Asymptotic expansion for Stokes prturbed problems - Évaluation of singular integrals in Electromagnetism.

Abstract : This thesis contains three main parts. The first part concerns the derivation of an asymptotic expansion for the solution of Stokes resolvent problem with a small perturbation of the domain. Firstly, we verify the continuity of the solution with respect to the small perturbation via the stability of the density function. Secondly, we derive the asymptotic expansion ofthe solution, after deriving the expansion of the density function. The procedure is based on potential theory for Stokes problem in connection with boundary integral equation method, and geometric properties of the perturbed boundary. The main objective of the second part on this report, is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbationsof the boundary. Also, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the Stokes eigenvalues due to interface changes of the inclusion. The implementation of the boundary element method requires the evaluation of integrals with a singular integrand. A reliable andaccurate calculation of these integrals can in some cases be crucial and difficult. In the third part of this report we propose a method of evaluation of singular integrals based on recursive reductions of the dimension of the integration domain. It leads to a representation of the integralas a linear combination of one-dimensional integrals whose integrand is regular and that can be evaluated numerically and even explicitly. The Maxwell equation is used as a model equation, but these results can be used for the Laplace and the Helmholtz equations in 3-D.For the discretization of the domain we use planar triangles, so we evaluate integrals over the product of two triangles. The technique we have developped requires to distinguish between several geometric configurations.
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Imen Balloumi. Asymptotic expansion for Stokes prturbed problems - Évaluation of singular integrals in Electromagnetism.. Algebraic Geometry [math.AG]. Université de Cergy Pontoise; Université de Carthage (Tunisie), 2018. English. ⟨NNT : 2018CERG0958⟩. ⟨tel-02284054⟩

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