2018/10/08 by Leo G. Rebholz, Rebholz, Leo G., Camille Zerfas +1 · 2 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1810.03512
openalex publication_date 2018/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce, analyze and test a new interpolation operator for use with\ncontinuous data assimilation (DA) of evolution equations that are discretized\nspatially with the finite element method. The interpolant is constructed as an\napproximation of the L2 projection operator onto piecewise constant functions\non a coarse mesh, but which allows nudging to be done completely at the linear\nalgebraic level, independent of the rest of the discretization, with a diagonal\nmatrix that is simple to construct. We prove the new operator maintains\nstability and accuracy properties, and we apply it to algorithms for both fluid\ntransport DA and incompressible Navier Stokes DA. For both applications we\nprove the DA solutions with arbitrary initial conditions converge to the true\nsolution (up to optimal discretization error) exponentially fast in time, and\nare thus long-time accurate. Results of several numerical tests are given,\nwhich both illustrate the theory and demonstrate its usefulness on practical\nproblems.\n