ch-ReconstructionSparse.tex

% \include{ch-ReconstructionSparse}

assimilation step, but having different meshes requires a projection of the vorticity to the WLSFEM mesh. It is also important to note that the WLSFEM does not place any requirements on the finite element space or smoothness for the given vorticity field. Previous studies have shown that when solving the Navier-Stokes equations with many common least-squares finite element methods, the standard LSFEM approach (or even a WLSFEM approach) can lead to an approximate solution with poor mass conservation depending on the method used to rewrite the momentum balance as a system of first-order equations and depending on the finite element approximation space that is used $[21,26,28,29]$. A number of approaches have been developed to alleviate poor mass conservation, including the use of higher-order polynomial basis functions, but these can lead to higher computational costs for some problems $[16,18$, 22, 26]. The approach demonstrated here largely avoids the mass conservation challenge because the Navier-Stokes equations are solved using alternative discretization methods to WLSFEM. The new approach is also computationally cheaper compared to previous WLSFEM approaches used for data assimilation problems $[9,12,15]$.

    \input{/home/chaztikov/git/modulus_examples/fosls_examples/utils/regularized_ldc_2d/l2_relative_error_u_128_10_adam.tex}