A unified construction of high order shape functions is given for all four classical energy spaces (H 1 , H(curl), H(div) and L 2 ) and for elements of "all" shapes (segment, quadrilateral, triangle, hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element. The shape functions are conforming, hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes. Expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape. The polynomial order is allowed to differ for each separate topological entity (vertex, edge, face or interior) in the mesh, so the shape functions can be used to implement local p adaptive finite element methods. Each topological entity may have its own orientation, and the shape functions can have that orientation embedded by a simple permutation of arguments.i

A spacetime discontinuous Petrov-Galerkin (DPG) method for the linear timedependent Schrödinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrödinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrödinger operator. Two variational formulations are proved to be well posed: a strong formulation (with no relaxation of the original equation) and a weak formulation (also called the "ultraweak formulation", which transfers all derivatives onto test functions). The convergence of the DPG method based on the ultraweak formulation is investigated using an interpolation operator. A stand-alone appendix analyzes the ultraweak formulation for general differential operators. Reports of numerical experiments motivated by pulse propagation in dispersive optical fibers are also included.Corresponding author: Sriram Nagaraj (sriram@ices.utexas.edu).

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