We present a comprehensive a priori error analysis of a practical energy based atomistic/continuum coupling method (Shapeev, arXiv:1010.0512) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. The majority of the work is devoted to the analysis of consistency and stability of the method. These yield a priori error estimates in the H1-norm and th...

In this paper, the C-conforming finite element method is analyzed for a class of nonlinear fourth-order hyperbolic partial differential equation. Some a priori bounds are derived using Lyapunov functional, and existence, uniqueness and regularity for the weak solutions are proved. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Keywords—Nonlinear fourth-ord...

This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit sp...

In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence b...

Abstract. Optimal control problems with convex but non-smooth cost functional are considered. The non-smoothness arises from a L-norm in the objective functional, which recently attracted much research effort in the context of inverse problems. The problem is regularized to permit the use of semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Mo...

Abstract. Semilinear elliptic optimal control problems involving the L1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discret...

We continue our work on adaptive nite element methods with a study of time discretization of analytic semigroups. We prove optimal a priori and a posteriori error estimates for the discontinuous Galerkin method showing, in particular, that analytic semigroups allow long-time integration without error accumulation. 1. Introduction This paper is a continuation of the series of papers 1], 2], 3], ...

We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discertization with the θ -scheme, and eigenva...

We present a comprehensive a priori error analysis of a practical energy based atomistic/continuum coupling method (Shapeev, arXiv:1010.0512) in two dimensions, for finite-range pairpotential interactions, in the presence of vacancy defects. We establish first-order consistency and stability of the method, from which we a priori error estimates in the H-norm and the energy in terms of the mesh ...