Multiscale enrichment of a finite volume element method for the stationary Navier–Stokes problem

In this paper, we introduce a finite volume element method for the Navier–Stokes problem. This method is based on the multiscale enrichment and uses the lowest finite element pair P ₁/P ₀. The stability and convergence of the optimal order in H ¹-norm for velocity and L ²-norm for pressure are obtained. Using a dual problem for the Navier–Stokes problem, we establish the convergence of the optimal order in L ²-norm for the velocity.     

Error Estimates of a FEM with Lumping for Parabolic PDEs

L ²-norm. This well-known FEM is given by the use of the vertical line method and conforming linear finite elements on a triangulation. The main result of the paper are new estimates in the L ²-norm for the additional error term originated by lumping. Using these ones, for the FEM with lumping we can apply directly the proof technique of error estimates known for conforming FEMs. Received May 17, 2001; revised November 2, 2001 Published online February 18, 2002     

The Streamline–Diffusion Method for Conforming and Nonconforming Finite Elements of Lowest Order Applied to Convection–Diffusion Problems

We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties are quite different. We give a detailed overview on the stability and the convergence properties in the L ²- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence properties are sharp. Received December 7, 1999; revised October 5, 2000     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

Gauss–Newton Multilevel Methods for Least-Squares Finite Element Computations of Variably Saturated Subsurface Flow

We apply the least-squares mixed finite element framework to the nonlinear elliptic problems arising in each time-step of an implicit Euler discretization for variably saturated flow. This approach allows the combination of standard piecewise linear H ¹-conforming finite elements for the hydraulic potential with the H(div)-conforming Raviart–Thomas spaces for the flux. It also provides an a posteriori error estimator which may be used in an adaptive mesh refinement strategy. The resulting nonlinear algebraic least-squares problems are solved by an inexact Gauss–Newton method using a stopping criterion for the inner iteration which is based on the change of the linearized least-squares functional relative to the nonlinear least-squares functional. The inner iteration is carried out using an adaptive multilevel method with a block Gauss–Seidel smoothing iteration. For a realistic water table recharge problem, the results of computational experiments are presented. Received January 4, 1999; revised July 19, 1999     

Error Estimates for Semidiscrete Finite Element Approximations of the Stokes Equations Under Minimal Regularity Assumptions

Semidiscrete (spatially discrete) finite element approximations of the Stokes equations are studied in this paper. Properties of L ², H ¹ and H ^–1 projections onto discretely divergence-free spaces are discussed and error estimates are derived under minimal regularity assumptions on the solution.     

A Coupled Multigrid Method for Nonconforming Finite Element Discretizations of the 2D-Stokes Equation

This paper investigates a multigrid method for the solution of the saddle point formulation of the discrete Stokes equation obtained with inf–sup stable nonconforming finite elements of lowest order. A smoother proposed by Braess and Sarazin (1997) is used and L ²-projection as well as simple averaging are considered as prolongation. The W-cycle convergence in the L ²-norm of the velocity with a rate independently of the level and linearly decreasing with increasing number of smoothing steps is proven. Numerical tests confirm the theoretically predicted results. Received January 19, 1999; revised September 13, 1999     

Error estimates for least squares finite element methods

A least squares finite element scheme for a boundary value problem associated with a second-order partial differential equation is considered. Previous work on this subject is generalized and improved by considering a larger class of equations, by working in the natural context, without additional smoothness conditions, and by deriving error estimates, not only in the H¹-norm and the H_div-norm, but also in the L₂-norm. Some of these estimates are sharpened by using finite element spaces with the grid decomposition property. The error estimates are supported by numerical results which extend previous numerical work.     

Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data

The purpose of this paper is to study the effect of numerical quadrature in the finite element analysis for a time dependent parabolic equation with nonsmooth initial data. Both semidiscrete and fully discrete schemes are analyzed using standard energy techniques. For the semidiscrete case, optimal order error estimates are derived in the L ² and H ¹-norms and quasi-optimal order in the L ^∞-norm, when the initial function is only in H ₀ ¹. Finally, based on the backward Euler method, a time discretization scheme is discussed and almost optimal rates of convergence in the L ², H ¹ and L ^∞-norms are established. Received: September 1997 / Accepted: October 1997     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-interpolation on quadrilateral and hexahedral meshes with hanging nodes

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H ¹-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H ¹. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.      

P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

In this paper we use P ₁-nonconforming quadrilateral finite volume methods with interpolated coefficients to solve the semilinear elliptic problems. Two types of control volumes are applied. Optimal error estimates in H ¹-norm on the quadrilateral mesh and superconvergence of derivative on the rectangular mesh are derived by using the continuity argument, respectively. In addition, numerical experiments are presented adequately to confirm the theoretical analysis and optimal error estimates in L ²-norm is also observed obviously. Compared with the standard Q ₁-conforming quadrilateral element, numerical results of the proposed finite volume methods show its better performance than others.     

Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations

In this paper, we prove that the Legendre tau method has the optimal rate of convergence in L ²-norm, H ¹-norm and H ²-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in C([0,T];L ²(I))-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in C([0,T];L ²(I))-norm. Finally, we give some numerical examples.     

A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations

Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L ² can be established, as well as one in a broken H ¹-norm for the velocity. Received: January 1999 / Accepted: April 1999     

A generalization of the vertex-centered finite volume scheme to arbitrary high order

A higher order finite volume method for elliptic problems is proposed for arbitrary order ${p \in \mathbb{N}}$ . Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. In these tests the optimal error is achieved in the H ¹-norm. The error in the L ²-norm is one order below optimal for even polynomial degrees and optimal for odd degrees.     

Three-points interfacial quadrature for geometrical source terms on nonuniform grids

This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells’ size, for which L ^p -error estimates, 1≤p<+∞, are proven.     

Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H¹-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L²-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L² for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

A numerical method for exact boundary controllability problems for the wave equation

The computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied. A numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem. The uniqueness of the discrete finite-difference solutions obtained in this manner is demonstrated. The convergence properties of the method are illustrated through computational experiments. Efficient implementation strategies for the method are also discussed. It is shown that for smooth, minimum L²-norm Dirichlet controls, the method results in convergent approximations without the need to introduce regularization. Furthermore, for the generic case of nonsmooth Dirichlet controls, convergence with respect to L² norms is also numerically demonstrated. One of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria; such generalizations are discussed. In particular, the minimum H¹-norm Dirichlet controllability problem is approximated and solved, as are minimum regularized L²-norm Dirichlet controllability problems with small penalty constants. Finally, a discussion is provided about the differences between our method and existing methods; these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner.