This paper deals with a fluid-solid interaction problem inspired by a biomechanical brain model. The problem consists of determining the response to prescribed static forces of an elastic solid containing a barotropic and inviscid fluid at rest. The solid is described by means of displacement variables, whereas displacement potential and pressure are used for the fluid. This approach leads to a well posed symmetric mixed problem, which is discretized by standard Lagrangian finite elements of arbitrary order for all the variables. Optimal order error estimates in H1 and L2 norms are proved for this method. A residual a posteriori error estimator is also proposed, for which efficiency and reliability estimates are proved. Finally, some numerical tests are reported to assess the performance of the method and that of an adaptive scheme based on the error estimator. Keywords: fluid-solid interaction, biomechanical brain model, finite elements, a priori and a posteriori error estimates, adaptive scheme.