The accelerated pseudo-transient (APT) method is an efficient way to solve partial differential equations, particularly well-suited for parallel computing. This paper explores the APT method's effectiveness in solving elastic, viscoelastic, and hydromechanical problems, focusing on quasi-static conditions in 1D, 2D, and 3D. The study examines the best numerical settings for fast and accurate solutions. The paper shows how the APT method can handle complex problems in high-resolution models.