Residual-free Bubble Research Articles

A residual-free bubble formulation for nonlinear elliptic problems with oscillatory coefficients

ABSTRACTWe present an investigation of the Residual-Free Bubble (RFB) finite element method for a class of multiscale nonlinear elliptic partial differential equations. After proposing a nonlinear version for the method, we address fundamental questions as existence and uniqueness of solutions. We also obtain a best approximation result and investigate possible linearizations that generate different versions for the method. All estimates are independent of eventual oscillations in the coefficients, and as far as we are aware, this is the first time that an analysis for the nonlinear RFB method is considered.

A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems

The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convection-dominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations.

Pointwise Error Estimation for the One-Dimensional Transport Equation Based on the Variational Multiscale Method

This paper presents an exact pointwise error estimator applied to the transport equation. The error estimate, which is represented by a combination of residual-free bubble and Green’s functions, is computed by postprocessing the finite element solution. The fine or unresolved scales are analyzed applying the variational multiscale method. The technology can be applied to linear and high-order elements. Numerical examples are presented in which the satisfactory results corroborate the theoretical foundations.

On the relation of the Embedded Discontinuous Galerkin method to the stabilized residual-based finite element methods

The paper traces the relation between the Embedded Discontinuous Galerkin method and the Streamline Upwind Petrov–Galerkin method and its generalizations, such as the Discontinuous Residual-free Bubble method, for the Euler and Navier–Stokes equations. Popular choices of the stabilization terms for both discretizations are related on the analytical level. The conservation property of the Embedded Discontinuous Galerkin discretization on dual volumes of the computational grid is established.

Stabilized finite element methods to simulate the conductances of ion channels

We have previously developed a finite element simulator, ichannel, to simulate ion transport through three-dimensional ion channel systems via solving the Poisson–Nernst–Planck equations (PNP) and Size-modified Poisson–Nernst–Planck equations (SMPNP), and succeeded in simulating some ion channel systems. However, the iterative solution between the coupled Poisson equation and the Nernst–Planck equations has difficulty converging for some large systems. One reason we found is that the NP equations are advection-dominated diffusion equations, which causes troubles in the usual FE solution. The stabilized schemes have been applied to compute fluids flow in various research fields. However, they have not been studied in the simulation of ion transport through three-dimensional models based on experimentally determined ion channel structures. In this paper, two stabilized techniques, the SUPG and the Pseudo Residual-Free Bubble function (PRFB) are introduced to enhance the numerical robustness and convergence performance of the finite element algorithm in ichannel. The conductances of the voltage dependent anion channel (VDAC) and the anthrax toxin protective antigen pore (PA) are simulated to validate the stabilization techniques. Those two stabilized schemes give reasonable results for the two proteins, with decent agreement with both experimental data and Brownian dynamics (BD) simulations. For a variety of numerical tests, it is found that the simulator effectively avoids previous numerical instability after introducing the stabilization methods. Comparison based on our test data set between the two stabilized schemes indicates both SUPG and PRFB have similar performance (the latter is slightly more accurate and stable), while SUPG is relatively more convenient to implement.

PSEUDO-MULTI-SCALE FUNCTIONS FOR THE STABILIZATION OF CONVECTION-DIFFUSION EQUATIONS ON RECTANGULAR GRIDS

We propose a finite element method of Petrov-Galerkin type for a singularly perturbed convection diffusion problem on a discretization consisting of rectangular elements. The method is based on enriching the finite-element space with a combination of multiscale and residual-free bubble functions. These functions require the solution of the original differential problem, which makes the method quite expensive, especially in two dimensions. Therefore, we instead employ their cheap, yet efficient approximations, using only a few nodes in each element. Several numerical tests confirm the good performance of the corresponding numerical method.

Enriched residual free bubbles for semiconductor device simulation

This article outlines a method for stabilising the current continuity equations which are used for semiconductor device simulation. Residual-free bubble functions (RfBF) are incorporated into a finite element (FE) implementation that are able to prevent oscillations which are seen when using the conventional Bubnov-Galerkin FE implementation. In addition, it is shown that the RfBF are able to provide stabilisation with very distorted meshes and curved interface boundaries. Comparison with the commonly used SUPG scheme is made throughout, showing that in the case of 2D problems the RfBF allow faster convergence of the coupled semiconductor device equations, especially in the case of distorted meshes.

A stabilized finite element method for convection–diffusion problems

AbstractA stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τK that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011

Residual-free bubble methods for numerical homogenization of elliptic problems

Residual-free bubble methods for numerical homogenization of elliptic problems

A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.

Nonconforming local projection stabilization for generalized Oseen equations

A new method of nonconforming local projection stabilization for the generalized Oseen equations is proposed by a nonconforming inf-sup stable element pair for approximating the velocity and the pressure. The method has several attractive features. It adds a local projection term only on the sub-scale (H ⩾ h). The stabilized term is simple compared with the residual-free bubble element method. The method can handle the influence of strong convection. The numerical results agree with the theoretical expectations very well.

A Novel Least-Squares Finite Element Method Enriched with Residual-Free Bubbles for Solving Convection-Dominated Problems

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient $\mathbf{p}:=-\kappa\nabla u$ as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables $u$ and $\mathbf{p}$ is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This enriched LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving stability parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed enriched LSFEM. The accuracy and computational cost of this enriched LSFEM are also compared with those of the primitive LSFEM. We find that for a small diffusivity $\kappa$, the enriched LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the enriched LSFEM provides a significant improvement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems.

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C 0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.

Two‐level finite element method with a stabilizing subgrid for the incompressible MHD equations

AbstractWe consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate solutions obtained show the well‐known characteristics of the MHD flow. Copyright © 2009 John Wiley & Sons, Ltd.

Residual-free bubbles for embedded Dirichlet problems

We examine a stabilized method employing residual-free bubbles for enforcing Dirichlet constraints on embedded finite element interfaces. In particular, we focus attention on problems where the underlying mesh is not explicitly “fitted” to the geometry of the interface. The residual-free bubble problem is derived for a simple case and extensions are discussed. We show that under certain conditions, stabilization only requires knowledge of the residual-free bubble on the interface. We then examine methods to approximate the residual-free bubble. A series of benchmark tests are used to demonstrate the accuracy and robustness of the method. Comparisons are made to Nitsche’s method employing an eigenvalue estimate for the global stability parameter. Particular emphasis is placed on the accuracy of flux evaluations on the interface. Finally, we employ the method to simulate an evolving interface problem motivated by resin transfer molding.

Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

AbstractWe consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.

An implicit finite element solution of thermal flows at low Mach number

Thermal flows at low Mach numbers are a basic problem in combustion, environmental pollution prediction and atmospheric physics areas. Most of the existing schemes for solving this problem treat convection explicitly, which confines time step width due to the CFL condition. In this paper, based on the pseudo residual-free bubble approach [F. Brezzi, L.P. Franca, T.J.R. Hughes, A. Russo, b = ∫ g , Methods Appl. Mech. Eng. 145 (1997) 329–339; T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilised methods, Method. Appl. Mech. Eng. 127 (1995) 387–401], we introduce an implicit finite element scheme for the thermal flow problem. We firstly give a low Mach number asymptotics of compressible Navier–Stokes equations for the thermal flows and then derive the numerical scheme for them in detail. Three representative case studies are used to investigate and to test the numerical performances of the proposed scheme.

An inf-sup Stable and Residual-Free Bubble Element for the Oseen Equations

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 is the case for the streamline upwind Petrov-Galerkin 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.

The Residual-Free-Bubble Finite Element Method on Anisotropic Partitions

The subject of this work is the analysis and implementation of stabilized finite element methods on anisotropic meshes. We develop the anisotropic a priori error analysis of the residual-free-bubble (RFB) method applied to elliptic convection-dominated convection-diffusion problems in two dimensions, with finite element spaces of type $Q_k$, $k \geq 1$. In the case of $P_1$ finite elements, relying on the equivalence of the RFB method to classical stabilized finite element methods, we propose a new rule, justified through the analysis of the RFB method, for selecting the stabilization parameter in classical stabilized methods on two-dimensional anisotropic triangulations.

ON THE STABILITY OF THE RESIDUAL-FREE BUBBLES FOR THE NAVIER-STOKES EQUATIONS

This paper considers the Galerkin finite element method for the incompressible Navier-Stokes equations in two dimensions, where the finite-dimensional spaces employed consist of piecewise polynomials enriched with residual-free bubble (RFB) functions. The stability features of the residual-free bubble functions for the linearized Navier-Stokes equations are analyzed in this work. It is shown that the enrichment of the velocity space by bubble functions stabilizes the numerical method for any value of the viscosity parameter for triangular elements and for values of the viscosity parameter in the vanishing limit case for quadrilateral elements.