Crank–Nicolson method in linear elasticity

[kamensky]

You don’t need to use a mixed space; you can formally eliminate the velocity unknowns at the current time level, and solve a problem in terms of displacement unknowns only. See my response to an older question here, which provides an implementation of the scalar wave equation:

Second order time discretization problem

You can convert to a first order system but then formally eliminate the extra velocity unknown to avoid solving in a mixed function space. The implicit midpoint rule gives you \frac{u^{n+1} - u^n}{\Delta t} = \frac{1}{2}(\dot{u}^{n+1} + \dot{u}^{n})~~~\Rightarrow~~~\dot{u}^{n+1} = 2(\frac{u^{n+1} - u^n}{\Delta t}) - \dot{u}^n which can be substituted into the left-hand side of \frac{\dot{u}^{n+1} - \dot{u}^n}{\Delta t} = F((u^{n+1} + u^{n})/2) to get a problem for u^{n+1} alone, given bot…