$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

Summary.   We study the -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let denote the `small scale' of such approximations (– the viscosity amplitude , the spatial grad-size , etc.), then our -error estimates are of , and are sharper than the classical -results of order one half, . The main building blocks of our theory are the notions of the semi-concave stability condition and -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain -bounds on their associated truncation errors; -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our -theory. Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000     

Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional grids

Summary. Based on Nessyahu and Tadmor's nonoscillatory central difference schemes for one-dimensional hyperbolic conservation laws [16], for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented. The experiments show the wide applicability of these multidimensional schemes. The theoretical arguments which support this are some maximum-principles and a convergence proof in the scalar linear case. A general proof of convergence, as obtained for the original one-dimensional NT-schemes, does not exist for any of the extensions to multidimensional nonlinear problems. For the finite volume extension on two-dimensional unstructured grids introduced by Arminjon and Viallon [3,4] we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law. Received April 8, 2000 / Published online December 19, 2000     

On the stability of the unsmoothed Fourier method for hyperbolic equations

Summary. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted -stability, the -degree Fourier solution is algebraically stable in the sense that its amplification does not exceed . Yet, the Fourier method is weakly -unstable in the sense that it does experience such amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their amplification. Happily, with enough resolution nothing worse can happen. Received December 14, 1992/Revised version received March 1, 1993     

Convergence of finite volume schemes for semilinear convection diffusion equations

The topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method. Dirichlet boundary conditions are considered here. A discrete Poincaré inequality is used, and estimates on the approximate solutions are proven. The convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions. Received January 16, 1998 / Revised version received June 19, 1998     

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Bilinear estimates in BMO and the Navier-Stokes equations

We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class. Received February 15, 1999; in final form October 11, 1999 / Published online July 3, 2000     

A priori error estimates for approximate solutions to convex conservation laws

Summary. We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of Engquist-Osher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introduced by Tadmor [SIAM J. Numer. Anal. 1991]. The error is related to the consistency error of cell averages of the entropy weak solution. This consistency error can be estimated by exploiting a regularity structure of the entropy weak solution. One ends up with optimal error estimates. Received December 21, 2001 / Revised version received February 18, 2002 / Published online June 17, 2002     

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation by a piecewise constant function using a discretization in space and time and a finite volume scheme. The convergence of to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on are used to prove the convergence, up to a subsequence, of to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of to{\it u}. Some on a model equation are shown. Received September 27, 2000 / Published online October 17, 2001     

Error indicators for the Navier-Stokes equations in stream function and vorticity formulation

This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second step, these results are extended to the Navier-Stokes equations. Received March 25, 1996 / Revised version received April 7, 1997     

Interpolation error estimates of a modified 8-node serendipity finite element

Summary. Interpolation error estimates for a modified 8-node serendipity finite element are derived in both regular and degenerate cases, the latter of which includes the case when the element is of triangular shape. For defined over a quadrilateral K, the error for the interpolant is estimated as , where in the regular case and in the degenerate case, respectively. Thus, the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for . Results for some related elements are also given. Received June 2, 1997 / Published online March 16, 2000     

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model. Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001     

Stability and error estimates for the $\theta$-method for strongly monotone and infinitely stiff evolution equations

Summary.   For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method , for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results. Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001     

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in in a finite volume sense. Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000     

On one approach to a posteriori error estimates for evolution problems solved by the method of lines

Summary. In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the -norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented. Received September 2, 1999 / Revised version received March 6, 2000 / Published online March 20, 2001     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization

Summary. The long-time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a Hopf bifurcation is studied in this paper. The framework includes reaction-diffusion and incompressible Navier-Stokes equations. It is shown that the phase portrait of a supercritical Hopf bifurcation is correctly represented by Runge-Kutta time discretization. In particular, the bifurcation point and the Hopf orbits are approximated with higher order. A basic tool in the analysis is the reduction of the dynamics to a two-dimensional center manifold. A large portion of the paper is therefore concerned with studying center manifolds of the discretization. Received March 18, 1997 / Revised version received February 19, 1998     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations

Summary. This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher-order schemes (filtered to ``preserve' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians. Received May 16, 1994     

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems. Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980