A SHARP ERROR ESTIMATE FOR THE NUMERICAL SOLUTION OF MULTIVARIATE DIRICHLET PROBLEM FOR THE HEAT EQUATION

Abstract

For the multidimensional Dirichlet problem of the heat equation on a cylinder, this study examines convergence properties with rates of approximate solutions, obtained by a naturally arising difference scheme over inscribed uniform grids. Sharp quantitative estimates are given by the use of first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is accomplished by using the probabilistic method an appropriate random walk.

Keywords:

• Primary: Dirichlet problem—continuous and discrete, heat equation, space-time Wiener process, space-time random walk, convergence with rates, first and second moduli of continuity, sharp inequality, approximate solution
• Secondary: Average operator, first exit time, uniform grid, Lipschitz class, parabolic function, cylinder
• 1991 AMS Subject Classification:
• Primary: 31C20, 35A40, 35G15, 41A17, 41A25, 60J15, 60J45, 65N15
• Secondary: 31B05, 35K20, 39A70, 41A63, 60J65, 65N06

Acknowledgments