On positivity preserving finite volume schemes for Euler equations

Summary. We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described. Received May 20, 1994     

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Summary. This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the -norm in the situation, where the diffusion parameter is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results. Received February 25, 1999 / Revised version received July 6, 1999 / Published online August 2, 2000     

A stabilized finite element method for the incompressible magnetohydrodynamic equations

Summary. We propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large. This observation is confirmed by our numerical tests. Received August 31, 1998 / Revised version received June 16, 1999 / Published online June 21, 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     

Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary. This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in for every . This also yields a new proof of the existence of an entropy weak solution. Received May 18, 2000 / Revised version received November 21, 2000 / Published online June 7, 2001     

A posteriori error analysis for numerical approximations of Friedrichs systems

The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments. Received January 10, 1997 / Revised version received March 5, 1998     

Numerical and theoretical study of a Dual Mesh Method using finite volume schemes for two phase flow problems in porous media

Summary. In this paper we are interested in two phase flow problems in porous media. We use a Dual Mesh Method to discretize this problem with finite volume schemes. In a simplified case (elliptic - hyperbolic system) we prove the convergence of approximate solutions to the exact solutions. We use the Dual Mesh Method in physically complex problems (heterogeneous cases with non constant total mobility). We validate numerically the Dual Mesh Method on practical examples by computing error estimates for different test-cases. Received March 21, 1997 / Revised version received October 13, 1997     

Convergence of Runge–Kutta approximations for parabolic problems with Neumann boundary conditions

Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations, with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded. Received June 25, 1994 / Revised version received February 26, 1996     

Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity

Summary. This work considers semi- and fully discrete approximations to the primal problem in elastoplasticity. The unknowns are displacement and internal variables, and the problem takes the form of an evolution variational inequality. Strong convergence of time-discrete, as well as spatially and fully discrete approximations, is established without making any assumptions of regularity over and above those established in the proof of well-posedness of this problem. Received June 8, 1998 / Published online July 12, 2000     

An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy

Summary. An error bound is proved for a fully practical piecewise linear finite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy. Received October 12, 1994     

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     

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     

Nonconforming finite element methods without numerical locking

We analyze the numerical approximation of a class of elliptic problems which depend on a small parameter . We give a generalization to the nonconforming case of a recent result established by Chenais and Paumier for a conforming discretization. For both the situations where numerical integration is used or not, a uniform convergence in and h is proved, numerical locking being thus avoided. Important tools in the proof of such a result are compactness properties for nonconforming spaces as well as the passage to the limit problem. Received October 7, 1997     

Grid modification for the wave equation with attenuation

Summary. The wave equation with attenuation due to a linear friction is approximated by a new mixed finite element method which allows one to use different grids and basis functions at different times when necessary. This method enables one to track sharp moving wave fronts more efficiently and accurately. Error estimates with optimal convergent rates are established. Unconditional stability is also proved for this method. Received March 27, 1992/Revised version received May 21, 1993     

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix

Summary. Using a slightly different discretization scheme in time and adapting the approach in Nochetto et al. (1998) for analysing the time discretization error in the backward Euler method, we improve on the error bounds derived in (i) Barrett and Blowley (1998) and (ii) Barrett and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix and (i) a logarithmic free energy, and (ii) a non-smooth free energy (the deep quench limit); respectively. Moreover, the improved error bound in the deep quench limit is optimal. Numerical experiments with three components illustrating the above error bounds are also presented. Received June 28, 1999 / Revised version received December 3, 1999 / Published online November 8, 2000     

A new class of particle methods

Summary. Particle methods are numerical methods designed to solve problems in fluid mechanics and related problems in continuum mechanics. A general approach to the construction of such particle methods is presented in this article. The particles are no mass points but possess a finite extension. They can rotate in space and have a spin. The conservation of mass is automatically guaranteed by the ansatz. The forces of interaction between the particles are derived in a canonical way from the force laws of continuum mechanics and are directly based on a regularized stress tensor. In the absence of external forces and of heat sources and sinks, momentum, angular momentum, and energy are conserved as in the continuum case. Received February 17, 1995 / Revised version received December 28, 1995     

Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix

Summary. In this paper we propose four algorithms to compute truncated pivoted QR approximations to a sparse matrix. Three are based on the Gram–Schmidt algorithm and the other on Householder triangularization. All four algorithms leave the original matrix unchanged, and the only additional storage requirements are arrays to contain the factorization itself. Thus, the algorithms are particularly suited to determining low-rank approximations to a sparse matrix. Received February 23, 1998 / Revised version received April 16, 1998     

A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension

Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown for a nonlinear parabolic initial-boundary value problem. Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999     

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes . Received November 27, 1995; in final form September 29, 1997     

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