Analysis and convergence of finite volume method using discontinuous bilinear functions

We develop finite volume method using discontinuous bilinear functions on rectangular mesh. This method is analyzed for the Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh‐dependent norm. First order L²‐error estimates are derived for the approximations of both velocity and pressure. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An analysis for the compressible Stokes equations by first‐order system of least‐squares finite element method

This article applies the first‐order system least‐squares (fosls) finite element method developed by Cai, Manteuffel and McCormick to the compressible Stokes equations. By introducing a new dependent velocity flux variable, we recast the compressible Stokes equations as a first‐order system. Then it is shown that the ellipticity and continuity hold for the least‐squares functionals employing the mixture of H^?1 and L², so that the fosls finite element methods yield best approximations for the velocity flux and velocity. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:689–699, 2001     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.     

A two-grid method with backtracking for the mixed Navier–Stokes/Darcy model

In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L² and H¹ optimal velocity and piezometric head approximations and an L² optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method.     

An algebraic multigrid method for Q2−Q1 mixed discretizations of the Navier–Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q ₂? Q ₁ mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q ₂? Q ₁ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN‐AMG) is utilized. EMIN‐AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Stability of time‐splitting schemes for the Stokes problem with stabilized finite elements

The time‐dependent Stokes problem is solved using continuous, piecewise linear finite elements and a classical stabilization procedure. Four order‐one methods are proposed for the time discretization. The first one is nothing but the Euler backward scheme and requires a large linear system involving the velocity and pressure unknowns to be solved. The other three schemes allow velocity and pressure computations to be decoupled, namely the pressure‐matrix method, a method based on an inexact LU factorization, and an operator splitting method. Stability and condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:632–656, 2001     

An error estimate for a finite volume scheme for a diffusion–convection problem on a triangular mesh

We study here a finite volume scheme for a diffusion-convection equation on an open bounded set Ω of ?², using a triangular mesh for the discretization of Ω. The 4-point numerical scheme is presented along with the geometrical assumptions on the mesh. An error estimate of order h on the discrete L² norm is obtained, where h denotes the “size” of the mesh. The proof uses an estimate of order h of the consistency error on the fluxes and an estimate of the number of edges of the mesh between one given triangle and the boundary Ω. © 1995 John Wiley & Sons, Inc.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.     

Associated consistency and Shapley value

In this work, a new axiomatization of the Shapley is presented. An associated game is constructed. We define a sequence of games, when the term of order n, in this sequence, is the associated game of the term of order (n−1). We show that the sequence converges and that the limit game is inessential. The solution is obtained using the inessential game axiom, the associated consistency axiom and the continuity axiom. As a by-product, we note that neither the additivity nor the efficiency axioms are needed. Accepted September 2001     

A dual iterative substructuring method with a penalty term

An iterative substructuring method with Lagrange multipliers is considered for second order elliptic problems, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with the saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter η, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of η, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. Computational issues and numerical results are presented. This work was partially supported by the SRC/ERC program of MOST/KOSEF(R11-2002-103).     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

Discretization of the Navier‐Stokes equations with slip boundary condition

We propose and analyze a two‐level method of discretizing the nonlinear Navier‐Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no‐slip condition u = 0 is not valid. The two‐level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two‐level method exploits the quadratic nonlinearity in the Navier‐Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 26–42, 2001     

Post-Coulombian Dynamics at Order c-3

Summary. We study the dynamics of N charges interacting with the Maxwell field. If their initial velocities are small compared to the velocity of light, c , then in lowest order their motion is governed by the static Coulomb Lagrangian. We investigate higher-order corrections with an explicit control on the error terms. The Darwin correction, order |v/c| ² , has been proved previously. In this contribution we obtain the dissipative corrections due to radiation damping, which are of order |v/c| ³ relative to the Coulomb dynamics. If all particles have the same charge-to-mass ratio, the dissipation would vanish at that order. Received February 7, 2001; accepted September 21, 2001 Online publication November 30, 2001     

Asymptotic analysis of the steady stokes equation with randomly perturbed viscosity in a thin tube structure

The Stokes equation with the nonconstant viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order ε ≪ 1 and bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed. In the case of random perturbations of the constant viscosity, we prove that the leading term for the velocity is deterministic, while for the pressure it is random, but the expectations of the pressure satisfies the deterministic Darcy equation. Estimates for the difference between the exact solution and its asymptotic approximation are proved. Bibliography: 11 titles. Illustrations: 3 figures.