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This study investigates multiphysics block preconditioners that are critical in devising scalable Newton--Krylov iterative solvers for longer time-scale fully implicit fluid plasma models. The specific model of interest is the visco-resistive, low Mach number, compressible magnetohydro-dynamics (MHD) model. This model describes the dynamics of conducting fluids in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics. The specific application of interest that motivates this study is the macroscopic simulation of longer time-scale stability and disruptions of magnetic confinement fusion devices, specifically the ITER Tokamak. The computational solution of the governing balance equations for mass, momentum, heat transfer, and magnetic induction for resistive MHD systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena as well as the significant range of time and length scales that the interactions of these physical mechanisms produce. To handle the range of time and spatial scales of interest, a fully implicit unstructured variational multiscale finite element formulation is employed. For the scalable solution of the Newton linearized systems, fully coupled block preconditioners are designed to leverage algebraic multigrid subsolves. Results are presented for the strong and weak scaling of the method as well as the robustness of these techniques for a large range of Lundquist numbers. [ABSTRACT FROM AUTHOR]

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/ h p stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/ h p framework. We introduce revised stabilization parameters, τ s u p g , with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ : the first is described with entropy variables, and the other is the Y Z β parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L 1 and L 2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod's shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries. [ABSTRACT FROM AUTHOR]

This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging. [ABSTRACT FROM AUTHOR]

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

• Novel robust and accurate finite element scheme for homogenous mixture cavitation. • Stable linearizations of finite mass transfer for cavitation transport equations. • Fully implicit partitioned iterative coupling for turbulent FSI and cavitation. • Assessment of convergence and accuracy through bubble collapse problem. • Demonstration to 3D pitching hydrofoil undergoing cloud cavitation. In the current work, we present a variational mechanics framework for the coupled numerical prediction of cavitating turbulent flow and structural motion via a stabilized finite element formulation. To model the finite mass transfer rate in cavitation phenomena, we employ the homogenous mixture-based approach via phenomenological scalar transport differential equations given by the linear and nonlinear mass transfer functions. Stable linearizations of the finite mass transfer terms for the mass continuity equation and the reaction term of the scalar transport equations are derived for the robust and accurate implementation. The linearized matrices for the cavitation equation are imparted a positivity-preserving property to address numerical oscillations arising from high-density gradients typical of two-phase cavitating flows. The proposed formulation is strongly coupled in a partitioned manner with an incompressible 3D Navier-Stokes finite element solver, and the unsteady problem is advanced in time using a fully-implicit generalized- α time integration scheme. We first verify the implementation on the benchmark case of the Rayleigh bubble collapse. We demonstrate the accuracy and convergence of the cavitation solver by comparing the numerical solutions with the analytical solutions of the Rayleigh-Plesset equation for bubble dynamics. We find our solver to be robust for large time steps and the absence of spurious oscillations/spikes in the pressure field. The cavitating flow solver is coupled with a hybrid URANS-LES turbulence model. We validate the coupled solver for a very high Reynolds number turbulent cavitating flow over a NACA0012 hydrofoil section. Finally, the proposed method is applied in an Arbitrary Lagrangian-Eulerian framework to study turbulent cavitating flow over a pitching hydrofoil section and the characteristic features of cavitating flows such as re-entrant jet and periodic cavity shedding are discussed. [ABSTRACT FROM AUTHOR]

The classical dual mixed finite element method for flow simulations is based on H (div , Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L 2 (Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of H (div , Ω) and H 1 (Ω). The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements. [ABSTRACT FROM AUTHOR]

We propose a novel framework for investigating fluid–membrane interactions involving fracture, and apply it to simulate initial stages during the bursting of a balloon. Fluid flow is computed using a stabilized space–time finite element method over a body-fitted mesh. The membrane is modelled as a hyperelastic material and the equations are solved using the standard Galerkin method. Structural failure is incorporated using the element deletion technique based on a simple strain based criterion. Change in topology of the fluid domain due to element deletion is handled without resorting to remeshing. The coincident fluid nodes on either side of the deleted membrane elements on the surface of the balloon are connected, thereby allowing the fluid to gush through the opening. First, fluid–membrane interaction, without failure, in a square sail with fixed edges is studied. The sail exhibits vibration owing to vortices shed from its edges. We then study the inflation and deflation of a spherical balloon equipped with an inlet pipe. A hydrostatic pressure gradient is applied to drive the flow into the balloon. Effect of different values of membrane density, stiffness and the hydrostatic pressure gradient on inflation rate and balloon shape is studied. The bursting of the inflated balloon is initiated by deleting one structural element on its surface. The proposed algorithm is then applied to simulate the subsequent dynamics of the fluid–membrane system. [ABSTRACT FROM AUTHOR]

Abstract Next generation, conservative finite element hydrodynamic and transport models are vital for accurate and efficient ocean, estuary and river simulation. Numerical models such as these have been developed for decades by the U.S. Army Corps of Engineers at the Engineering Research and Development Center (ERDC). This paper focuses on recently developed implicit, multi-dimensional finite element 3D shallow water and transport models included in the Adaptive Hydraulics (AdH) numerical suite. These AdH 3D models benefit from their adaptive meshing capabilities to resolve sharp solution gradients, such as those often encountered with baroclinic wedges traveling up an estuary channel. This paper presents the AdH 3D mathematical formulation and solution procedure used to solve the weak finite element equations for these models, along with results for several common verification cases and an AdH Galveston Bay validation study. A novel Streamline Upwind Petrov–Galerkin (SUPG) method for 3D shallow water models is described which reduces to the AdH 2D shallow water SUPG formulation under certain conditions. Careful attention is placed on ensuring discrete consistency in the equation set. [ABSTRACT FROM AUTHOR]

Adjoint-based error estimation provides the ability to approximate the discretization error for a functional quantity of interest, such as point-wise displacements or stresses. Mesh adaptation provides the ability to control the discretization error to obtain more accurate solutions while still remaining computationally feasible. In this paper, we develop an approach for adjoint-based error estimation and mesh adaptation for nonlinear finite deformation elasticity using a mixed stabilized finite element method. We apply our developed technique to a well known test case, the Cook’s membrane problem, to validate and demonstrate its effectiveness. We then demonstrate the utility of adjoint-based error estimation and mesh adaptation for a three-dimensional example motivated by the study of a cell embedded in a matrix. [ABSTRACT FROM AUTHOR]

In this paper, an adjoint-based error estimation and mesh adaptation framework is developed for the compressible inviscid flows. The algorithm employs the Finite Calculus (FIC) scheme for the numerical solution of the flow and discrete adjoint equations in the context of the Galerkin finite element method (FEM) on triangular grids. The FIC scheme treats the instabilities normally generated in the numerical solution of the fluid equations through adding two stabilization terms, called streamline term and transverse term, to the original central-based discretized formulation. The non-linear system of equations resulting from the flow problem is solved implicitly using a damped Newton's method accompanied with the exact Jacobian matrix. A defect corrected scheme is implemented to iteratively solve the linear system of equations related to the adjoint problem benefiting from the transpose of the Jacobian matrix. At each iteration, the linear systems of equations resulting from the fluid and adjoint problems are solved using a preconditioned GMRES method. Having calculated the error of a specified output functional locally, an h -refinement methodology based on the element subdivision is performed to refine the candidate elements. The quality of the numerical results proves the capability of the presented approach for the adjoint-based error estimation and mesh adaptation problems in different flow regimes. [ABSTRACT FROM AUTHOR]

A new implicit stabilized formulation for the numerical solution of the compressible Navier-Stokes equations is presented. The method is based on the finite calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors Kouhi, Oñate (Int J Numer Methods Fluids 74:872-897, ). In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented. [ABSTRACT FROM AUTHOR]

A new finite element method is proposed to simulate two-phase mixture problems with free surfaces and/or large moving boundaries. The two phases are treated as interpenetrating continua within a two-fluid model. In contrast to the conventional two-fluid model formulations, in which the governing equations are written in the Eulerian-Eulerian description, the modified two-fluid model has the governing equations of one fluid phase written in the Lagrangian description while the respective equations of the other phase is formulated in the Arbitrary Lagrangian-Eulerian (ALE) description with relative velocities defined with respect to the moving mesh associated with the first fluid phase. The Finite Element Method (FEM) together with stabilization techniques-the Characteristic-Based Split (CBS) and the Algebraic-Flux Correction (AFC) method-is employed to solve the equations numerically in space and time. The nodes of the computational mesh are updated together with the flow of the phase described in the Lagrangian description. The computational mesh is updated by applying techniques used in the so-called Particle Finite Element Method (PFEM), characterized by regenerating the mesh in each computational step according to the updated nodal positions while the current boundary of the domain is identified using αα-shape method. The performance of the method is demonstrated by means of three 2D and 3D examples, including a mixing process of two fluids in a stirred tank, a free surface flow, a 3D two-phase channel flow, a phase separation problem and the flow of two immiscible fluids. [ABSTRACT FROM AUTHOR]

SUMMARY The hydrodynamics of fluid mixtures is receiving more and more attention in many science and engineering applications. Within the techniques for dealing with front displacements and moving boundaries between different density and/or viscosity fluids, phase fields are a class of models in which a diffusive transition region is taken into account instead of a steep interface. Although these models have a physical motivation, they require the definition of extra parameters. In order to make it less parameter dependent, the classic Allen-Cahn phase field model is modified, exploring its similarities with residual-based discontinuity-capturing schemes, making the phase field equation dependent on its own residual. We solve the coupling between incompressible viscous fluid flow and the phase field advective-diffusive-reactive transport to simulate the main processes in interface tension and/or buoyancy driven problems. For the solution of the Navier-Stokes and transport equations, we use a stabilized finite element formulation. The implementation has been performed using the libMesh finite element library, written in C++ , which provides support for adaptive mesh refinement and coarsening. A chemical convection benchmark problem is used to validate the proposed model, and then we solve two bubble interaction problems. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space-time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space-time domain. An explicit fourth-order Runge-Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high-gradient zones. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we describe a new method for the three-dimensional steady incompressible Navier-Stokes equations, which is called the dimension split method (DSM). The basic idea of DSM is that the three-dimensional space is split up into a cluster of two-dimensional manifolds and then the three-dimensional solution is approximated by the solutions on these two-dimensional manifolds. Through introducing some technologies, such as SUPG stabilization, multigrid method, and such, we firstly make DSM feasible in the computation of real flow. Because of split property of DSM, all computation is carried out on these two-dimensional manifolds, namely, a series of two-dimensional problems only need to be solved in the computation of three-dimensional problem, which greatly reduces the difficulty and the computational cost in the mesh generation. Moreover, these two-dimensional problems can be computed simultaneously and a coarse-grained parallel algorithm would be constructed, whereas the two-dimensional manifold is considered as the computation unit. In the last, we explore the behavior and the accuracy of the proposed method in two numerical examples. Firstly, error estimates, performance of multigrid method, and parallel algorithm are well-demonstrated by the known analytical solution case. Secondly, the computations of three-dimensional lid-driven cavity flows with different Reynolds numbers are compared with other numerical simulations. Results show that the present implementation is able to exhibit good stability and accuracy properties for real flows. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The one-fluid visco-resistive MHD model provides a description of the dynamics of a charged fluid under the influence of an electromagnetic field. This model is strongly coupled, highly nonlinear, and characterized by physical mechanisms that span a wide range of interacting time scales. Solutions of this system can include very fast component time scales to slowly varying dynamical time scales that are long relative to the normal modes of the model equations. Fully implicit time stepping is attractive for simulating this type of wide-ranging physical phenomena. However, it is essential that one has effective preconditioning strategies so that the overall fully implicit methodology is both efficient and scalable. In this paper, we propose and explore the performance of several candidate block preconditioners for this system. One of these preconditioners is based on an operator-split approximation. This method reduces the 3x3 system (momentum, continuity, and magnetics) into two 2x2 operators: a Navier-Stokes operator (momentum and continuity) and a magnetics-velocity operator (momentum and magnetics) which takes into account the critical Lorentz force coupling. Using previously developed preconditioners for Navier-Stokes, and an initial Schur-complement approximation for the magnetics-velocity system, we show that the split preconditioner is scalable and competitive with other preconditioners, including a fully coupled algebraic multigrid method. [ABSTRACT FROM AUTHOR]

Abstract: This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier–Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier–Stokes. [Copyright &y& Elsevier]

Abstract: The two-level pressure projection stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier–Stokes type variational inequality problem of the second kind. Based on the triangular element and using the pressure projection stabilized finite element method, we solve a small Navier–Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if , the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier–Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method to simulate an incompressible fluid through an immersed porous interface. The interface is modeled by a surface measure term in the Navier-Stokes equations and it is characterized by a resistance parameter. This approach can be used, for example, to model valves or to simulate blood flood through an immersed stent. Starting from a monolithic formulation proposed recently, a fractional step algorithm is derived. The difficult point is that this formulation is singular when the resistance vanishes, which can be a serious issue in some applications. We show that an appropriate Nitsche treatment of the interface condition fixes this problem and ensures uniform energy stability in time for any nonnegative value of the resistance. The theoretical stability and convergence results are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: This paper explores the development of a scalable, nonlinear, fully-implicit stabilized unstructured finite element (FE) capability for 2D incompressible (reduced) resistive MHD. The discussion considers the implementation of a stabilized FE formulation in context of a fully-implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton–Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, including both low-Lundquist number (e.g., an MHD Faraday conduction pump and a hydromagnetic Rayleigh–Bernard linear stability calculation) and moderately-high Lundquist number (magnetic island coalescence problem) examples. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented. [ABSTRACT FROM AUTHOR]

The moment-of-fluid (MOF) method is a new volume-tracking method that accurately treats evolving material interfaces. The MOF method uses moment data, namely the material volume fraction, as well as the centroid, for a more accurate representation of the material configuration, interfaces and concomitant volume advection. In contrast, the volume-of-fluid method uses only volume fraction data for interface reconstruction and advection. Based on the moment data for each material, the material interfaces are reconstructed with second-order spatial accuracy in a strictly conservative manner. The MOF method is coupled with a stabilized finite element incompressible Navier–Stokes solver for two materials. The effectiveness of the MOF method is demonstrated with a free-surface dam-break and a two-material Rayleigh–Taylor problem. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a parallel implementation of fractional solvers for the incompressible Navier–Stokes equations using an algebraic approach. Under this framework, predictor–corrector and incremental projection schemes are seen as sub-classes of the same class, making apparent its differences and similarities. An additional advantage of this approach is to set a common basis for a parallelization strategy, which can be extended to other split techniques or to compressible flows. The predictor–corrector scheme consists in solving the momentum equation and a modified “continuity” equation (namely a simple iteration for the pressure Schur complement) consecutively in order to converge to the monolithic solution, thus avoiding fractional errors. On the other hand, the incremental projection scheme solves only one iteration of the predictor–corrector per time step and adds a correction equation to fulfill the mass conservation. As shown in the paper, these two schemes are very well suited for massively parallel implementation. In fact, when compared with monolithic schemes, simpler solvers and preconditioners can be used to solve the non-symmetric momentum equations (GMRES, Bi-CGSTAB) and to solve the symmetric continuity equation (CG, Deflated CG). This gives good speedup properties of the algorithm. The implementation of the mesh partitioning technique is presented, as well as the parallel performances and speedups for thousands of processors. [Copyright &y& Elsevier]

Abstract: In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier–Stokes equations. It focuses on the lowest equal-order velocity–pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure. [Copyright &y& Elsevier]

The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection–diffusion–reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ‘stabilization parameter’ τ, in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green's method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Solving Stokes–Darcy coupled flow using stable Galerkin formulations for both subproblems usually leads to unbalanced rates of convergence. We present a stabilized mixed finite element method for Darcy flow, compatible with Galerkin stable elements for the Stokes problem, with balanced rates of convergence in norm for velocity and for pressure. The discontinuities of the solutions on the interface of the free-fluid and the porous medium are incorporated in the finite element approximation by simple linear transformations derived from the interface conditions. The resulting global finite element system can be assembled and directly solved using standard finite element implementation. Results of some numerical experiments are presented to demonstrate the flexibility and the robustness of the proposed finite element formulation. [Copyright &y& Elsevier]

The aim of this study is to numerically predict the aerodynamic behaviour of rectangular buildings. Simulations were made for rectangles of different side coefficients and different angles of attack. The finite element method is used to simulate fluid flow considered Newtonian and incompressible. We adopt the LES method acronym Large Eddy Simulation using the Smagorinsky model. We use the Fast Fourier Transformation (FFT) method to be able In achieve a frequency analysis of the calculated wind loads. Regarding the flow around a bluff body of square section, the Strouhal number and the drag average value are calculated for different Reynolds numbers. The obtained results show a good agreement with available experimental and numerical results. According to the evolution of the RMS of lift and moment coefficients as well as average values of the drag coefficients depending on various side factors, it was possible to conclude that the shape of square cross section is more appropriate. It was also numerically shown the influence of the change in the angle of attack of airflow on the frequency evolution of aerodynamic loading. [ABSTRACT FROM AUTHOR]

Abstract: Unconditionally stable finite element methods for Darcy flow are derived by adding least squares residual forms of the governing equations to the classical mixed formulations. The proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous Lagrangian finite element spaces of any order for the velocity and the potential. Stability, convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence. [Copyright &y& Elsevier]

Abstract: A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal error estimate of the finite volume solution and error estimate for are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM. [Copyright &y& Elsevier]

Stable and accurate finite element methods are presented for Darcy flow in heterogeneous porous media with an interface of discontinuity of the hydraulic conductivity tensor. Accurate velocity fields are computed through global or local post-processing formulations that use previous approximations of the hydraulic potential. Stability is provided by combining Galerkin and least squares (GLS) residuals of the governing equations with an additional stabilization on the interface that incorporates the discontinuity on the tangential component of the velocity field in a strong sense. Numerical analysis is outlined and numerical results are presented to illustrate the good performance of the proposed methods. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions, for both global and local post-processings. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Linear equilibrium and non-equilibrium models for leaching of solutes in soils give rise to unsteady linear convection–diffusion–reaction problems. We present several numerical schemes to approximate the solution of this kind of problems based on Stabilized Finite Element Methods, including the recent Link-Cutting Bubbles strategy adapted to deal with unsteady problems, which gives the best numerical results. [Copyright &y& Elsevier]

Abstract: The population dynamics problems considered here describe propagation of travelling population fronts that represent either single-species invasion or single-species retreat. The influence of the weak and strong Allee effect is analyzed on the spread of a single species invading organism in a one-dimensional domain. This problem is mathematically modelled by a non-linear advection–diffusion–reaction transport equation, which is numerically solved by using the finite element method. The resulting numerical model is stable and accurate for a wide range of parameters. Some scenarios of single species invasion are investigated so as to evaluate the interaction of travelling population fronts under different parameter values, allowing to derive numerically whether they can reverse or block the species invasion, and to predict the speed of an invasion. [Copyright &y& Elsevier]

Abstract: Based on the Stokes coupling method for solving the two dimensional exterior unsteady Navier–Stokes equations, the stabilized finite element approximation combining the boundary element with the stabilized finite element for solving the reduced Stokes coupling equations is presented. Finally, the optimal error estimates for the finite element solution are derived. [Copyright &y& Elsevier]

Abstract: The objective of this work is to study the role of some wavelike motion observed in the interior of a conical hydrocyclone used in mineral processing operations. In this paper, we obtained a stable numerical solution of the equations of motion, using a streamfunction-vorticity formulation on non-structured mesh. The numerical solution is obtained in a finite element context, where a stabilization is imposed through an unusual finite element method (USFEM) scheme. Finally, an adaptive mesh refinement is used for to enhance the solution. [Copyright &y& Elsevier]

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q1− P0 quadrilateral element and the P1− P0 triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h= O( H2) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h= O(|log h|1/2 H3). The methods we study provide an approximate solution ( u h, p h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time. [ABSTRACT FROM AUTHOR]

This paper focuses on adaptive finite element (FE)-methods for computation of the motion of viscous fluid interfaces fundamentally encountered in multiphase flow problems in porous media. An interface capturing technique (ICT)-procedure is formulated with a stabilized finite element scheme in a Eulerian framework to solve the two-dimensional (2D) and three-dimensional (3D) Navier–Stokes equation in porous media. Global mesh refinements of the discretized domain and local mesh refinements in the vicinity of the interface are used for the spatial discretization. The ICT is embedded into the finite element scheme by adding an extra advection equation and an additional unbounded degree of freedom to the number of the unknowns. Problems of non-linear free surface seepage flow in earth-fill dams are simulated in order to validate the performance of the FE-ICT. Computations for steady non-linear seepage flows in 2D and 3D are obtained for homogenous, isotropic and isothermal porous media. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]