We present the optimal local truncation error method (OLTEM) for PDEs (e.g., see [1-2]). Similar to the finite difference method, the structure and the width of discrete equations are assumed. The unknown coefficients of the discrete system are calculated by minimization of the order of the local truncation error. The boundary and interface conditions on irregular geometry are imposed as constraints without the introduction of unknowns on boundaries and interfaces. OLTEM provides an optimal high accuracy of discrete equations on trivial unfitted Cartesian meshes (no need in complicated mesh generators) for irregular domains and interfaces (composite materials). OLTEM rigorously calculates the lumped mass matrix for time dependent PDEs. A new OLTEM post-processing procedure for the calculation of the spatial derivatives (e.g., stresses or heat fluxes) that is based on the use of original PDEs significantly increases the accuracy of the spatial derivatives (it is applicable to FEM). For example, 9-th order of accuracy for stresses is obtained by OLTEM with ‘quadratic elements’ for elastostatics [2]. Currently, OLTEM has been applied to the solution of the wave, heat, elastodynamics, Helmholtz, Poisson, Stoke’s and elastostatics equations. The theoretical and numerical results show that at the computational costs of linear finite elements, OLTEM yields the 4th order of accuracy for the considered scalar PDEs on irregular domains and interfaces. At the computational costs of quadratic finite elements, OLTEM yields 10th order of accuracy for elastostatics and 11th order of accuracy for the Poisson equation with complex irregular interfaces (see [1,2]), i.e., the increase in accuracy by 7 and 8 orders compared to FEM. OLTEM reduces the computation time by a factor of 103 - 106 and more [1-2] compared to existing methods and will be effective for the solution of PDEs with stationary and evolving geometry (e.g., crack propagation) on stationary unfitted meshes. [1] Idesman A. V., Optimal Local Truncation Error Method for Solution of Partial Differential Equations on Irregular Domains and Interfaces Using Unfitted Cartesian Meshes: Review, Archives of Computational Methods in Engineering, 2023 30, 4517–4564. [2] Idesman, A., Mobin M., Bishop J., 10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes. Computational Mechanics, 2025, 76, pp. 1085–1115.