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Singular perturbations without limit in the energy space. Convergence and computation of the associated layers. (English) Zbl 1142.35325
Cioranescu, Doina (ed.) et al., Nonlinear partial differential equations and their applications. Collège de France seminar. Vol. XIV. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1997–1998. Amsterdam: Elsevier (ISBN 0-444-51103-2/hbk). Stud. Math. Appl. 31, 489-507 (2002).
The first part of this paper concerns one-dimensional singular perturbation problems \(P_{\varepsilon}\) of the form: \[ -\frac{d^2u_{\varepsilon}}{dx^2}+\varepsilon^2\frac{d^4u_{\varepsilon}}{dx^4}=f \] in \((0,1), u_{\varepsilon}(0)=u_{\varepsilon}(1)=u_{\varepsilon}'(0)=u_{\varepsilon}'(1)=0,\) in two cases: (1) \(f=\delta'(\frac{1}{2}),\) where \(\delta\) is the Dirac measure at \(x=\frac{1}{2},\) and (2) \(f(x)=x^{-p-2} + (1-x)^{-p-2}\), with \(p\in (0,\frac{1}{2}).\) For \(\varepsilon >0,\) the variational formulation of \(P_{\epsilon}\) makes sense. For \(\varepsilon =0,\) the limit problem is a Lions-Magenes problem in case \((1)\); in case \((2)\), the Lions-Magenes theory does not apply. In both cases, the asymptotic behavior of \(u_{\varepsilon}\), as \(\varepsilon \to 0,\) is investigated by means of the method of matched asymptotic expansions. The layer phenomena are intensively studied. These results were announced earlier by D. Leguillon, J. Sanchez-Hubert and É. Sanchez-Palencia in [C. R. Acad. Sci. Paris, Sér. IIB Méc. Phys. Astron. 327, No. 5, 485–492 (1999; Zbl 0932.35064)]. The last section of the paper is devoted to numerical computations for an analogous problem in dimension 2, namely: \(\varepsilon^{2}\Delta^{2}u_{\epsilon}-\Delta u_{\varepsilon}=f\) in \(\Omega=(0,1)\times(0,1)\), with suitable boundary conditions, when \(f=\delta'(C)\) on a curve \(C\), so that \(\langle f,v\rangle=-\int_{c}\frac{\partial v}{\partial n}\,ds\). An adapted mesh procedure is presented.
For the entire collection see [Zbl 0992.00032].
35B25 Singular perturbations in context of PDEs
34E15 Singular perturbations, general theory for ordinary differential equations
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs