The theory of approximate energy conservation for symplectic discretizations of ordinary differential equations (ODEs) is a well-established research field, but poses a challenging open problem for partial differential equations (PDEs). We are extending our work on exponential error estimates for approximate momentum conservation of discretized semilinear wave equations to study approximate energy conservation of structure preserving discretizations of PDEs in continuum mechanics. We expect that the techniques we use can also be applied to locking phenomena of discrete travelling waves, to error estimates for almost invariant slow manifolds of PDEs and to infinite order homogenization of nonlinear elliptic PDEs. We also work on fractional trajectory and energy error estimates for symplectic time semidiscretizations of semilinear Hamiltonian PDEs with non-smooth initial data.