A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

Abstract

SUMMARYIn this paper, we combine the usual ÿnite element method with a Dirichlet-to-Neumann (DtN) mapping, derived in terms of an inÿnite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non-linear incompressible 2d-elasticity. We show that the variational formulation can be written in a Stokes-type mixed form with a linear constraint and a non-linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea-type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding ÿnite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given.

A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate

“…To this end, we follow the method proposed in [12] (which extends previous results given in [9], [2] and [3]) and combine it with the ideas from [14] and [20] (for dealing with nonlinear operators).…”

“…In order to illustrate our results, we consider the PEERSspace introduced in [7] as a particular example. In addition, we follow [12] and [13], together with the ideas from [14] and [20], to derive a reliable aposteriori error estimate for the resulting Galerkin scheme. We remark that the extensions to our dual-dual case of the a-posteriori error estimates of explicit residual type or of those given in terms of hierarchical bases, have not been obtained yet.…”

Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries

“…Du and Yu ( 2001 ), Hu and Yu ( 2001 ), Gatica et al. ( 2003 ), Koyama ( 2007 ), Koyama ( 2009 ), Das and Mehrmann ( 2016 ), Das and Natesan ( 2014 ), Das ( 2015 ) and references therein present the applications of this methods.…”

Implementing FEM-DtN for an incompressible material on an unbounded domain

“…This article presents a procedure for studying the Galerkin approximation to an incompressible material on unbounded exterior domain Ω. This procedure used Dirichlet-to-Neumann mapping (DtN), (Han and Bao, 1997;Gatica, Gatica and Stephan, 2003;Kako and Touda, 2004), consisting of introducing an artificial boundary by drawing circumference Γ in ℝ with radius in domain Ω; Ω was then divided in bounded part Ω and unbounded part Ω . Exact and approximate boundary conditions were given on artificial boundary Γ to solve the problem on bounded domain (Figure 1).…”