Continuous data assimilation for problems with limited regularity using non-interpolant observables

Continuous data assimilation addresses time-dependent problems with unknown initial conditions by incorporating observations of the solution into a nudging term. For the prototypical heat equation with variable conductivity and the Neumann boundary condition, we consider data assimilation schemes with non-interpolant observables unlike previous studies. These generalized nudging strategies are notably useful for problems which possess limited or even no additional regularity beyond the minimal framework. We demonstrate that a spatially discretized nudged solution converges exponentially fast in time to the true solution with the rate guaranteed by the choice of the nudging strategy independent of the discretization. Furthermore, the long-term discrete error is optimal as it matches the estimates available for problems of limited regularity with known initial conditions. Three particular strategies -- nudging by a conforming finite element subspace, nudging by piecewise constants on the boundary mesh, and nudging by the mean value -- are explored numerically for three test cases, including a problem with Dirac delta forcing and the Kellogg problem with discontinuous conductivity.