Boujemâa Achchab

Strengthened Cauchy–Bunyakowski–Schwarz inequality for a three‐dimensional elasticity system

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three-dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large-scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real-life elasticity problems. Copyright

Estimate of the Constant in Two Strengthened C.B.S. Inequalities for F.E.M. Systems of 2D Elasticity: Application to Multilevel Methods and a Posteriori Error Estimators

The constant γ in the strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems. We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ2≤ 3/4

A posteriori error estimates for subgrid viscosity stabilized approximations of convection–diffusion equations☆

uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2

A posteriori error analysis using the constitutive law for the Crouzeix–Raviart element☆

Abstract We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Peclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Peclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.

Upper Bound of the Constant in Strengthened C.B.S. Inequality for Systems of Linear Partial Differential Equations

Abstract In this work, we study the error in the approximation of the solution of elliptic partial differential equations obtained with the nonconforming finite elements method; we adopt the error in a constitutive law approach.

Robust a posteriori error estimates for subgrid stabilization of non-stationary convection dominated diffusive transport

The constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is the framework of finite element approximations of systems of partial differential equations. We consider an approximation of general systems of linear partial differential equations in R3. Concerning a multilevel convergence rate corresponding to nested general tetrahedral meshes of size h and 2h, we give an estimate of this constant for general three-dimensional cases.

On a posteriori error estimator for primal, equilibrium and mixed approximation of diffusion equations

Abstract We derive a robust residual a posteriori error estimator for time-dependent convection–diffusion–reaction problem, stabilized by subgrid viscosity in space and discretized by Crank–Nicolson scheme in time. The estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time. Numerical experiments illustrate the theoretical performance of the error estimator.

Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes

In this paper, we present an a posteriori error estimator for diffusion equation with the solution obtained by Black-box solver. The estimator is valid for classical primal, equilibrium and mixed finite element approximations with or without numerical integration.