- Survey on open problems and recent results
- Stabilization techniques for Oseen and Navier-Stokes equations
- SOLD schemes and other discretizations for steady-state convection-diffusion equations
- A posteriori parameter choice of stabilization parameters
- Time-dependent convection-diffusion-reaction equations
- Variational multiscale (VMS) approaches for convection-diffusion equations

Survey on open problems and recent results (2018) Volker John, Petr Knobloch, Julia Novo A Finite Elements for Scalar Convection-Dominated Equations and Incompressible Flow Problems: a Never Ending Story?, Comput. Visual. Sci. 19, 47 - 63, 2018 The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important \revi{open} problems in these fields are discussed. The exposition concentrates on $H^1$-conforming finite elements. |

Optimal error bounds for LPS methods and equal order spacesJavier de Frutos, Bosco Garcia-Archilla, Volker John, Julia Novo Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier--Stokes Equations with Local Projection Stabilization, IMA J. Numer. Anal. 39, 1747 - 1786, 2019 This paper studies non inf-sup stable finite element approximations to the evolutionary Navier--Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty(0,T;L^2(\Omega))$ decays with rate $l+1/2$ in the case that $\nu\le h$, with $\nu$ being the dimensionless viscosity and $h$ the mesh width. In the analysis of another method, it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results. |

Optimal error bounds for a combination of LPS method and cGP method for
the evolutionary Oseen equationsNaveed Ahmed, Volker John, Gunar Matthies, Julia Novo A Local Projection Stabilization/Continuous Galerkin-Petrov Method for Incompressible Flow Problems, Appl. Math. Comp. 333, 304--324, 2018 A local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous-in-time case for both the velocity and pressure approximation. In addition, the fully discrete case in combination with higher order continuous Galerkin--Petrov (cGP) methods is studied. Error estimates of order $k+1$ are proved, where $k$ denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms. |

Uniform (semi-robust) finite element error analysis for time-dependent Navier-Stokes
equations and grad-div stabilizationJavier de Frutos, Bosco Garcia-Archilla, Volker John, Julia Novo Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math. 44, 195 - 225, 2018 This paper studies inf-sup stable finite element discretizations of the evolutionary Navier--Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier- Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order $\mathcal O(h^2)$ in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed. |

Analysis of a fully discrete LPS method for the Navier-Stokes equations Naveed Ahmed, Tomas Chacon Rebollo, Volker John, Samuele Rubino Analysis of Local Projection Stabilization Methods for the Fully Discrete Evolutionary Navier-Stokes Equations, IMA J. Numer. Anal. 37, 1437 - 1467, 2017 A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy. |

Isogeometric analysis for the Navier-Stokes equations Jannis Bulling, Volker John, Petr Knobloch Isogeometric analysis for flows around a cylinder, Appl. Math. Lett. 63, 65 - 70, 2017 This note studies the accuracy of Isogeometric Analysis (IGA) applied in the simulation of incompressible flows around a cylinder in two and three dimensions. Quantities of interest, like the drag coefficient, the lift coefficient, and the difference of the pressure between the front and the back of the cylinder are monitored. Results computed with standard finite element methods are used for comparison. |

WENO finite difference scheme for time-dependent Navier-Stokes equationsJavier de Frutos, Volker John, Julia Novo, Projection Methods for Incompressible Flow Problems with WENO Finite Difference Schemes , J. Comput. Phys. 309, 368 - 386, 2016 Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary con\-vec\-tion-diffusion equations \cite{JN12}. This paper explores the applicability of these sche\-mes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier--Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov--Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious. |

Crouzeix-Raviart FE and residual-free bubblesLeopoldo P. Franca, Volker John, Gunar Matthies and Lutz Tobiska, An inf-sup stable and residual-free bubble element for the Oseen equations , SIAM J. Numer. Anal. 45, 2392 - 2407, 2007 We investigate the residual-free bubble method for the linearized incompressible Navier-Stokes equations. Starting with a nonconforming inf-sup stable element pair for approximating the velocity and pressure, we enrich the velocity space by discretely divergence-free bubble functions to handle the influence of strong convection. An important feature of the method is that the stabilization does not generate an additional coupling between the mass equation and the momentum equation as it is the case for the streamline upwind Petrov Galerkin (SUPG) method applied to equal order interpolation. Furthermore, the discrete solution is piecewise divergence-free, a property which is useful for the mass balance in transport equations coupled with the incompressible Navier-Stokes equations. |

ReviewM.Braack, E.Burman, V.John and G.Lube "Stabilized finite element methods for the generalized Oseen problem" Comput. Meth. Appl. Mech. Engrg. 196, 853 - 866, 2007 The numerical solution of the non-stationary, incompressible Navier-Stokes model can be split into linearized auxiliary problems of Oseen type. We present in a unique way different stabilization techniques of finite element schemes on isotropic meshes. First we describe the state-of-the-art for the classical residual-based SUPG/PSPG method. Then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method. These methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation. Finally, we give a critical comparison of these methods. |

Solvers for the nonlinear problems in AFC schemes, part II Abhinav Jha, Volker John A Study of Solvers for Nonlinear AFC Discretizations of Convection-Diffusion Equations, Computers and Mathematics with Applications 78, 3117 - 3138, 2019 Nonlinear discretizations are necessary for convection-diffusion equations for obtaining accurate solutions that satisfy the discrete maximum principle. The numerical solution of the arising nonlinear problems is often difficult. This paper presents several approaches for solving the nonlinear problems of algebraic flux correction (AFC) schemes for the Kuzmin limiter and the BJK limiter. Comprehensive numerical studies are performed at examples that model the transport of energy from a body in a flow field in two and three dimensions. It turns out that the most efficient approach, from the point of view of computing times, is a simple fixed point iteration, because the iteration matrix possesses properties that can be exploited by the solvers of the arising linear systems of equations. |

Solvers for the nonlinear problems in AFC schemes, part I Abhinav Jha, Volker John On Basic Iteration Schemes for Nonlinear AFC Discretizations, Proceedings of BAIL 2018, in press, 2019 Algebraic flux correction (AFC) finite element discretizations of steady-state convection-diffusion-reaction equations lead to a nonlinear problem. This paper presents first steps of a systematic study of solvers for these problems. Two basic fixed point iterations and a formal Newton method are considered. It turns out that the fixed point iterations behave often quite differently. Using a sparse direct solver for the linear problems, one of them exploits the fact that only one matrix factorization is needed to become very efficient in the case of convergence. For the behavior of the formal Newton method, a clear picture is not yet obtained. |

Algebraic flux correction schemes: unified analysis Gabriel R. Barrenechea, Volker John, Petr Knobloch, Richard Rankin A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations, SeMa Journal 75, 655 - 685, 2018 Recent results on the numerical analysis of Algebraic Flux Correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme. |

Algebraic flux correction schemes: linearity preservation Gabriel R. Barrenechea, Volker John, Petr Knobloch A linearity preserving algebraic flux correction scheme satisfying the discrete maximum principle on general meshes, Mathematical Models and Methods in Applied Sciences (M3AS), 27, 525 - 548, 2017 This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection-diffusion problems confirm the theory. |

Algebraic flux correction schemes: first convergence analysis Gabriel R. Barrenechea, Volker John, Petr Knobloch Analysis of algebraic flux correction schemes, SIAM J. Numer. Analysis 54, 2427 - 2451, 2016 A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection--diffusion--reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness. |

Algebraic flux correction schemes Gabriel R. Barrenechea, Volker John, Petr Knobloch Some analytical results for an algebraic flux correction scheme for a steady convection--diffusion equation in 1D , IMA J. Numer. Anal. 35, 1729 - 1756, 2015 Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection--diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method. |

Isogeometric analysis for the Hemker problem Volker John, Liesel Schumacher Study of Isogeometric Analysis for Scalar Convection-Diffusion Equations , Appl. Math. Lett. 27, 43 - 48, 2014 Isogeometric Analysis (IGA), in combination with the Streamline Upwind Petrov--Galerkin (SUPG) stabilization, is studied for the discretization of steady-state con\-vection-diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers. |

LPS schemes with crosswind diffusion for steady-state and time-dependent problem Gabriel R. Barrenechea, Volker John, Petr Knobloch A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations , ESAIM M2AN 47, 1335 - 1366, 2013 An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations. |

Comparison of a lot of discretizations at the Hemker example Matthias Augustin, Alfonso Caiazzo, Andre Fiebach, Jürgen Fuhrmann, Volker John, Alexander Linke, Rudolf Umla An assessment of discretizations for convection-dominated convection-diffusion equations , Comput. Meth. Appl. Mech. Engrg. 200, 3395 - 3409, 2011 The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galer\-kin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented. |

Analysis of best methods from review on selected examples V.John and P.Knobloch "On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II - analysis for P1 and Q1 finite elements", Comput. Meth. Appl. Mech. Engrg. 197, 1997 - 2014, 2008 An unwelcome feature of the popular streamline upwind/Petrov--Galerkin (SUPG) stabilization of convection--dominated convection--diffusion equations is the presence of spurious oscillations at layers. A review and a comparison of the most methods which have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers are given in Part~I,~\cite{JohnKnobloch}. In the present paper, the most promising of these SOLD methods are investigated in more detail for $P_1$ and $Q_1$ finite elements. In particular, the dependence of the results on the mesh, the data of the problems and parameters of the methods are studied analytically and numerically. Furthermore, the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined. Our conclusion is that, also for simple problems, any of the SOLD methods can generally provide solutions with non--negligible spurious oscillations. |

Review and first numerical results V.John and P.Knobloch "On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I - a review", Comput. Meth. Appl. Mech. Engrg. 196, 2197 - 2215, 2007 An unwelcome feature of the popular streamline upwind/Petrov-Galerkin (SUPG) stabilization of convection-dominated convection-diffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980-ies, a number of methods have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers. The paper gives a review and state of the art of these methods, discusses their derivation, proposes some alternative choices of parameters in the methods and categorizes them. Some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods. |

Numerical results for nonconstant convection fieldV.John and P.Knobloch "On the performance of SOLD methods for convection-diffusion problems with interior layers", Int. J. Comp. in Science and Mathematics 1, 245 - 258, 2007 Numerical solutions of convection-diffusion equations obtained using the streamline-upwind Petrov-Galerkin (SUPG) stabilization typically possess spurious oscillations at layers. Spurious oscillations at layers diminishing (SOLD) methods aim to suppress or at least to diminish these oscillations without smearing the layers extensively. In the recent review by \cite{JK06}, numerical studies at convection-diffusion problems with constant convection whose solutions have boundary layers led to a pre-selection of the best available SOLD methods with respect to the two goals stated above. The behaviour of these methods is studied in the present paper for a convection-diffusion problem with a nonconstant convection field whose solution possesses an interior layer. |

More numerical results Volker John, Petr Knobloch, A computational comparison of methods diminishing spurious oscillations in finite element solutions of convection-diffusion equations" , in the Proceedings of the International Conference on Programs and Algorithms of Numerical Mathematics 13, J. Chleboun, K. Segeth, T. Vejchodsky (eds.), 122 - 136, Academy of Science of the Czech Republic, 2006 This paper presents a review and a computational comparison of various stabilization techniques developed to diminish spurious oscillations in finite element solutions of scalar stationary convection-diffusion equations. All these methods are defined by enriching the popular SUPG discretization by additional stabilization terms. Although some of the methods can substantially enhance the quality of the discrete solutions in comparison to the SUPG method, any of the methods can fail in very simple situations and hence none of the methods can be regarded as reliable. We also present results obtained using the improved Mizukami-Hughes method which is often superior to techniques based on the SUPG method. |

Basic idea and first numerical results Volker John, Petr Knobloch, Simona B. Savescu A posteriori optimization of parameters in stabilized methods for convection-diffusion problems - Part I , Comput. Meth. Appl. Mech. Engrg. 200, 2916 - 2929, 2011 Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator, an error indicator, and a functional including the crosswind derivative of the computed solution. Benefits of the basic approach are demonstrated by means of numerical results. |

Coupling with Stokes equations in mantle convection problem Volker John, Julia Novo, Songul Kaya Finite Element Error Analysis of a Mantle Convection Model, Int. J. Num. Anal. Model. 15, 677 - 698, 2018 A mantle convection model consisting of the stationary Stokes equations and a time-dependent convection-diffusion equation for the temperature is studied. The Stokes problem is discretized with a conforming inf-sup stable pair of finite element spaces and the temperature equation is stabilized with the SUPG method. Finite element error estimates are derived which show the dependency of the error of the solution of one problem on the error of the solution of the other equation. The dependency of the error bounds on the coefficients of the problem is monitored. |

A posteriori error estimation and adaptive grid refinement Javier de Frutos, Bosco Garcia-Archilla, Volker John, Julia Novo An adaptive SUPG method for evolutionary convection-diffusion equations, Comput. Methods Appl. Mech. Engrg. 273, 219 - 237, 2014 An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from John, Novo (2013) will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in de Frutos, Garcia-Archilla, Novo (2011). |

Comparison of finite difference and finite element schemes Volker John, Julia Novo On (Essentially) Non-Oscillatory Discretizations of Evolutionary Convection-Diffusion Equations , J. Comput. Phys. 231, 1570 - 1586, 2012 Finite element and finite difference discretizations for evolutionary convection-dif\-fusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods. |

Error analysis of SUPG method with parameter independent of length of the time step Volker John, Julia Novo Error Analysis of the SUPG Finite Element Discretization of Evolutionary Convection-Diffusion-Reaction Equations , SIAM J. Numer. Analysis 49, 1149 - 1176, 2011 Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the SUPG finite element stabilization of time-dependent convection-diffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experience, this seems not to be the correct behavior. For this reason, the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step. It is shown that such estimates can be obtained in the case of time-independent convection and reaction. An error estimate for the time-continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady-state problem. Analogous estimates are obtained for the fully discrete case using the backward Euler method and the Crank--Nicolson scheme. Numerical studies support the analytical results. |

Numerical assessment of finite element methods for a 3D problem Volker John, Ellen Schmeyer On finite element methods for 3D time-dependent convection-diffusion-reaction equations with small diffusion , BAIL 2008 - Boundary and Interior Layers, Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008 Series: Lecture Notes in Computational Science and Engineering 69, Hegarty, A.; Kopteva, N.; O' Riordan, E.; Stynes, M. (Eds.), 173 - 181, 2009 The paper studies finite element methods for the simulation of time--dependent convection--diffusion--reaction equations with small diffusion: the SUPG method, a SOLD method and two types of FEM--FCT methods. The methods are assessed, in particular with respect to the size of the spurious oscillations in the computed solutions, at a 3D example with nonhomogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions. |

Numerical assessment of finite element methods for 2D problems Volker John, Ellen Schmeyer Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion , Comput. Meth. Appl. Mech. Engrg., 198, 475 - 494, 2008 Numerical studies of stabilized finite element methods for solving scalar time--dependent convection--diffusion--reaction equations with small diffusion are presented in this paper. These studies include the Streamline--Upwind Petrov--Galerkin (SUPG) method with different parameters, various Spurious Oscillations at Layers Diminishing (SOLD) methods, a Local Projection Stabilization (LPS) scheme based on enrichment and two Finite Element Method Flux--Corrected--Transport (FEM--FCT) methods. The focus of the evaluation of the numerical results is on the reduction of spurious oscillations. |

Implementation and numerical studies of a 2-level approachV. John, S. Kaya, W. Layton, "A Two-Level Variational Multiscale Method for Convection-Diffusion Equations", Comput. Meth. Appl. Mech. Engrg. 195, 4594 - 4603, 2006 This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convection-dominated problems. The VMS studied uses a fine mesh $C^0$ finite element space $X^h$ to approximate the concentration and a coarse mesh discontinuous vector finite element space $L^H$ for the large scales of the flux in the two scale discretization. Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) $L^2$-orthogonal basis for $L^H$ is used. A fully implicit and a semi-implicit treatment of the terms which link effects across scales are tested and compared. The semi-implicit VMS was much more efficient. The observed global accuracy of the most straightforward VMS implementation was much better than the artificial diffusion stabilization and comparable to a streamline-diffusion finite element method in our tests. |