Application of nonconforming quadtree grids in the finite element method

In the standard finite element method, the edges of the adjacent elements are aligned to each other, and the corner of an element does not located on the edges of another one. If this constraint is violated it is said that the mesh is non-conforming and the use of such meshes in the finite element method requires specific techniques. In the present paper, a new method is suggested for the use of non-conforming meshes which are generated using quadtree algorithm. Quadtree is a data structure that has a very fast recursive algorithm and can be used to divide a two-dimensional domain into sub-regions or elements. The elements produced by this method are non-conforming and are not usable in the standard finite element method. In general, six types of elements with different arrangement of nodes may be appeared in a quadtree mesh and the conventional methods which are using the polynomials cannot be used to derive the shape functions. In the present paper, a new idea is proposed to obtain the shape functions of such elements. In this method, a boundary value problem is defined and its solution is used as the shape functions of the non-conforming elements. To evaluate the applicability and accuracy of the proposed method, two numerical examples are solved and the results are presented. The results show that the proposed method can be used to effectively apply the non-conforming meshes in the finite element method.