Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.     

Analysis of a stabilized finite element approximation of the transient convection-diffusion-reaction equation using orthogonal subscales

In this paper we analyze a stabilized finite element method to solve the transient convection-diffusion-reaction equation based on the decomposition of the unknowns into resolvable and subgrid scales. We start from the time-discrete form of the problem and obtain an evolution equation for both components of the decomposition. A closed-form expression is proposed for the subscales which, when inserted into the equation for the resolvable scale, leads to the stabilized formulation that we analyze. Optimal error estimates in space are provided for the first order, backward Euler time integration. Received: 31 January 2001 / Accepted: 30 September 2001     

A Parallel Subgrid Stabilized Finite Element Method Based on Two-Grid Discretization for Simulation of 2D/3D Steady Incompressible Flows

Based on domain decomposition and two-grid discretization, a parallel subgrid stabilized finite element method for simulation of 2D/3D steady convection dominated incompressible flows is proposed and analyzed. In this method, a subgrid stabilized nonlinear Navier–Stokes problem is first solved on a coarse grid where the stabilization term is based on an elliptic projection defined on the same coarse grid, and then corrections are calculated in overlapped fine grid subdomains by solving a linearized problem. By the technical tool of local a priori estimate for finite element solution, error bounds of the approximate solution are estimated. Algorithmic parameter scalings of the method are derived. Numerical results are also given to demonstrate the effectiveness of the method.     

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.     

A stabilizing augmented grid for rectangular discretizations of the convection–diffusion–reaction problems

We propose a numerical method for approximate solution of the convection–diffusion–reaction problems in the case of small diffusion. The method is based on the standard Galerkin finite element method on an extended space defined on the original grid plus a subgrid, where the original grid consists of rectangular elements. On each rectangular elements, we construct a subgrid with few points whose locations are critical for the stabilization of the problem, therefore they are chosen specially depending on some specific conditions that depend on the problem data. The resulting subgrid is combined with the initial coarse mesh, eventually, to solve the problem in the framework of Galerkin method on the augmented grid. The results of the numerical experiments confirm that the proposed method shows similar stability features with the well-known stabilized methods for the critical range of problem parameters.     

The dissipative structure of variational multiscale methods for incompressible flows

In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.     

Analysis of a variational multiscale method for Large–Eddy simulation and its application to homogeneous isotropic turbulence

In this paper a variational multiscale method based on local projection and grad–div stabilization for Large–Eddy simulation for the incompressible Navier–Stokes problem is considered. An a priori error estimate is given for a case with rather general nonlinear (piecewise constant) coefficients of the subgrid models for the unresolved scales of velocity and pressure. Then the design of the subgrid scale models is specified for the case of homogeneous isotropic turbulence and studied for the standard benchmark problem of decaying homogeneous isotropic turbulence.     

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.     

Spatial approximation of the radiation transport equation using a subgrid-scale finite element method

In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.     

On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes

In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated. In order to incorporate the distortion of the mesh into this approximation, we transform the equation for the subgrid scale within each element to the shape-regular reference domain. The expression for the subgrid scale arises from an approximate Fourier analysis and the identification of the wave number direction where instabilities are most likely to occur. The final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.     

The variational multiscale method for laminar and turbulent flow

Summary  The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.     

Solving inverse problems involving the Navier–Stokes equations discretized by a Lagrange–Galerkin method

In this article, we are investigating the numerical approximation of an inverse problem involving the evolution of a Newtonian viscous incompressible fluid described by the Navier–Stokes equations in 2D. This system is discretized using a low order finite element in space coupled with a Lagrange–Galerkin scheme for the nonlinear advection operator. We introduce a full discrete linearized scheme that is used to compute the gradient of a given cost function by ensuring its consistency. Using gradient based optimization algorithms, we are able to deal with two fluid flow inverse problems, the drag reduction around a moving cylinder and the identification of a far-field velocity using the knowledge of the fluid load on a rectangular bluff body, for both fixed and prescribed moving configurations.     

On a numerical subgrid upscaling algorithm for Stokes–Brinkman equations

This paper discusses a numerical subgrid resolution approach for solving the Stokes–Brinkman system of equations, which is describing coupled flow in plain and in highly porous media. Various scientific and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil filtration. In many complicated filters, the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid. The subgrid approach presented in this paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and careful procedure of modifying and updating the coefficients of the Stokes–Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so-called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm’s efficiency, both in terms of computational time and memory usage, are presented. Comparison of the full fine grid solution (when possible) of the Stokes–Brinkman system with the subgrid solution of the upscaled Stokes–Brinkman system (including effective permeabilities for the quasi-porous cells), are presented in order to evaluate the accuracy and the efficiency. Advantages and limitations of the considered subgrid approach are discussed.     

Semiconductor device simulation using a viscous-hydrodynamic model

In this article we deal with a hydrodynamic model of Navier–Stokes (NS) type for semiconductors including a physical viscosity in the momentum and energy equations. A stabilized finite difference scheme with upwinding based on the characteristic variables is used for the discretization of the NS equations, while a mixed finite element scheme is employed for the approximation of the Poisson equation. A consistency result for the method is established showing that the scheme is first-order accurate in both space and time. We also perform a stability analysis of the numerical method applied to a linearized incompletely parabolic system in two space dimensions with vanishing viscosity. A thorough numerical parametric study as a function of the heat conductivity and of the momentum viscosity is carried out in order to investigate their effect on the development of shocks in both one and two space dimensional devices.     

Convergence Analysis of Primal–Dual Based Methods for Total Variation Minimization with Finite Element Approximation

We consider a minimization model with total variational regularization, which can be reformulated as a saddle-point problem and then be efficiently solved by the primal–dual method. We utilize the consistent finite element method to discretize the saddle-point reformulation; thus possible jumps of the solution can be captured over some adaptive meshes and a generic domain can be easily treated. Our emphasis is analyzing the convergence of a more general primal–dual scheme with a combination factor for the discretized model. We establish the global convergence and derive the worst-case convergence rate measured by the iteration complexity for this general primal–dual scheme. This analysis is new in the finite element context for the minimization model with total variational regularization under discussion. Furthermore, we propose a prediction–correction scheme based on the general primal–dual scheme, which can significantly relax the step size for the discretization in the time direction. Its global convergence and the worst-case convergence rate are also established. Some preliminary numerical results are reported to verify the rationale of considering the general primal–dual scheme and the primal–dual-based prediction–correction scheme.     

On the choice of a stabilizing subgrid for convection–diffusion problems

SUPG and residual-free bubbles are closely related methods that have been used with success to stabilize a certain number of problems, including advection-dominated flows. In recent times, a slightly different idea has been proposed: to choose a suitable subgrid in each element, and then solving Standard Galerkin on the Augmented Grid. For this, however, the correct location of the subgrid node(s) plays a crucial role. Here, for the model problem of linear advection–diffusion equations, we propose a simple criterion to choose a single internal node such that the corresponding plain-Galerkin scheme on the augmented grid provides the same a priori error estimates that are typically obtained with SUPG or RFB methods.     

A mixed–primal finite element method for the Boussinesq problem with temperature-dependent viscosity

In this paper we focus on the analysis of a mixed finite element method for a class of natural convection problems in two dimensions. More precisely, we consider a system based on the coupling of the steady-state equations of momentum (Navier–Stokes) and thermal energy by means of the Boussinesq approximation (coined the Boussinesq problem), where we also take into account a temperature dependence of the viscosity of the fluid. The construction of this finite element method begins with the introduction of the pseudostress and vorticity tensors, and a mixed formulation for the momentum equations, which is augmented with Galerkin-type terms, in order to deal with the non-linearity of these equations and the convective term in the energy equation, where a primal formulation is considered. The prescribed temperature on the boundary becomes an essential condition, which is weakly imposed, leading us to the definition of the normal heat flux through the boundary as a Lagrange multiplier. We show that this highly coupled problem can be uncoupled and analysed as a fixed-point problem, where Banach and Brouwer theorems will help us to provide sufficient conditions to ensure well-posedness of the problems arising from the continuous and discrete formulations, along with several applications of continuous injections guaranteed by the Rellich–Kondrachov theorem. Finally, we show some numerical results to illustrate the performance of this finite element method, as well as to prove the associated rates of convergence.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.