Finite element approximation of multi-scale elliptic problems using patches of elements

Abstract

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented. Mathematics Subject Classifications (1991) 65N55 · 65N30 · 65N12

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

“…We emphasize here that the term "patch" will only refer to the concept of IGA patch in the following. Thus, the term patch will never refer to the local model (in opposition to [32] for example).…”

“…This strategy has been applied in FEM for the modelling of crack propagation [11], for the modelling of localized uncertainties [28], for 3D-plate coupling [29] and for nonlinear domain decomposition [30]. Let us note that this methodology, involving the coupling of a global model and a local model in an iterative manner, has similarities with some hierarchical global/local methods in FEM: for example, the Chimera method [31], the method of finite element patches [32], numerical zoom [33] or the hp − d method [34][35][36]. However, the difference of the strategy considered here is that the contribution of the global solution in the local area is totally replaced by the local solution while, in the hierarchical strategy, an approximate solution is sought as the sum of the global coarse contribution and a local fine one.…”

Dynamical iteration schemes for multiscale simulation in nanoelectronics

“…Notice that the sum in (1) need not be a direct sum, therefore the representation (3) could, in principle, be non unique. The description of the method we use to overcome this difficulty is beyond the the scope of this paper, we only wish to point out that an iterative method proposed in [2][3][4] would not be convenient in our context. We introduce the vector T of all the degrees of freedom involved in the definifion of T Hh…”

“…The presentation of such model, especially the part in [1] Sec.3, was quite abstract in the sense that the required discretization method was not described in detail, the present paper is intended to address this topic. The scheme we have chosen to adopt for the spatial discretization of our model is the method of Patches of Finite Elements (PFE) introduced in [2,3]. This method is based on the use of completely overlapping non-conforming meshes which has two main advantages for the application at hand:…”

On the use of XFEM within the Arlequin framework for the simulation of crack propagation

“…Moreover, the resulting problem in the s-method is solved globally. An approach similar to the s-method is one that involves the use of local meshes (also called patches), but whose resulting system of equations is solved by a relaxed Chimera algorithm [23,24] (see also the Schwarz algorithm (see e.g [37])). Our ultimate goal being the simulation of physically-based damage, localisations and crack propagation, we propose a novel and alternative approach, experienced in this paper on basic tests.…”