The present study is related to the utilization of the mixed Meshless Local PetrovGalerkin (MLPG) methods for solving problems in gradient elasticity, which are governed by fourth-order differential equations. Here, three different numerical MLPG methods are presented, where the continuity requirements for the approximation functions are lowered by applying different mixed procedures to improve the numerical accuracy and efficiency. The first one is based on the direct solution of the problem, where the primary variable (displacement) and its independently chosen higher-order variables are approximated separately. The global discretized system of equations consists of appropriate equilibrium and compatibility equations written for each node and the solution vector contains all unknown independent nodal variables. Such approach demands only the first-order continuity of meshless approximation functions. The second and third procedures are both based on the displacement-based operator-split approach, where the original gradient elasticity problem is solved as two uncoupled problems governed by the second-order differential equations. Herein, in both uncoupled problems only primary variable (displacement) and its first derivative (strain) are approximated independently. In these procedures the original problem is solved by a staggered approach, where the solution of the first uncoupled equation is utilized as an input in the second equation. The main difference in the second and third procedure is that the one is based on the solution of the local weak forms of the governing equations, while the other is based on solution of the strong forms of the same equations. The accuracy of the presented computational methods is compared to analytical solutions and demonstrated on a one-dimensional benchmark problem of axial bar in gradient elasticity.
