Simple and efficient continuous data assimilation of evolution equations via algebraic nudging

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.