In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a partial differential equation (or, more generally, of a differential inequality) in a domain D are said to satisfy the maximum principle if they achieve their maxima at the boundary of D. Harmonic functions and, more generally, solutions of elliptic partial differential equations satisfy the maximum principle.
The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations.
