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This paper introduces a new, reduced-order NS-α (rNS-α) model for the purpose of efficient, stable, and accurate simulations of incompressible flow problems on coarse meshes. We motivate the new model by discussing the difficulties in efficient and stable algorithm construction for the usual NS-α model, and then derive rNS-α by using deconvolution as an approximation to the filter inverse, which reduces the fourth order NS-α formulation to a second order model. After proving the new model is well-posed, we propose a C0 finite element spatial discretization together with an IMEX BDF2 timestepping to create a linearized algorithm that decouples the conservation of mass and momentum equations from the filtering. We rigorously prove the algorithm is well-posed, and provided a very mild timestep restriction, is also stable and converges optimally to the model solution. Finally, we give results of several benchmark computations that confirm the theory and show the proposed model/scheme is effective at efficiently finding accurate coarse mesh solutions to flow problems. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H log (H) where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space. [ABSTRACT FROM AUTHOR]