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.

翻译：连续数据同化通过将解的观测值融入"微调项"来处理初始条件未知的时变问题。针对具有可变导热系数和诺伊曼边界条件的典型热方程，我们考虑了与以往研究不同的非插值观测量数据同化方案。这些广义微调策略特别适用于在最小理论框架之外仅具有有限甚至无额外正则性的问题。我们证明，空间离散化的微调解随时间以指数速度收敛至真实解，其收敛速率由微调策略的选择保证，且与离散化方式无关。此外，长期离散误差是最优的，因为它与已知初始条件的有限正则性问题的现有估计相匹配。针对三个测试案例（包括狄拉克δ函数强迫问题和具有不连续导热系数的凯洛格问题），我们对三种具体策略——通过协调有限元子空间进行微调、通过边界网格上的分段常数进行微调以及通过平均值进行微调——进行了数值模拟研究。