•A new and robust numerical method based on use of isoparametric elements is proposed for solving general thermal and mechanical problems.•Only one element is used for each collocation node and the element can be freely formed by itself and surround nodes.•The bandwidth of the formed final system coefficient matrix is extremely narrow.•A new 21-node quadratic element for 3D problems is constructed in this paper for the first time. In this paper, a new numerical method, named as the Free Element Collocation Method (FECM), is proposed for solving general engineering problems governed by the second order partial differential equations (PDEs). The method belongs to the group of the collocation method, but the spatial partial derivatives of physical quantities are computed based on the isoparametric elements as used in FEM. The key point of the method is that the isoparametric elements used can be freely formed by the nodes around the collocation node. To achieve a narrow bandwidth of the final system of equations, elements with a central node are recommended. For this purpose, a new 21-node quadratic element for 3D problems is constructed for the first time. Attributed to the use of isoparametric elements, FECM can result in more stable results than the traditional collocation method. In addition, the elements can be freely formed by local nodes, FECM has the advantage of mesh-free methods to fit complicated geometries of engineering problems. A number of numerical examples of 2D and 3D thermal and mechanical problems are given to demonstrate the correctness and efficiency of the proposed method.