Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. (English) Zbl 1124.65103
The author treats the construction of convergent finite difference methods for a class of nonlinear partial differential equations, the class of degenerate elliptic equations with unique viscosity solutions, already defined by M. G. Crandall, H. Ishii and P.-L. Lions [Bull. Am. Math. Soc., New Ser. 27, No. 1, 1–67 (1992; Zbl 0755.35015)].
To get convergence results, theorems about monotonicity and nonexpansivity in the maximum norm are provided and proved for the class of the developed schemes, the so-called degenerate elliptic schemes. The techniques involve schemes for Hamilton-Jacobi equations, free boundary problems and obstacle problems using building blocks of schemes for simpler equations. For time-dependent equations explicit schemes with a nonlinear Courant-Friedrichs-Lewy condition are presented.
To get convergence results, theorems about monotonicity and nonexpansivity in the maximum norm are provided and proved for the class of the developed schemes, the so-called degenerate elliptic schemes. The techniques involve schemes for Hamilton-Jacobi equations, free boundary problems and obstacle problems using building blocks of schemes for simpler equations. For time-dependent equations explicit schemes with a nonlinear Courant-Friedrichs-Lewy condition are presented.
Reviewer: Georg Hebermehl (Berlin)
MSC:
65N06 | Finite difference methods for boundary value problems involving PDEs |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
35B50 | Maximum principles in context of PDEs |
35R35 | Free boundary problems for PDEs |
35K65 | Degenerate parabolic equations |
49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
35J70 | Degenerate elliptic equations |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
35R25 | Ill-posed problems for PDEs |