Piecewise polynomial shape functions for hp-finite element methods

High-order piecewise polynomial shape functions for quadrilateral finite elements are proposed. The shape functions are tensor products of one-dimensional piecewise functions based on Legendre polynomials. This allows the construction of H1H1 conforming approximations on meshes with arbitrary levels of hanging nodes without the necessity for constrained approximation techniques. Furthermore, meshes can be refined simply by inserting nodes on element edges. Application examples including the L-shape domain, a plate problem and a tie rod connection plate illustrate the applicability of the method.