We study equations to describe incompressible generalized Newtonian fluids, where the extra stress tensor satisfies a nonstandard anisotropic asymptotic growth condition. An implicit finite element discretization, and a simple, fully practical fixed-point scheme with proper thresholding criterion are proposed, and convergence towards weak solutions of the limiting problem is shown. Computational experiments are included, which motivate nontrivial fluid flow behavior.
Abstract. Weak solutions for nonlinear wave equations involving the p(x)-Laplacian, for p : Ω → (1, ∞) are constructed as appropriate limits of solutions of an implicit finite element discretization of the problem. A simple fixed-point scheme with appropriate stopping criteria is proposed to conclude convergence for all discretization, regularization, perturbation, and stopping parameters tending to zero. Computational experiments are included to motivate interesting dynamics, such as blowup, and asymptotic decay behavior.
Two numerical approximation schemes for minimising the Mumford-Shah functional for unit vector fields are proposed, analysed, and compared. The first uses a projection strategy, the second a penalisation strategy to enforce the sphere constraint. Both schemes are then applied to the segmentation of colour images using the Chromaticity and Brightness colour model.
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