A two-level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation

In this paper, we consider the well known Morley nonconformingelement approximation of a nonlinear biharmonic equation whichis related to the well-known two-dimensional Navier–Stokesequations. Firstly, optimal energy and H¹-norm estimates areobtained. Secondly, a two-level additive Schwarz method is presentedfor the discrete nonlinear algebraic system. It is shown thatif the Reynolds number is sufficiently small, the two-levelSchwarz method is optimal, i.e. the convergence rate of theSchwarz method is independent of the mesh size and the numberof subdomains.     

Two-level iteration penalty and variational multiscale method for steady incompressible flows

In this paper, we study two-level iteration penalty and variational multiscale method for the approximation of steady Navier-Stokes equations at high Reynolds number. Comparing with classical penalty method, this new method does not require very small penalty parameter $\varepsilon$. Moreover, two-level mesh method can save a large amount of CPU time. The error estimates in $H^1$ norm for velocity and in $L^2$ norm for pressure are derived. Finally, two numerical experiments are shown to support the efficiency of this new method.     

Navier-Stokes方程流函数形式两重网格算法的误差分析

对定常Navier-Stokes方程流函数形式两重网格有限元算法进行了误差分析。此方法包括在粗网格上求解一个非线性问题，在细网格上求解一个线性问题，然后再在粗网格上求解一个线性校正问题。分析了包括校正项和不包括校正项两种方法的误差，得出对于任意固定的Beynolds数，能达到最优逼近阶。     

A two-grid method based on Newton iteration for the Navier–Stokes equations

In this paper, we consider a two-grid method for resolving the nonlinearity in finite element approximations of the equilibrium Navier–Stokes equations. We prove the convergence rate of the approximation obtained by this method. The two-grid method involves solving one small, nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right-hand side. The algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any Reynolds number. Numerical example is given to show the convergence of the method.     

A new methodology for anisotropic mesh refinement based upon error gradients

We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered.     

A new stabilized finite element method for advection‐diffusion‐reaction equations

In this article, a new stabilized finite element method is proposed and analyzed for advection‐diffusion‐reaction equations. The key feature is that both the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity‐norm of the exact solution, the explicit‐dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 616–645, 2016     

Navier-Stokes方程带Backtracking技巧的两重网格算法

1 引 言考虑二维不可压 Navier-Stokes方程:     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Graded Meshes for Higher Order FEM

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.     

A nearly optimal preconditioner for the Navier–Stokes equations

We present a preconditioner for the linearized Navier–Stokes equations which is based on the combination of a fast transform approximation of an advection diffusion problem together with the recently introduced ‘BFBT^T’ preconditioner of Elman (SIAM Journal of Scientific Computing, 1999; 20 :1299–1316). The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh Péclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary flow direction. Copyright © 2001 John Wiley & Sons, Ltd.     

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.     

Interior maximum norm estimates for finite element discretizations of the Stokes equations

Interior estimates are proved in the L ^∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.     

Subgrid scale eddy viscosity finite element method on optimal control of system governed by unsteady Oseen equations

In this paper we focus on numerical analysis of finite element methods with stabilizations for the optimal control of system governed by unsteady Oseen equations. Using continuous equal-order finite elements for both velocities and pressure, two fully discrete schemes are proposed. Convective effects and pressure are stabilized by adding a subgrid scale eddy viscosity term and a pressure stabilized term. Convergence of the approximate solution is proved. A-Priori error estimates are obtained uniformly with Reynolds number, especially the \(L^2\) -error estimates of numerical solution are independent of Reynolds number. The numerical experiments are shown to be consistent with our theoretical analysis.     

Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier-Stokes equations

Summary In this article we study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for and a coarser one for . Error estimates show that boundary layers do not require to refine the mesh for the stream function as much as for the vorticity when the Reynolds number is large. We prove estimates and study implementation problems.     

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Galerkin Methods for Nonlinear Ordinary Differential Equation with Impulses

Methods based on fixed mesh variational formulations for ordinary differential equations in presence of a possibly infinite number of impulses on the righthand side are presented. A simple transformation allows us to show that the problem can be treated as an ordinary differential equation. Existence and uniqueness results for the solution and approximation schemes with their error estimates are obtained.     

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     

New interpolants for asymptotically correct defect control of BVODEs

The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h →0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interpolant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the user-prescribed tolerance. In this paper we employ a boot-strapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach.     

A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMP(E)RE EQUATION

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.     

Numerical study of spherical Couette flows for certain zenith-angle-dependent rotations of boundary spheres at low Reynolds numbers

The numerical method with splitting of boundary conditions developed previously by the first and third authors for solving the stationary Dirichlet boundary value problem for the Navier-Stokes equations in spherical layers in the axisymmetric case at low Reynolds numbers and a corresponding software package were used to study viscous incompressible steady flows between two con-centric spheres. Flow regimes depending on the zenith angle ?? of coaxially rotating boundary spheres (admitting discontinuities in their angular velocities) were investigated. The orders of accuracy with respect to the mesh size of the numerical solutions (for velocity, pressure, and stream function in a meridional plane) in the max and L ₂ norms were studied in the case when the velocity boundary data have jump discontinuities and when some procedures are used to smooth the latter. The capabilities of the Richardson extrapolation procedure used to improve the order of accuracy of the method were investigated. Error estimates were obtained. Due to the high accuracy of the numerical solutions, flow features were carefully analyzed that were not studied previously. A number of interesting phenomena in viscous incompressible flows were discovered in the cases under study.