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# Singular Perturbations with Nonsmooth Limit and Finite Element

DOI link for Singular Perturbations with Nonsmooth Limit and Finite Element

Singular Perturbations with Nonsmooth Limit and Finite Element book

# Singular Perturbations with Nonsmooth Limit and Finite Element

DOI link for Singular Perturbations with Nonsmooth Limit and Finite Element

Singular Perturbations with Nonsmooth Limit and Finite Element book

## ABSTRACT

J.SANCHEZ-HUBERT Laboratoire de Mécanique, Université dc Caen. Caen, France E.SANCHEZ PALENCIA Laboratoire de Modélisation en Mécanique, Université Paris

VI, Paris, France

1Introduction

We consider perturbation problems depending on a small parameter ε ∈(0, 1] for ε>0 which are variational problems in the energy space V. The energy space of the limit problem as ε 0 is V a V. The corresponding duals are such that . When the

right hand side classical singular perturbation theory shows that the solution u ε converges to the solution u 0 of the limit problem. We are here concerned with the case

but . This situation often appears in shell theory where the space V a ′ is, in some cases, so small that it does not contain usual loadings. In this case obviously the limit problem does not make sense as a variational problem. Nevertheless it may be shown [3] that the solutions u ε converge in some abstract topology which goes out of the variational framework. In usual partial differential equations, the convergence may not take place in the distribution sense. Two examples of such a situation in one space dimension are handled in Sect.3. In the first one (section 3.1) the limit exists in the distribution sense but the limit solution is not variational. In the second one (section 3.2) there is no limit in the usual sense (even as distributions). In any case when the energy of the solution uε tends to infinity as ε 0 (see theorem 2). It will be seen from examples that in such cases the energy is concentrated in boundary or internal layers. In fact a part of the present work is concerned with the asymptotic description of the boundary and internal layers using the classical formal method of matched asymptotic expansions (see for instance [17]). We shall not give here a complete description of the asymptotic expansions which may be found in [10] and [9].