Estimates of the discrete van Cittert deconvolution error in approximate deconvolution models of turbulence in bounded domains

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Recent turbulence models such as the Approximate Deconvolution Model (ADM) or the Leray-deconvolution model are derived from the Navier–Stokes equations using the van Cittert approximate deconvolution method. As a consequence, the numerical error in the approximation of the Navier–Stokes weak solution with discrete solutions of the above models is influenced also by the discrete deconvolution error u−DNu‾h caused by the approximate deconvolution method. Here u is the flow field, u‾ is its average and DN is the N-th order van Cittert deconvolution operator.It is therefore important to analyze the deconvolution error u−DNu‾h in terms of the mesh size h, the filter radius α and the order N of the deconvolution operators used in the computation.This problem is investigated herein in the case of bounded domains and zero-Dirichlet boundary conditions. It is proved that on a sequence of quasiuniform meshes the L2 norm of the discrete deconvolution error convergences to 0 in the order of hk+1+KNh provided that the filter radius is in the order of the mesh size and the flow field has enough regularity. Here K<1 is a parameter that depends on the ratio (filter radius)/(mesh size) (which is assumed constant) and k is the order of the FE velocity space. The rate of convergence of the H1 norm of the error is also provided. The estimates are valid for the differential and Stokes filters. The result proves that higher order deconvolution operators decrease the deconvolution error and can be used to increase accuracy in approximate deconvolution models of flow problems.