Newton's method in constructing a singular set of a minimax solution in a class of boundary value problems for the Hamilton~--- Jacobi equations

P. D. Lebedev ; A. A. Uspenskii

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 1, pp. 63-76.

Résumé

The non-smooth features of the minimax (generalized) solution of the considered class of Dirichlet problems for equations of Hamiltonian type are due to the existence of pseudo-vertices — singular points of the boundary of the boundary set. The paper develops analytical and numerical methods for constructing pseudo-vertices and their accompanying constructive elements, which include local diffeomorphisms generating pseudo-vertices, as well as markers — numerical characteristics of these points. For markers, an equation with a characteristic structure inherent in equations for fixed points is obtained. An iterative procedure based on Newton's method for the numerical construction of its solution is proposed. The convergence of the procedure to the pseudovertex marker is proved. An example of the numerical-analytical construction of a minimax solution is given, illustrating the effectiveness of the developed approaches for constructing nonsmooth solutions of boundary value problems.

Détail

Keywords: Hamilton — Jacobi equation, minimax solution, speed, singular set, wave front, diffeomorphism, eikonal, pseudovertex.

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