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     

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L²-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.     

Stokes方程的压力梯度局部投影间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过将通常的间断Galerkin有限元法与压力梯度局部投影相结合,建立了一个稳定的间断有限元格式,对速度和压力的任意分片多项式空间P_l(K),P_m(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L~2范数的最优误差估计.     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

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.     

Analysis and convergence of a MAC-like scheme for the generalized Stokes problem

We introduce a MAC-like scheme (a covolume method on rectangular grids) for approximating the generalized Stokes problem on an axiparallel domain. Two staggered grids are used in the derivation of the discretization. The velocity is approximated by conforming bilinears over rectangular elements, and the pressure by piecewise constants over macro-rectangular elements. The error in the velocity in the H¹ norm and the pressure in the L² norm are shown to be of first order, provided that the exact velocity is in H² and the exact pressure in H¹, and that the partition family of the domain is regular. © 1997 John Wiley & Sons, Inc.     

Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L². Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016     

A posteriori error estimates for mixed finite element approximations of parabolic problems

We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L ²(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.     

A discontinuous mixed covolume method for elliptic problems

We develop a discontinuous mixed covolume method for elliptic problems on triangular meshes. 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.     

Least‐squares method for the Oseen equation

This article studies the least‐squares finite element method for the linearized, stationary Navier–Stokes equation based on the stress‐velocity‐pressure formulation in d dimensions (d = 2 or 3). The least‐squares functional is simply defined as the sum of the squares of the L² norm of the residuals. It is shown that the homogeneous least‐squares functional is elliptic and continuous in the norm. This immediately implies that the a priori error estimate of the conforming least‐squares finite element approximation is optimal in the energy norm. The L² norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right‐hand side f belongs only to , we derive an a priori error bound in a weaker norm, that is, the norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1289–1303, 2016     

A partially penalized P1/CR immersed finite element method for planar elasticity interface problems

We propose a partially penalized P₁/CR immersed finite element (IFE) method with midpoint values on edges as degrees of freedom for CR elements to solve planar elasticity interface problems. Optimal approximation errors in L² norm and H¹ semi‐norm are obtained for the P₁/CR IFE spaces. Moreover, by adding some stabilization terms on the edges of interface elements, we derive an optimal error estimate for the P₁/CR IFE method. Our method differs from the method with average values on edges as degrees of freedom for P₁/CR elements in Qin et al.'s study, where no approximation theoretical result was presented. Numerical examples confirm our theoretical results.     

Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L²$L^2$ norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.     

Inf-sup stable non-conforming finite elements of arbitrary order on triangles

We introduce a family of scalar non-conforming finite elements of arbitrary order k≥1 with respect to the H¹-norm on triangles. Their vector-valued version generates together with a discontinuous pressure approximation of order k−1 an inf-sup stable finite element pair of order k for the Stokes problem in the energy norm. For k=1 the well-known Crouzeix-Raviart element is recovered.     

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a posteriori estimates.     

On computing the pressure by thep version of the finite element method for Stokes problem

Summary This paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving Stokes problem using thep version of the finite element method. When one uses a localH ¹ projection, we show that optimal rates of convergence for the pressure approximation is achieved. When the pressure is not inH ¹. or the value of the pressure is only needed at a few points, one may extract the pressure pointwise using e.g. a single layer potential recovery. Negative, zero, and higher norm estimates for the Stokes velocity are derived within the framework of thep version of the F.E.M.Partially supported by ONR grants N00014-87-K-0427 and N00014-90-J-1238     

New error estimates of nonconforming mixed finite element methods for the Stokes problem

In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi‐interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high‐order term, especially which can be of arbitrary order provided that f in the right‐hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L² norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L² norms under the regularity assumption ( u ,p) ∈ H ^{1 + s}(Ω) × H^s(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity . Furthermore, a robust convergence is proved with minimal regularity assumption s = 0. These results seem to be missing in the literature. Numerical tests are provided, confirming the analysis, especially the new results on the L² convergence. Copyright © 2013 John Wiley & Sons, Ltd.     

Spectral Tau approximation of the two-dimensional stokes problem

Summary We analyse the Spectral Tau method for the approximation of the Stokes system on a square with Dirichlet boundary conditions. We provide an error estimate, in the norm of the Sobolev spaceH ^s, for the approximation of a divergence free vector field with polynomial divergence free vector fields. We apply this result to prove some convergence estimates for the solution of the discrete Stokes problem.This work has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C 0035     

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.     

Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H¹(Ω)‐norm, L²(Ω)‐norm and L²(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

The numerical approximation by a lower‐order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi‐optimal‐order error estimates are proved in the ε‐weighted H¹‐norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε‐weighted H¹‐norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.