Adaptive mirrors are used in laser material processing machines in order to control the propagation of the raw beam through the beam-guiding system as well as the geometry of the focused beam. As modern machine concepts for laser material processing tend to operate at increasing working speed, fast and lightweight adaptive mirrors with excellent optical properties and a large range of radius variation are needed. In this paper, a new approach for a fast, compact and lightweight adaptive mirror is discussed. In the approach the deformation of a circular plate under pressure is used. The pressure is generated by a small piezoelectric actuator which presses against a small water layer behind the mirror plate, thereby deforming it. The deformation of the mirror surface is almost parabolic across a large area of the overall surface, which results in a high optical quality even at large deformations. Based on this concept, an adaptive mirror was designed that can be operated at a bandwidth up to 400 Hz at maximum deformation (which results in a range of radius-variation from -20 m to +20 m) and about 2 kHz at smaller deformations. Furthermore, extensive investigations have been performed using FEM analysis of the adaptive mirrors in order to understand the influence of mirror thickness, diameter and operating pressure on the deformation and the resulting radii of curvature. The result is an analytical method which allows to calculate the mirror dimensions in order to achieve a desired characteristic of deformation for given beam diameters.