Optimal Estimate for the Numerical Solution of Multidimensional Dirichlet Problem for the Heat Equation

Abstract

For the multidimensional Dirichlet problem of the heat equation on a cylinder, this chapter examines convergence properties with rates of approximate solutions, obtained by a naturally arising difference scheme over inscribed uniform grids. Sharp quantitative estimates are presented by the use of first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is achieved by using the probabilistic method of an appropriate random walk. This chapter is based on [64].