We derive a priori error estimates for three prototypical energy-based quasicontinuum (QC) methods: the local QC method, the energy-based QC method, and the quasinonlocal QC method. Our analysis clearly decomposes the consistency error into modelling and coarsening errors. While previous results on estimating the modelling error exist, we give a new and simpler proof based on negative-norm esti...

In this article a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable a...

A priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models. Particular attention is devoted to the dependence of the estimates on a small parameter. For typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightfor...

We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of EngquistOsher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introducedbyTadmor [SIAMJ.Numer.Anal. 1991]. The error is related to the consistency error ...

In this article we introduce a new locking-free completely discontinuous formulation for Reissner–Mindlin plates that combines the discontinuous Galerkin methods with weakly over-penalized techniques. We establish a new discrete version of Helmholtz decomposition and some important residual estimates. Combining the residual estimates with enriching operators we derive an optimal a priori error ...