The Newton–Raphson method or a different fixed-point iteration can be used to solve FSI problems. Methods based on Newton–Raphson iteration are used in both the monolithic 
 and the partitioned  approach. These methods solve the nonlinear flow equations and the structural equations in the entire fluid and solid domain with the Newton–Raphson method. The system of linear equations within the Newton–Raphson iteration can be solved without knowledge of the Jacobian with a matrix-free iterative method, using a finite difference approximation of the Jacobian-vector product.
Whereas Newton–Raphson methods solve the flow and structural problem for the state in the entire fluid and solid domain, it is also possible to reformulate an FSI problem as a system with only the degrees of freedom in the interface’s position as unknowns. This domain decomposition condenses the error of the FSI problem into a subspace related to the interface. The FSI problem can hence be written as either a root finding problem or a fixed point problem, with the interface’s position as unknowns.
Interface Newton–Raphson methods solve this root-finding problem with Newton–Raphson iterations, e.g. with an approximation of the Jacobian from a linear reduced-physics model. The interface quasi-Newton method with approximation for the inverse of the Jacobian from a least-squares model couples a black-box flow solver and structural solver  by means of the information that has been gathered during the coupling iterations. This technique is based on the interface block quasi-Newton technique with an approximation for the Jacobians from least-squares models which reformulates the FSI problem as a system of equations with both the interface’s position and the stress distribution on the interface as unknowns. This system is solved with block quasi-Newton iterations of the Gauss–Seidel type and the Jacobians of the flow solver and structural solver are approximated by means of least-squares models.
The fixed-point problem can be solved with fixed-point iterations, also called (block) Gauss–Seidel iterations, which means that the flow problem and structural problem are solved successively until the change is smaller than the convergence criterion. However, the iterations converge slowly if at all, especially when the interaction between the fluid and the structure is strong due to a high fluid/structure density ratio or the incompressibility of the fluid. The convergence of the fixed point iterations can be stabilized and accelerated by Aitken relaxation and steepest descent relaxation, which adapt the relaxation factor in each iteration based on the previous iterations.
If the interaction between the fluid and the structure is weak, only one fixed-point iteration is required within each time step. These so-called staggered or loosely coupled methods do not enforce the equilibrium on the fluid–structure interface within a time step but they are suitable for the simulation of aeroelasticity with a heavy and rather stiff structure.
Several studies have analyzed the stability of partitioned algorithms for the simulation of fluid-structure interaction