Residual-free Bubble Research Articles

Proper intrinsic scales for a-posteriori multiscale error estimation

Recently the multiscale a-posteriori error estimator has been introduced, showing excellent robustness for fluid mechanics problems. In this paper, a theoretical analysis for element edge exact solutions is conducted, in which case, the error constant is the norm of a Green’s function or a residual-free bubble. This finds application when the solution is computed with a stabilized method. One of the features of the technique is that it gives the proper scales for a-posteriori error estimation in any norm of interest, such as the L2, H1, energy and L∞ norms. For fluid transport problems it is shown that the constant for predicting the error in the H1 seminorm is unbounded as the element Peclet number tends to infinity, making Lp norms more suitable for this type of problems. Furthermore, it is shown that the flow intrinsic time scale parameter represents the L1 norm of the error as a function of the L∞ norm of the residual. When these scales are employed for one-dimensional nodally-exact solutions, piecewise linear finite element spaces and piecewise constant residuals, the multiscale error estimator is shown to be exact.

Finite element method solution of electrically driven magnetohydrodynamic flow

The magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field.

Enhanced RFB method

The residual-free bubble method (RFB) is a parameter-free stable finite element method that has been successfully applied to a wide range of boundary-value problems exhibiting multiple-scale behaviour. If some local features of the solution are known a priori, the approximation properties of the RFB finite element space can be improved by enriching it on selected edges of the partition by edge-bubbles that are supported on pairs of neighbouring elements. Motivated by this idea, we define and analyse an enhanced residual-free bubble method for the solution of convection-dominated convection-diffusion problems in 2-D. Our a priori analysis highlights the limitations of the RFB method and the improved global approximation properties of the new method. The theoretical results are supported by detailed numerical experiments.

Enhanced residual‐free bubble method for convection–diffusion problems

AbstractWe analyse the performance of the enhanced residual‐free bubble (RFBe) method for the solution of elliptic convection‐dominated convection–diffusion problems in 2‐D, and compare the present method with the standard residual‐free bubble (RFB) method. The advantages of the RFBe method are two‐fold: it has better stability properties and it can be used to resolve boundary layers with high accuracy on globally coarse meshes. Copyright © 2005 John Wiley & Sons, Ltd.

Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions

In this paper we propose a novel way, via finite elements to treat problems that can be singular perturbed, a reaction–diffusion equation in our case. We enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem, as in the residual free bubble (RFB) setting, but do not require these functions to vanish on each element edge, a departure from the RFB paradigm. Such multiscale functions have an analytic expression, for triangles and rectangles. Bubbles are the choice for the test functions allowing static condensation, thus our method is of Petrov–Galerkin type. We perform several numerical validations which confirm the good performance of the method.

The finite element method for MHD flow at high Hartmann numbers

A stabilized finite element method using the residual-free bubble functions (RFB) is proposed for solving the governing equations of steady magnetohydrodynamic duct flow. A distinguished feature of the RFB method is the resolving capability of high gradients near the layer regions without refining mesh. We show that the RFB method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the Hartmann number. Numerical results confirming theoretical findings are presented for several configurations of interest. The approximate solution obtained by the RFB method is also compared with the analytical solution of Shercliff’s problem.

A discontinuous residual-free bubble method for advection-diffusion problems

A discontinuous finite-element method is presented for solving the linear advection-diffusion equation, based on the Residual-Free Bubble (RFB) finite-element formulation. After the macro-scales (usual piecewise-polynomials elements) are separated from the micro-scales (the bubble part), they are computed by a standard Galerkin formulation, while the bubble part is approximated by a discontinuous Galerkin method. The advantage of this approach, as compared to other implementations of the Residual-Free Bubble formulation, is that the macro-scales are computed accurately, at least for the model problem presently considered. Numerical tests are performed to confirm the validity of the proposed approach.

THE RESIDUAL-FREE BUBBLE NUMERICAL METHOD WITH QUADRATIC ELEMENTS

In this paper we analyze the Residual-Free Bubble (RFB) method applied to the linear diffusion–advection–reaction problem. We propose a new a priori error analysis for the method and for its practical implementation in a quite general context, which allows, e.g. linear or quadratic elements on the resolvable scales. We also perform some numerical tests, showing in both cases the advantages of the method.

Analysis of a Two-Scale, Locally Conservative Subgrid Upscaling for Elliptic Problems

We present a two-scale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We consider the elliptic problem in mixed variational form over $W \times {\bf V} \subset L^2 \times H({\rm div})$. We base our scale expansion on local mass conservation over the coarse grid. It is used to define a direct sum decomposition of $W \times {\bf V}$ into coarse and subspaces $W_c \times {\bf V}_c$ and $\delta W \times \delta{\bf V}$ such that (1) $\nabla \cdot {\bf V}_c=W_c$ and $\nabla \cdot \delta{\bf V} = \delta W$, and (2) the space $\delta{\bf V}$ is locally supported over the coarse mesh. We then explicitly decompose the variational problem into coarse and subgrid scale problems. The subgrid problem gives a well-defined operator taking $W_c \times {\bf V}_c$ to $\delta W \times \delta{\bf V}$, which is localized in space, and it is used to upscale, that is, to remove the subgrid from the coarse-scale problem. Using standard mixed finite element spaces, two-scale mixed spaces are defined. A mixed approximation is defined, which can be viewed as a type of variational multiscale method or a residual-free bubble technique. A numerical Green's function approach is used to make the approximation to the subgrid operator efficient to compute. A mixed method $\pi$-operator is defined for the two-scale approximation spaces and used to show optimal order error estimates.

Stability of the residual free bubble method for bilinear finite elements on rectangular grids

We consider the nature of the stabilizing term arising in the residual free bubble approach for piecewise bilinear functions on rectangular grids. We show, that on the subspace of piecewise linear functions the stabilizing term is identical to that in the streamline diffusion approach. However, on the space of piecewise bilinear functions there is a case in which the stabilizing term is weaker compared to the term used in the streamline diffusion method. In the particular case when the direction of the convection is directed parallel to a diagonal of the quadrilateral, control is lost over the mixed derivatives in the convection-dominated limit.

On a two‐level finite element method for the incompressible Navier–Stokes equations

AbstractWe consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.

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.

Convergence Analysis of the Residual Free Bubble Method for Bilinear Elements

We present an a priori error analysis for the residual free bubble (RFB) method applied to diffusion-convection equations with a dominating and antisymmetric convection operator. Special attention is paid to the case of piecewise bilinears in ${\mathbb R}^2$ and the theoretical comparison with the streamline diffusion finite element (SDFE) (equivalently, streamline upwind Petrov--Galerkin (SUPG) method).

Residual-free bubbles for advection-diffusion problems: the general error analysis

We develop the general a priori error analysis of residual-free bubble finite element approximations to linear elliptic convection-dominated diffusion problems subject to homogeneous Dirichlet boundary condition. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree greater than or equal to 1.

Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions

For high wave numbers, the Helmholtz equation suffers the so-called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p-method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual-free bubble [3,4] is effective in one dimension but not in higher dimensions. The least-square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss-cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.

A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems

We develop an a priori error analysis of a finite element approximation to the elliptic advection-diffusion equation $-\eps \Delta u + \conv\cdot \nabla u = f$ subject to a homogeneous Dirichlet boundary condition, based on the use of residual-free bubble functions. An optimal order error bound is derived in the so-called stability-norm \[ \biggl(\eps \|\nabla v\|^2_{L_2(\Omega)} + \sum_{T} h_T\|\conv\cdot \nabla v\|^2_{L_2(T)}\biggr)^{1/2},\] where hT denotes the diameter of element T in the subdivision of the computational domain.

Unusual stabilized finite element methods and residual free bubbles

An overview of the unusual stabilized finite element method and of the standard Galerkin method enriched with residual free bubble functions is presented. For the first method a concrete model problem illustrates its application in advective–diffusive–reactive equations and for the second method it is shown how static condensation of residual free bubbles gives rise to mass lumping and selective reduced integration, which are viewed as numerical tricks and can now be derived by the standard Galerkin method without tricks. © 1998 John Wiley & Sons, Ltd.