We introduce, analyze, and test an interpolation operator designed 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; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient 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 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.

中文翻译：

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通过代数微分对演化方程进行简单而有效的连续数据同化

我们介绍，分析和测试插值算子，该算子设计用于与有限元方法在空间上离散化的演化方程的连续数据同化（DA）。插值构造为L ²的近似值将运算符投影到粗网格上的分段常数函数上，但可以使用线性对数矩阵构造简单的对角线，而无需依赖离散化的其余部分，就可以完全在线性代数水平上进行微调；它甚至可以完全消除对显式构造粗网格的需要。我们证明了插值算子具有足够的稳定性和准确性，并将其应用于流体传输DA和不可压缩的Navier–Stokes DA的算法。对于这两种应用，我们证明了具有任意初始条件的DA解在时间上均以指数级速度快速收敛到真实解（直至离散化误差），因此具有长期精确性。给出了一些数值测试的结果，它们既说明了理论，又证明了其在实际问题上的有用性。