On the approximation of 3D hyperbolic boundary integral equations A. Temponi, A. Salvadori, F. Mordenti, A. Carini, P. Pelizzari CeSiA - Centro di studio e ricerca di sismologia applicata e dinamica strutturale, DICATA - Dipartimento di Ingegneria Civile, Architettura, Territorio e Ambiente Universit` a di Brescia, via Branze 43, 25123 Brescia, Italy E-mail: alessandro.temponi@ing.unibs.it, alberto.salvadori@ing.unibs.it Keywords: Hyperbolic problems, boundary integral equations, computational complexity. SUMMARY. The present note summarizes some new results for hyperbolic problems involving 3D scalar fields modeled by integral equations extensively reported in [1]. Classical approximation schemes as well as recently published energetic weak forms are considered; algorithms for the nu- merical solution are formulated adopting polynomial shape functions of arbitrary degree (in space and time) on trapezoidal flat tessellations of polynomial domains. Analytical integrations are per- formed both in space and time for Lebesgue integrals working in a local coordinate system; for singular integrals, both a limit to the boundary as well as the finite part of Hadamard approach have been pursued.. 1 INTRODUCTION Modeling hyperbolic problems by means of boundary integral equations (BIEs) and approximat- ing their solution through boundary element methods (BEM) is firmly established in the academic community as well as in industry. Such methods have been successfully used for decades in the propagation and scattering of acoustics, electromagnetic [2] and elastic waves [3]. Several modern research and applications topics are dealt with them: see [1] for a short review. The integral formulation of the scalar wave problem can be formulated (see e.g. [3, 4]) stemming from Graffi’s [5] generalization of steady state reciprocity theorem to dynamics. Under the hypoth- esis of vanishing initial conditions and no external body forces, the boundary integral representation (BIR) of the primal field u in the interior of the open domain Ω at time t reads: u(x,t)=  Σ G uu (r,t − τ )p(y,τ ) dΣ τ,y −  Σ G up (r, l(y),t − τ )u(y,τ ) dΣ τ,y (1) Here, Σ is the lateral boundary Σ = (0,T ) × Γ and r = x − y stands for the vector that joins point y to x. Identity (1) is based on Green’s functions (also called kernels) G uu and G up . An additional integral equation can be provided by the application of the co-normal derivative operator to identity (1): the BIR of the dual field p(x)= σ(x) · n(x) on a surface of normal n(x) in the interior of the domain, i.e. {t, x}∈ Σ turns out to be: p(x,t)=  Σ G pu (r, n(x),t − τ )p(y,τ )dΣ τ,y −  Σ G pp (r, n(x), l(y),t − τ )u(y,τ )dΣ τ,y (2) Such a BIR involves Green’s functions G pu and G pp . A set of two BIEs can be derived from BIRs (1) (thus obtaining the so-called “primal equation”) and (2) (thus obtaining the so-called “dual equation”) by performing the space boundary limit Ω ∋ x → x ∈ Γ. In the limit process, after integration in time, singularities of Green’s functions are 1