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In recent years, two new numerical methods were proposed based on a direct element differentiation scheme, which are called as element differential method (EDM) [1] and free element method (FrEM) [2]. The former is a type of the finite element method (FEM) and the latter belongs to the category of the mesh free method (MFM). Comparing to the conventional FEM, EDM has the advantage that no variational principles are needed to set up a solution scheme and the system of equations are established based on the conservative (balance) condition of fluxes. FrEM is a very robust method, which has the distinct feature that only one independent local element is needed for each point, and the element is completely free, that is, the element not only can be formed by freely selecting certain surround nodes, but also is not connected to any other points’ elements. This feature can make the mesh generation much easy, and give us the convenience to form the global system without need to consider the C1-continuity of physical variables across the elements’ interfaces as required in the standard FEM.
This study presents some recent advances made in both EDM and FrEM, which can improve their stability, accuracy, and efficiency. The first is the use of the discontinuous elements in both methods to make the algorithms rigorous in mathematics and easy to keep the conservativeness of fluxes as well as to embody the discontinuity of some variables at boundary corners or edges. The second advance is the introduction of the zonal technique to FrEM, which can make FrEM easy to utilize high order isoparametric elements to improve the stability and accuracy. A number of numerical examples will be given to demonstrate the improvements made in both methods.