  We aim at the development and analysis of the numerical schemes for
approximately solving the backward diffusion-wave problem, which involves a
fractional derivative in time with order $\alpha\in(1,2)$. From terminal
observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously
recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for
all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the
backward diffusion-wave problem were established under some conditions about
$T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value
scheme to regularize the "mildly" ill-posed problem, and show the convergence
of the regularized solution. Next, to numerically solve the regularized
problem, a fully discrete scheme is proposed by applying finite element method
in space and convolution quadrature in time. We establish error bounds of the
discrete solution in both cases of smooth and nonsmooth data. The error
estimate is very useful in practice since it indicates the way to choose
discretization parameters and regularization parameter, according to the noise
level. The theoretical results are supported by numerical experiments.
