  We introduce, analyze and test a new interpolation operator for use with
continuous data assimilation (DA) of evolution equations that are discretized
spatially with the finite element method. The interpolant is constructed as an
approximation of the L2 projection operator onto piecewise constant functions
on a coarse mesh, but which allows nudging to be done completely at the linear
algebraic level, independent of the rest of the discretization, with a diagonal
matrix that is simple to construct. We prove the new operator maintains
stability and accuracy properties, and we apply it to algorithms for both fluid
transport DA and incompressible Navier Stokes DA. For both applications we
prove the DA solutions with arbitrary initial conditions converge to the true
solution (up to optimal discretization error) exponentially fast in time, and
are thus long-time accurate. Results of several numerical tests are given,
which both illustrate the theory and demonstrate its usefulness on practical
problems.
