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This article develops, analyzes, and tests a finite element method for approximating solutions to the Leraydeconvolution regularization of the Navier-Stokes equations. The scheme combines three ideas to create an accurate and effective algorithm: the use of an incompressible filter, a linearization that decouples the velocity-pressure system from the filtering and deconvolution operations, and a stabilization that works well with the linearization. A rigorous and complete numerical analysis of the scheme is given, and numerical experiments are presented that show clear advantages of the scheme.

SUMMARY We study the adaptive nonlinear filtering in the Leray regularization model for incompressible, viscous Newtonian flow. The filtering radius is locally adjusted so that resolved flow regions and coherent flow structures are not ‘filtered out’, which is a common problem with these types of models. A numerical method is proposed that is unconditionally stable with respect to time step and decouples the problem so that the filtering becomes linear at each time step and is decoupled from the system. Several numerical examples are given that demonstrate the effectiveness of the method. Copyright © 2012 John Wiley & Sons, Ltd.

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