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

TL;DR

This paper presents a new algebraic interpolation operator for continuous data assimilation in evolution equations, enabling stable, accurate, and efficient nudging at the linear algebraic level, applicable to fluid transport and Navier-Stokes equations.

Contribution

The paper introduces a novel algebraic interpolation operator for data assimilation that simplifies implementation and maintains stability and accuracy for evolution equations discretized with finite elements.

Findings

Proves exponential convergence of DA solutions to true solutions.

Demonstrates stability and accuracy of the new operator.

Shows practical effectiveness through numerical tests.

Abstract

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.