Residual-free Bubbles Research Articles

Anisotropic meshes and streamline‐diffusion stabilization for convection–diffusion problems

AbstractWe study a convection‐diffusion problem with dominant convection. Anisotropic streamline aligned meshes with high aspect ratios are recommended to resolve characteristic interior and boundary layers and to achieve high accuracy. We address the question of how the stabilization parameter in the streamline‐diffusion FEM (SDFEM) and the Galerkin least‐squares FEM (GLSFEM) should be chosen inside the layers. Using a residual free bubbles approach, we show that within the layers the stabilization must be drastically reduced. Copyright © 2005 John Wiley & Sons, Ltd.

Multiscale and Residual-Free Bubble Functions for Reaction-Advection-Diffusion Problems

We propose a finite element method based on enriching the Galerkin approximation spaces with a combination of multiscale functions and residual-free bubbles (RFB). This approach is presented as a Petrov-Galerkin method, and applied to the singularly perturbed reaction-advection-diffusion model. Numerical tests confirm that switching RFB by suitable multiscale functions in the elements connected to the outflow boundaries of the domain improves the accuracy of the solution in this region. ∗Address all correspondence to lfranca@math.cudenver.edu 0731-8898/05/$35.00 c © 2005 by Begell House, Inc. 1 2 Franca, Ramalho, and Valentin

Large eddy simulation of turbulent incompressible flows by a three-level finite element method

The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in the form of a three-level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual-free bubbles are used for the numerical approximation of the small-scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small-scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small-scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modelling procedure. The method is tested for two different turbulent flow situations.

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 three-level finite element method for the instationary incompressible Navier–Stokes equations

We present a two- and a three-level finite element method for the numerical simulation of incompressible flow governed by the Navier–Stokes equations. Within the theoretical framework of the variational multiscale method and exploiting the concept of residual-free bubbles we propose a separation of three different scale groups, namely large (resolved) scales, small (resolved) scales, and unresolved scales. The resolution of the large and small scales takes place on levels 1 and 2 with the aid of diverse approaches. The dynamic calculation of a subgrid viscosity representing the effect of the unresolved scales constitutes level 3 of our algorithm. The proposed algorithm is tested for various laminar flow situations and compared to the results obtained via an unusual stabilized finite element method. It is supposed to be the basic scheme also for turbulent flow in a subsequent study.

LINK-CUTTING BUBBLES FOR THE STABILIZATION OF CONVECTION-DIFFUSION-REACTION PROBLEMS

It is known that the addition and elimination of suitable bubble functions can result in a stabilized scheme of the SUPG-type. Residual-Free Bubbles (RFB), in particular, can assure a quasi-optimal stabilized scheme, but they are difficult to compute in one dimension and nearly impossible to compute in two and three dimensions, unless in special limit cases. Strongly convection-dominated problems (without reaction terms) are one of these cases, where it is possible to find reasonably simple computable bubbles that provide a stabilizing effect as good as that of true RFB. Here, although in a one-dimensional framework, we analyze the case in which a non-negligible reaction term is present, and provide a simple recipe for spotting a suitable bubble space (adding two bubbles to each element) that provides a very good stabilizing effect. The method adapts very well to all regimes with continuous transitions from one regime to another. It is clear that the one-dimensional case, in itself, has no real interest. We believe, however, that the discussion can cast some light on the interaction between convection and reaction that could be useful in future works dealing with multidimensional, more realistic problems.

Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method

In this work we study the residual-free bubbles (RFB) finite element method for solving second order elliptic equations with rapidly varying coefficients. The RFB technique is closely related to both the multiscale finite element method (MsFEM) introduced by Hou, Wu, and Cai [Math. Comp., 68 (1999), pp. 913--943] and the upscaling procedures which are very common in the engineering literature for solving this kind of partial differential equation. We also introduce a variation of the RFB method, based on macrobubbles and referred to as the residual-free macrobubbles (RFMB) method, which gives more accurate numerical solutions. In the case of periodic coefficients we are able to prove a priori error estimates for the methods. Eventually, we test the numerical methods on model problems.

A note on practical bubbles for advection-diffusion problems

In this note, we show that, in the one-dimensional case, as an approximation to residual-free bubbles (RFB), certain practical bubbles can be applied to obtain a scheme which is uniformly convergent with respect to small viscosity in the energy norm for advection-diffusion problems.

Augmented spaces, two‐level methods, and stabilizing subgrids

AbstractStarting from the already known relationship between stabilized methods, augmented spaces and residual‐free bubbles (RFB), the paper introduces a possible way of mimicking the effect of RFB just by constructing a suitable subgrid and then solving the standard Galerkin equations on the modified grid. Concentrating on the model problem of linear convection‐dominated equations, we give sufficient conditions on the subgrid that ensure stability, and error bounds of the same type of standard stabilizing procedures. Copyright © 2002 John Wiley & Sons, Ltd.

Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation

AbstractWe introduce a new submesh strategy for the two‐level finite element method. The numerical results show that the new submesh is able to better capture the boundary layer which is caused by the choice of bubble functions. The effect of an improved approximation of the residual free bubbles is studied for the advective–diffusive equation. Copyright © 2002 John Wiley & Sons, Ltd.

Recent results in the treatment of subgrid scales

Recent results in the treatment of subgrid scales

Nearly H1-optimal finite element methods

We examine the problem of finding the H 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem. Solving the projection problem typically requires knowing the exact solution. We circumvent this issue and obtain a Petrov–Galerkin formulation which achieves H 1 optimality. Requiring weighting functions to be defined locally on the element level permits only approximate H 1 optimality in multi-dimensional configurations. We investigate the relation between our formulation and other stabilized FEM formulations. We show, in particular, that our formulation leads to a derivation of the SUPG method. In special cases, the present formulation reduces to that of residual-free bubbles. Finally, we present guidelines for obtaining the Petrov weight functions, and include a numerical example for the Helmholtz equation.

Approximation of the arch problem by residual-free bubbles

We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed nite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.

Recovering SUPG using Petrov–Galerkin formulations enriched with adjoint residual-free bubbles

We consider conforming Petrov–Galerkin formulations for the advective and advective–diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different “bubble” enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first-order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data are assumed to be piecewise constant. Generalization to include diffusion is also presented.

Global and Local Error Analysis for the Residual-Free Bubbles Method Applied to Advection-Dominated Problems

We prove the stability and a priori global and local error analysis for the residual-free bubbles (RFB) finite element method applied to advection-dominated advection-diffusion problems.

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems

Residual-free bubbles have been recently introduced in order to compute optimal values for the stabilization methods á la Hughes-Franca. However, unless in very special situations (one-dimensional problems, limit cases, etc.) they require the actual solution of PDE problems (the bubble problems) in each element. Thus, they are very difficult to be used in practice. In this paper we present, for the special case of convection-dominated elliptic problems, a cheap way to compute approximately the solution of the bubble problem in each element. This provides, as a consequence, a cheap way to compute good approximations for the optimal values of the stabilization parameters.

Stabilized spectral methods for the Navier-Stokes equations: residual-free bubbles and preconditioning

We study a stabilized spectral method for the incompressible Navier-Stokes equations; stabilization is achieved by using the residual-free bubbles approach. Numerical results for the regularized driven cavity and the backward-facing step are presented.

Further considerations on residual-free bubbles for advective-diffusive equations

We further consider the Galerkin method for advective-diffusive equations in two dimensions. The finite dimensional space employed is of piecewise polynomials enriched with residual-free bubbles (RFB). We show that, in general, this method does not coincide with the SUPG method, unless the piecewise polynomials are spanned by linear functions. Furthermore, a simple stability analysis argument displays the effect of the RFB on the reduced space of piecewise polynomials, which, in some situations, is not equivalent to streamline diffusion for bilinears.

The variational multiscale method—a paradigm for computational mechanics

We present a general treatment of the variational multiscale method in the context of an abstract Dirichlet problem. We show how the exact theory represents a paradigm for subgrid-scale models and a posteriori error estimation. We examine hierarchical p-methods and bubbles in order to understand and, ultimately, approximate the ‘fine-scale Green's function’ which appears in the theory. We review relationships between residual-free bubbles, element Green's functions and stabilized methods. These suggest the applicability of the methodology to physically interesting problems in fluid mechanics, acoustics and electromagnetics.

On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method

We consider the Galerkin finite element method for partial differential equations in two dimensions, where the finite-dimensional space used consists of piecewise (isoparametric) polynomials enriched with bubble functions. Writing L for the differential operator, we show that for elliptic convection-diffusion problems, the component of the bubble enrichment that stabilizes the method is equivalent to a Petrov-Galerkin method with an L-spline (exponentially fitted) trial space and piecewise polynomial test space; the remaining component of the bubble influences the accuracy of the method. A stability inequality recently obtained by Brezzi, Franca and Russo for a limiting case of bubbles applied to convection-diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the SDFEM/SUPG method, thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method. When the trial functions are piecewise linear, we show that residual-free bubbles are as stable as SDFEM/SUPG, and we extend this stability inequality to include positive mesh-Peclet numbers in the convection-dominated regime. Approximate computations of the residual-free bubbles are performed using a two-level finite element method.