Error Estimates For Numerical Approximation Research Articles

Well-posedness and error estimates for coupled systems of nonlocal conservation laws

Abstract This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: (1) Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform $\operatorname {BV}$ bound on the numerical approximations; (2) Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; (3) Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.

Error Estimates for Numerical Approximation of Hamilton–Jacobi Equations Related to Hybrid Control Systems

Hybrid control systems are dynamical systems that can be controlled by a combination of both continuous and discrete actions. In this paper we study the approximation of optimal control problems associated to this kind of systems, and in particular of the Quasi-Variational Inequality which characterizes the value function. Our main result features the error estimates between the value function of the problem and its approximation. We also focus on the hypotheses describing the mathematical model and the properties defining the class of numerical scheme for which the result holds true.

Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.

Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems

Convection-diffusion problems of practical interest are often convection-dominated in the sense that the diffusion parameter e satisfies 0<e≪1. Standard error estimates for numerical approximations of such problems mostly contain constants which depend either reciprocally on e or at least on higher order norms of the solution. These norms possibly increase to infinity in the hyperbolic limit e→0. For the Lagrange-Galerkin scheme such error estimates have been proven by Douglas and Russell, Pironneau or Suli. In the particular case of the Navier-Stokes equations Johnson, Rannacher and Boman have observed that some of those constants which arise in existing analyses depend exponentially on the Reynolds number. Consequently, standard error analyses have no real meaning in the case of convection-dominance. Recently, Bause and Knabner have developed a rigorous e-uniform convergence theory for finite element and Lagrange-Galerkin approximations of convection-dominated diffusion problems. The error analysis is heavily based on e-uniform a priori estimates for the solution of the continuous problems. In their work such e-uniform estimates are established in the Sobolev spacesL2(Ω) andH2(Ω) in a Lagrangian framework by transforming the convection-diffusion problem into subcharacteristic coordinates. To obtain the e-uniformH2-bound, strong conditions about the right-hand side which cannot realistically be assumed in practice are supposed. In this paper we demonstrate how the method of complex interpolation of Banach spaces can be employed to derive elegantly e-uniform a priori estimates in intermediate Sobolev spaces of fractional order. We will observe that in particularHα-regularity of the data has to be supposed to get an e-uniformHα-bound for the solution. To derive the desired estimates we use several complex interpolation results which we provide first.