A-priori analysis of the quasicontinuum method in one dimension

The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we give an a-priori error analysis for the quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long range interactions and a practical QC formulation.\ud \ud First, we prove the existence, the local uniqueness and the stability with respect to discrete W1,∞-norm of elastic and fractured atomistic solutions. We then used a fixed point technique to prove the existence of quasicontinuum approximation which satisfies an optimal a-priori error bound.\ud \ud The first-named author acknowledges the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina (University of Pavia)

Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bound

Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class {of} polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an \emph{arbitrary} number of faces for polynomial basis with degree $p=2,3$. The key feature of the proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}_p$, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes

Finite element heterogeneous multiscale method for time-dependent Maxwell’s equation

We propose a Finite Element Heterogeneous Multiscale Method (FEHMM) for time dependent Maxwell’s equations in second-order formulation. This method can approximate the effective behavior of an electromagnetic wave traveling through a highly oscillatory material without the need to resolve the microscopic details of the material. To prove an a-priori error bound for the semi-discrete FE-HMM scheme, we need a new generalization of a Strang-type lemma for second-order hyperbolic equations. Finally, we present a numerical example that is in accordance with the theoretical results

An extended model order reduction technique for linear delay systems

\u3cp\u3eThis paper presents a model reduction technique for linear delay differential equations that, first, preserves the infinite-dimensional nature of the system, and, second, enables the preservation of additional properties such as physical interconnection structures or uncertainties. This structured/robust reduction of delay systems is achieved by allowing additional degrees of freedom in the characterization of (bounds on) controllability and observability energy functionals, leading to a so-called extended balancing procedure. In addition, the proposed technique preserves stability properties and provides an a priori error bound. The relevance of the method in controller reduction is discussed and illustrative numerical examples are presented.\u3c/p\u3