Volker John, Julia Novo Analysis of the PSPG Stabilization for the Evolutionary Stokes Equations Avoiding Time-Step Restrictions , SIAM J. Numer. Anal. 53, 1005 - 1031, 2015 Optimal error estimates for the pressure stabilized Petrov--Galerkin (PSPG) method for the evolutionary Stokes equations are proved, in the case of regular solutions, without restriction on the length of the time step. These results clarify the instability of the discrete pressure for small time steps'' reported in the literature. First, the limit situation of the continuous-in-time discretization is considered and second the numerical analysis for the backward Euler scheme is carried out. The main results are applicable to higher order finite elements. The analytical results are strongly based on an appropriate approach for computing the initial velocity, which is suggested in this paper. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition. Grad-div for transient Oseen Javier de Frutos, Bosco Garcia-Archilla, Volker John, Julia Novo Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements , J. Sci. Comp. 66, 991 - 1024, 2016 The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank--Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.

Conservation Properties of Discretizations

 Volker John, Alexander Linke, Christian Merdon, Michael Neilan, Leo R. Rebholz On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Review 59, 492 - 544, 2017 The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $\bH(\mathrm{div})$-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations. E.W. Jenkins, V. John, A. Linke, L.G. Rebholz On the parameter choice in grad-div stabilization for the Stokes equations , Adv. Comput. Math. 40, 491 - 516, 2014 Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the $H^1(\Omega)$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the $L^2(\Omega)$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the $H^1(\Omega)$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. \tred{Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.

Higher Order Finite Elements for the Stokes and Navier-Stokes Equations

 Drag and lift coefficients in a time dependent 2d flow through a channel around a cylinder V. John "Reference values for drag and lift of a two-dimensional time dependent flow around a cylinder", Int. J. Num. Meth. Fluids 44, 777 - 788, 2004 This paper presents a numerical study of a two-dimensional time dependent flow around a cylinder. Its main objective is to provide accurate reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time. In addition, the accuracy of these values obtained with different time stepping schemes and different finite element methods is studied. Drag and lift coefficients in a 3d flow through a channel around a cylinder V. John "Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations",, Int. J. Num. Meth. Fluids 40, 775 - 798, 2002 This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier-Stokes equations within the DFG high priority research program Flow Simulation with High--Performance Computers by Sch\"afer and Turek (1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid methods proves to be the best solver. Analysis of the multiple discretization multigrid method for higher order finite elements and for symmetric positive definite saddle point problems V. John, P. Knobloch, G. Matthies and L. Tobiska , "Non-nested multi-level solvers for finite element discretizations of mixed problems" , Computing 68, 313 - 341, 2002 We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations. Drag and lift coefficients in a 2d flow through a channel around a cylinder V. John and G. Matthies, " Higher Order Finite Element Discretizations in a Benchmark Problem for Incompressible Flows", Int. J. Num. Meth. Fluids 37, 885 - 903, 2001 We present a numerical study of several finite element discretizations applied to a benchmark problem for the 2d steady state incompressible Navier--Stokes equations defined in Schäfer and Turek (1996). The discretizations are compared with respect to the accuracy of the computed benchmark parameters. Higher order isoparametric finite element discretizations turned out to be by far most accurate. The discrete systems obtained with higher order discretizations are solved with a modified coupled multigrid method whose behaviour within the benchmark problem is also studied numerically.

Higher Order Time Integrators

 Variational time stepping schemes and adaptive time step control for convection-diffusion equations Naveed Ahmed, Volker John Adaptive time step control for the incompressible Navier-Stokes equations Comput. Methods Appl. Mech. Engrg. 285, 83 - 101, 2015 Higher order variational time stepping schemes allow an efficient post-processing for computing a higher order solution. This paper presents an adaptive algorithm whose time step control utilizes the post-processed solution. The algorithm is applied to convection-dominated convection-diffusion-reaction equations. It is shown that the length of the time step properly reflects the dynamics of the solution. With respect to the performance (accuracy, efficiency), the variational time stepping schemes are compared with an adaptive Crank--Nicolson scheme, whose time step control relies on comparing two solutions computed with schemes of the same order. Adaptive time step control for the 2d Navier-Stokes Equations V. John, J. Rang Adaptive time step control for the incompressible Navier-Stokes equations Comput. Meth. Appl. Mech. Engrg. 199, 514 - 524, 2010 Adaptive time stepping is an important tool in Computational Fluid Dynamics for controlling the accuracy of simulations and for enhancing their efficiency. This paper presents a systematic study of three classes of implicit and linearly implicit time stepping schemes with adaptive time step control applied to a 2D laminar flow around a cylinder: $\theta$-schemes, diagonal-implicit Runge-Kutta (DIRK) methods and Rosenbrock-Wanner (ROW) methods. The time step is controlled using embedded methods. It is shown that several ROW methods clearly outperform the more standard $\theta$-schemes and the DIRK methods. The results depend on a prescribed tolerance in the time step control algorithm, whose appropriate choice varies from scheme to scheme. 2d Navier-Stokes Equations V. John, G. Matthies, J. Rang A comparison of time-discretization/linearization approaches for the incompressible Navier-Stokes equations Comput. Meth. Appl. Mech. Engrg. 195, 5995 - 6010, 2006 This paper presents a numerical study of two ways for discretizing and linearizing the time-dependent incompressible Navier-Stokes equations. One approach consists in first applying a semi-discretization in time by a fully implicit $\theta$-scheme. Then, in each discrete time, the equations are linearized by a fixed point iteration. The number of iterations to reach a given stopping criterion is a priori unknown in this approach. In the second approach, Rosenbrock schemes with $s$ stages are used as temporal discretization. The non-linearity of the Navier-Stokes equations is treated internally in the Rosenbrock methods. In each discrete time, exactly $s$ linear systems of equations have to be solved. The numerical study considers five two-dimensional problems with distinct features. Four implicit time stepping schemes and five Rosenbrock methods are involved.

Pressure Separation for the Navier-Stokes Equations

 Steady state Navier-Stokes Equations S. Ganesan, V. John "Pressure Separation - a Technique for Improving the Velocity Error in Finite Element Discretisations of the Navier-Stokes Equations", Appl. Math. Comp. 165, 275 - 290, 2005 This paper presents a technique to improve the velocity error in finite element solutions of the steady state Navier-Stokes equations. This technique is called pressure separation. It relies upon subtracting the gradient of an appropriate approximation of the pressure on both sides of the Navier-Stokes equations. With this, the finite element error estimate can be improved in the case of higher Reynolds numbers. For practical reasons, the pressure separation can be applied above all for finite element discretisations of the Navier-Stokes equations with piecewise constant pressure. This paper presents a computational study of five ways to compute an appropriate approximation of the pressure. These ways are assessed on two- and three-dimensional examples. They are compared with respect to the error reduction in the discrete velocity and the computational overhead.

Stabilization Techniques for Non-Conforming Discretizations of Convection-Diffusion Equations

 Quadrilateral finite elements V. John, G. Matthies, F. Schieweck, L. Tobiska, "A Streamline-Diffusion Method for Nonconforming Finite Element Approximations Applied to Convection-Diffusion Problems", Comput. Methods Appl. Mech. Engrg. 166: 85 - 97, 1998 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. Triangular finite elements V. John, J.M. Maubach, L. Tobiska, "Nonconforming Streamline-Diffusion-Finite-Element-Methods for Convection-Diffusion Problems", Numerische Mathematik 78(2): 165-188, 1997 We analyze nonconforming finite element approximations of stream\-line-diffusion type for solving convection-diffusion problems. Both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same $O(h^{k+1/2})$ order of convergence in L$^2$ as in the conforming case for convection dominated problems. A rigorous error analysis supported by numerical experiments is given.