Iterative methods for stabilized mixed velocity–pressure finite elements

This paper is concerned with iterative techniques for the solution of the linear system of equations arising from a finite element approximation of an elliptic partial differential equation by a mixed method. Three types of iterative algorithms are investigated. Applications to the Stokes equations are discussed and the results of numerical experiments given.     

Parallel conjugate gradient performance for least-squares finite elements and transport problems

In this study we consider parallel conjugate gradient solution of sparse systems arising from the least-squares mixed finite element method. Of particular interest are transport problems involving convection. The least-squares approach leads to a symmetric positive system and the conjugate gradient scheme is directly applicable. The scheme is applied to both the convection–diffusion equation and to the stationary Navier–Stokes equations. Here we demonstrate parallel solution and performance studies for a representative MIMD parallel computer with hypercube architecture. © 1998 John Wiley & Sons, Ltd.     

An all-at-once algebraic multigrid method for finite element discretizations of Stokes problem

We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform-then-solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over-relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform-then-solve approach is also more robust. In particular, for problems with variable viscosity, the transform-then-solve approach demonstrates significant speed-up with respect to the block diagonal preconditioner. The method is also particularly robust for time-dependent problems whatever the time step size.     

流体饱和两相多孔介质拟静态问题的混合有限元方法

针对基于混合物理论的两相多孔介质模型,采用Galerkin加权残值有限元法,导出求解所静态问题的基于us-uF-P变量的混合有限元方程,由于系统方程的系数矩阵非定,进而针对该方程组提出了一种失代求解方法,并由分片试验得出节点压力插值函数的阶须低于固体相节点的位移插值函数的阶的结论,算例结果表明,采用基于u2-uF-p变量的混合法计算所得的固体相和流体相速度以及固体相的有效应力与罚方法一致,而压力值的粗度高于罚方法。     

Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

In this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley & Sons, Ltd.     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, 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 new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

A nodal‐based implementation of a stabilized finite element method for incompressible flow problems

The objective of this paper is twofold. First, a stabilized finite element method (FEM) for the incompressible Navier–Stokes is presented and several numerical experiments are conducted to check its performance. This method is capable of dealing with all the instabilities that the standard Galerkin method presents, namely the pressure instability, the instability arising in convection‐dominated situations and the less popular instabilities found when the Navier–Stokes equations have a dominant Coriolis force or when there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal‐based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements, thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this gives rise to questions regarding the consistency and the conservation properties of the final scheme, which are addressed in this paper. Copyright © 2000 John Wiley & Sons, Ltd.     

An Uzawa-type algorithm for the coupled Stokes equations

An Uzawa-type algorithm is designed for the coupled Stokes equations discretized by the mixed finite element method. The velocity solved by the presented algorithm is weakly divergence-free, which is different from the one solved by the common Uzawa method. Besides, an optimal relaxation parameter of the presented algorithm is provided.     

Numerical simulation of natural and mixed convection flows by Galerkin‐characteristic method

A numerical investigation is performed to study the solution of natural and mixed convection flows by Galerkin‐characteristic method. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization in primitive variables. It can be interpreted as a fractional step technique where convective part and Stokes/Boussinesq part are treated separately. The main feature of the proposed method is that, due to the Lagrangian treatment of convection, the Courant–Friedrichs–Lewy (CFL) restriction is relaxed and the time truncation errors are reduced in the Stokes/Boussinesq part. Numerical simulations are carried out for a natural convection in squared cavity and for a mixed convection flow past a circular cylinder. The computed results are compared with those obtained using other Eulerian‐based Galerkin finite element solvers, which are used for solving many convective flow models. The Galerkin‐characteristic method has been found to be feasible and satisfactory. Copyright © 2006 John Wiley & Sons, Ltd.     

Approximation of generalized Stokes problems using dual‐mixed finite elements without enrichment

In this work a finite element method for a dual‐mixed approximation of generalized Stokes problems in two or three space dimensions is studied. A variational formulation of the generalized Stokes problems is accomplished through the introduction of the pseudostress and the trace‐free velocity gradient as unknowns, yielding a twofold saddle point problem. The method avoids the explicit computation of the pressure, which can be recovered through a simple post‐processing technique. Compared with an existing approach for the same problem, the method presented here reduces the global number of degrees of freedom by up to one‐third in two space dimensions. The method presented here also represents a connection between existing dual‐mixed and pseudostress methods for Stokes problems. Existence, uniqueness, and error results for the generalized Stokes problems are given, and numerical experiments that illustrate the theoretical results are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Least-squares mixed finite element method for a class of stokes equation

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A mixed finite element method for the generalized Stokes problem

We present and analyse a new mixed finite element method for the generalized Stokes problem. The approach, which is a natural extension of a previous procedure applied to quasi‐Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two‐fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babu?ka–Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to compute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities. Copyright © 2005 John Wiley & Sons, Ltd.     

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。     

On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal‐order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid‐based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions. Copyright © 2008 John Wiley & Sons, Ltd.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

A parallel solver based on the dual Schur decomposition of general finite element matrices

A parallel solver based on domain decomposition is presented for the solution of large algebraic systems arising in the finite element discretization of mechanical problems. It is hybrid in the sense that it combines a direct factorization of the local subdomain problems with an iterative treatment of the interface system by a parallel GMRES algorithm. An important feature of the proposed solver is the use of a set of Lagrange multipliers to enforce continuity of the finite element unknowns at the interface. A projection step and a preconditioner are proposed to control the conditioning of the interface matrix. The decomposition of the finite element mesh is formulated as a graph partitioning problem. A two-step approach is used where an initial decomposition is optimized by non-deterministic heuristics to increase the quality of the decomposition. Parallel simulations of a Navier–Stokes flow problem carried out on a Convex Exemplar SPP system with 16 processors show that the use of optimized decompositions and the preconditioning step are keys to obtaining high parallel efficiencies. Typical parallel efficiencies range above 80%. © 1998 John Wiley & Sons, Ltd.