In this paper interpolation techniques based on unstructured adaptive cartesian mesh are studied. It is very difficult that shape functions for Finite Element Method (FEM) directly are used for interpolation because the adaptive cartesian mesh have n-faced cells called polyhedral elements. we firest present an overview of Natural Neighbor Interpolation in Meshless FEM. and then it show that non-Sibsonian Interpolation which is more simple to implement is applied on the adaptive cartesian mesh.the non-Sibsonian method use area or volume rate of neighbor cells that is based on Voronoi cells. Futhermore we consider high-order interpolation based on non-Sibsonian method that is related to Harmonic Interpolation. finally the interpolations are applied to problem that is particle motion in prescribed magnetic fields on adaptive cartesian mesh. and we review performance and accuracy of the interpolations.