Mathematical and Numerical Study of Transient Wave Scattering by Obstacles with a New Class of Arlequin Coupling

Abstract

In this work we extend the Arlequin method to overlapping domain decomposition technique for transient wave equation scattering by obstacles. The main contribution of this work is to construct and analyze from the continuous level up to the fully discrete level some variants of the Arlequin method. The constructed discretizations allow to solve wave propagation problems while using non-conforming and overlapping meshes for the background propagating medium and the surrounding of the obstacle respectively. Hence we obtain a flexible and stable method in terms of the space discretization -an inf-sup condition is proven -while the stability of the time discretization is ensured by energy identities.

Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

“…For the sake of completeness we now give the proof of Theorem 2.1 that is rather standard and is inspired from the one given in [20].…”

“…However the setting we consider is rather general and for instance elastodynamics equations could also enter the abstract framework by writing transmission problems (continuity of displacements and stresses) and using vectorial forms of all the spaces and scalar products introduced. Moreover the abstract structure is also adapted to the domain decomposition method with overlapping introduced in [20] to deal with scattering problems in transient acoustics.…”

“…We do not provide here proofs of existence and uniqueness for such problems -they rely on energy analysis and/or Laplace transform -and refer, for instance to the work of [33] and [20] for related analysis. See also the Appendix 8 of this manuscript for more details on the matter.…”

“…In fact, ( 10) is not a consequence of ( 6) and this question requires dedicated analysis. On this specific topic, we refer the reader to [34] and [35] for reference work concerning the mortar finite element method, that is a domain decomposition method without overlapping, and to [20], for a work concerning a domain decomposition method with overlapping.…”

“…This is clearly in improvement compared to the stability condition (20). Finally, in practice, to ensure that (66) holds one can choose α ≤ 1, set ∆t = 2α/ √ ρ c,h and set P p (x) ≡ P p, (x) where is the smallest integer such that α 2 ρ f,h ≤ β 2 ρ c,h .…”