  The finite-element approach to lattice field theory is both highly accurate
(relative errors $\sim1/N^2$, where $N$ is the number of lattice points) and
exactly unitary (in the sense that canonical commutation relations are exactly
preserved at the lattice sites). In this talk I construct matrix elements for
dynamical variables and for the time evolution operator for the anharmonic
oscillator, for which the continuum Hamiltonian is $H=p^2/2+\lambda q^4/4.$
Construction of such matrix elements does not require solving the implicit
equations of motion. Low order approximations turn out to be extremely
accurate. For example, the matrix element of the time evolution operator in the
harmonic oscillator ground state gives a result for the anharmonic oscillator
ground state energy accurate to better than 1\%, while a two-state
approximation reduces the error to less than 0.1\%.
