Interfaces and Free Boundaries
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Stable discretization of scalar and constrained vectorial Perona–Malik equation
Sören Bartels (1) and Andreas Prohl (2)
(1) Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, BONN, GERMANY(2) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, TÜBINGEN, GERMANY
We survey recent results on analysis and numerics of the scalar Perona–Malik equation. A vectorial Perona–Malik equation is introduced to evolve unit vector fields for directional diffusion. For both cases, scalar and vectorial, fully discrete schemes are proposed which fulfill a discrete energy law, and satisfy a discrete sphere constraint in the vectorial case. Computational experiments are provided to illustrate quantitative behaviors, and compare with scalar total variation flow and heat flow of p-harmonic maps.
Keywords: Total variation, Perona–Malik, p-harmonic map, finite elements, full discretization, discrete energy law p-harmonic map, finite elements, full discretization, discrete energy law