We propose a strongly coupled approach within the Arbitrary Lagrangian–Eulerian (ALE) framework for solving Fluid–Structure Interaction (FSI) problems in the context of inviscid compressible fluids and geometrical nonlinear structure. The coupling of the Finite Volume Method (FVM) for the fluid and the Finite Element Method (FEM) or Spectral Element Method (SEM) for the structure is achieved by introducing a third field of Lagrange multipliers (LM) at the fluid–structure interface. Although the use of LM for FSI problems has already been investigated in the literature, it remains poorly explored in the context of FEM/FVM or SEM/FVM coupling within ALE framework. In this context, we employ the Gravouil–Combescure method, which enables heterogeneous time steps between the fluid and structural subdomains. Originally developed for structural dynamics, it has been adapted to SPH-based FSI formulations. In addition, we use the Mortar method to handle non-conforming meshes at the fluid–structure interface. Initially introduced in a purely FEM structural context using subdomain decomposition, this method has recently been extended to fully FEM/FEM FSI coupling framework. The proposed approach yields a strongly coupled formulation that enforces the continuity of normal velocities at the interface, while preserving a partitioned treatment of the fluid and structure solvers, without the need of coupling sub-iterations - as classically done in the literature. It therefore provides an attractive compromise between loosely coupled partitioned approaches and monolithic formulations. Moreover, the use of heterogeneous time steps and non-conforming meshes allows for selective time step size and mesh refinement according to each subdomain. The method has been tested using various 2D and 3D cases and temporal schemes (third-order accurate Runge-Kutta and first-order accurate Backward Euler for the fluid subdomain and second-order accurate Newmark for the structure subdomain). It suits particularly well to coupling scenarios involving large, implicitly integrated structure with large time steps, combined with smaller explicit fluid time steps. Two-dimensional results obtained with conforming mesh and linear structural models have been previously presented. The present work focuses on an extension of the method towards SEM space discretization and geometrical non-linearities, as well as its application on challenging FSI situations.