2007/12/17 by John W. Barrett, Xiaobing Feng, Barrett, John W. +3
Computer Science · Engineering · Mathematics · #35K65 #35Q80 #58E20 #65M12 #65M60 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · doi:10.48550/arxiv.0712.2528
openalex publication_date 2007/12/17 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Motivated by emerging applications from imaging processing, the heat flow of a generalized p-harmonic map into spheres is studied for the whole spectrum, 1≤ p<∞, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a BV-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms \cA⋅\nab\bv and \cA\wedge\nab\bv; which pair a divergence-L1, or a divergence-measure, tensor field \cA, and a BV-vector field \bv. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the p-harmonic map heat flow, and the convergence of the proposed numerical method is also established.