Symplectic-mixed Finite Element Approximation of Linear Wave Equations

We apply mixed finite element approximations to the first-order form of the acoustic wave equation. Our semidiscrete method exactly conserves the system energy, and we show that with a symplectic Euler time discretization, our method exactly conserves a perturbed energy quantity that is positive-definite and equivalent to the actual energy under a CFL condition. In addition to proving optimal-order L8pL2q estimates for our methods, we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions.