EVOLUTION EQUATIONS AND doi:10.3934/eect.2019024 CONTROL THEORY Volume 8, Number 3, September 2019 pp. 489–502 A DYNAMIC PROBLEM INVOLVING A COUPLED SUSPENSION BRIDGE SYSTEM: NUMERICAL ANALYSIS AND COMPUTATIONAL EXPERIMENTS Marco Campo Departamento de Matem´aticas, Universidade da Coru˜ na ETS de Ingenieros de Caminos, Canales y Puertos, Campus de Elvi˜ na 15071 A Coru˜ na, Spain Jos´ e R. Fern´ andez * Departamento de Matem´ atica Aplicada I, Universidade de Vigo ETSI Telecomunicaci´on, Campus As Lagoas Marcosende s/n 36310 Vigo, Spain Maria Grazia Naso Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica Universit` a degli Studi di Brescia Via Valotti 9, 25133 Brescia, Italy (Communicated by Josef Malek ) Abstract. In this paper we study, from the numerical point of view, a dy- namic problem which models a suspension bridge system. This problem is written as a nonlinear system of hyperbolic partial differential equations in terms of the displacements of the bridge and of the cable. By using the respec- tive velocities, its variational formulation leads to a coupled system of parabolic nonlinear variational equations. An existence and uniqueness result, and an exponential energy decay property, are recalled. Then, fully discrete approx- imations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property is shown and a priori error estimates are proved, from which the linear convergence of the algorithm is de- duced under suitable additional regularity conditions. Finally, some numerical results are shown to demonstrate the accuracy of the approximation and the behaviour of the solution. 1. Introduction. During the last decades the study of the so-called suspension bridges has received a large attention because this kind of bridges is a common type of civil engineering structure. It is well-known that these bridges may display certain oscillations under external aerodynamic forces like, for instance, it occurred in the famous Tacoma’s bridge (see [2, 5]), in which a strong wind caused the collapse of a narrow and very flexible suspension bridge. 2000 Mathematics Subject Classification. Primary: 74B20, 65M60, 65M15; Secondary: 74K10, 74H15. Key words and phrases. Coupled bridge system, finite elements, a priori error estimates, nu- merical simulations. This work has been supported by Ministerio de Econom´ ıa y Competitividad under the project MTM2015-66640-P (with the participation of FEDER). * Corresponding author: Jos´ eR.Fern´andez. 489