With the in-depth development of mechanical research towards interdisciplinary and cross-scale correlation, strong nonlinearity caused by extreme environment, intrinsic instability, and coupling/correlation of different factors/scales has become a significant mathematical feature of this new type of mechanical problems, which inevitably poses severe challenges to related researches on quantitative modelling and solution. In view of this situation, based on the multi-resolution analysis theory of function space decomposition, using the mathematical framework of wavelet analysis, and starting from the construction of the underlying basis function, we put forward a highly efficient and high-precision approximation theory for numerical approximation of general functions, their derivatives and integrals, and developed a new system of numerical methods for solving general strong nonlinear initial and boundary value problems in mechanics. The proposed solution methods have been extensively and successfully verified in the solution of strong nonlinear mechanics problems with different characteristics.