Two-dimensional variable-node elements compatible with quadratic interpolation are developed using the moving least-square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. It is shown that the present MLS-based variable-node elements meet the patch test if a sufficiently large number of integration points are employed for numerical integration. The cantilever problem with non-matching meshes is chosen to check the feasibility of the present MLS-based variable-node elements, and the result is compared with that from the lower-order case compatible with linear interpolation. Copyright (c) 2005 John Wiley & Sons, Ltd.