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     

Exact and approximate pulsed beam solutions of Maxwell's equations

An exact solution of Maxwell's equations is constructed which represents a pulse of finite energy propagating along a given axis. To simplify the computation of such a pulse, an approximate solution is constructed as a superposition of solutions of the paraxial equation. This approximation is vectorial in nature and satisfies the divergence-free condition. The difference between the exact and approximate solutions is estimated relative to the total energy transfer of the exact solution.     

Nonlinear projection methods for multi-entropies Navier-Stokes systems

This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

     

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     

$L^1$-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     

Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.     

Numerical analysis of finite element methods for miscible displacements in porous media

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element ( SUPG ) method is applied to the convection–diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 519–548, 1998     

Difference methods of order two and four for systems of mildly nonlinear biharmonic problems of the second kind in two space dimensions

In this article, we report two sets of finite difference methods of order two and four over a rectangular domain for the efficient numerical integration of the system of two-dimensional nonlinear elliptic biharmonic problems of the second kind. Second-order derivatives of the solutions are obtained as byproducts of the methods. We use only 9 grid points and do not require fictitious points in order to approximate the boundary conditions. In numerical experiments, the new second- and fourth-order formulas are compared with the exact solutions both in singular and nonsingular cases. © 1996 John Wiley & Sons, Inc.     

一类逼近l1精确罚函数的罚函数

本文对可微非线性规划问题提出了一个渐近算法,它是基于一类逼近l1精确罚函数的罚函数而提出的,我们证明了算法所得的极小点列的聚点均为原问题的最优解,并在Mangasarian-Fromovitz约束条件下,证明了有限次迭代之后,所有迭代均为可行的,即迭代所得的极小点为可行点.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Convergence of the finite element method for an elliptic equation with strong degeneration

Under consideration are the questions of the numerical solution by the finite element method (FEM) of the first boundary value problem for an elliptic equation with degeneration on a part of the boundary. The weak and strong variational statements are posed in the function spaces with the coordinated weights that correspond to the problem. Using the method of the multiplicative extraction of singularities for the finite element method that utilizes piecewise linear elements, we prove that the convergence of the approximate solutions to the exact solution in the weighted norm is not worse than in the case of an elliptic equation without degeneration.     

Difference inequalities generated by impulsive parabolic differential-functional problems

The paper deals with parabolic differential-functional equations. Initial-boundary value problems are considered with impulses given in fixed points. We prove theorems on difference-functional impulsive inequalities generated by original problems.Explicit finite difference schemes are used to approximate the solutions of the original problems. We give sufficient conditions for the convergence of sequences of approximate solutions under the assumptions that the right-hand sides satisfy the nonlinear estimates of the Perron type with respect to the functional argument. In proof of the convergence of difference methods we apply theorems on difference-functional impulsive inequalities.     

Fast-slow dynamics in Logistic models with slowly varying parameters

Fast-slow behaviors in the Logistic models with slowly varying parameters are revealed by using singular perturbation method. We first rewrite the Logistic models with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then we apply matching to obtain the approximate solutions, which are explicit and analytical and compare very well with the numerically integrated ones. More importantly, we prove the uniform validity of the approximate solutions rigorously and give the error estimate between the approximate solutions and the exact solutions via the way of upper and lower solutions.     

Energy Error Estimates for the Projection-Difference Method with the Crank–Nicolson Scheme for Parabolic Equations

We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.     

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.     

Superconvergence for rectangular mixed finite elements

Summary In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL ² between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.     

On the Existence of Solutions and the Convergence of Approximations to Scalar Conservation Laws

This paper deals with a new proof of the existence of weak solutions to scalar conservation laws. Our approach relies on the use of a particular finite difference scheme for time discretization which introduces a viscous term. The approximate solutions can be computed explicitly by solving a set of linear ordinary differential problems. We prove that they converge towards a weak solution which is, in a certain sense, unique and stable.     

On the sharpness of error bounds in connection with finite difference schemes on uniform grids for boundary value problems of ordinary differential equations

For linear two-point boundary value problems of ordinary differential equations, some convergence properties of approximate solutions Y_h obtained by standard finite difference schemes on uniform grids are discussed. By means of discrete Green's functions a representation of the error Y_h –Y in functional dependence on the exact solution Y is employed to prove the sharpness (with regard to the order) of well-known error estimates in terms of moduli of smoothness of derivatives of Y.     

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Preldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.