  In conventional spectral/finite element methods, the
triangulation/quadrilateralization of the domain produces many interior edges
which require additional DOF. What if we could directly use the original hull
without going to triangulation/quadrilateralization? There are three major
difficulties with this type of approach that are addressed: 1- How can a convex
hull tessellation be obtained for complex geometries encountered in practical
engineering applications? 2- How can basis functions and quadrature points be
defined on these hulls? 3- How can this type of grid and corresponding basis
functions be used in practice to yield accurate discretization of nonlinear
conservation laws? Geometrical approaches to tackle the first challenge are
discussed . To solve the second challenge, for the first time in the
literature, a closed form relation is proposed to approximate Fekete points (a
Nondeterministic Polynomial (NP) problem) on a general convex/concave
polyhedral. The approximate points are used to generate basis functions using
the SVD of the Vandermonde matrix. The type of basis functions derived include
Lagrange basis, orthogonal and orthonormal hull basis and radial basis. It is
shown that the hull basis is the best choice to enforce minimum DOF while
maintaining a small Lebesgue constant when very high p-refinement is done. The
proposed hull basis is rigorously proven to achieve arbitrary order of accuracy
by satisfying Weierstrass approximation theorem in $\mathbb{R}^d$. The third
challenge is met using a standard Discontinuous Galerkin (DG) and Discontinuous
Least-Squares (DLS) spectral hull formulations. The accuracy and stability of
the formulation is demonstrated for the linearized acoustics and
two-dimensional compressible Euler equations on some benchmark problems
including a cylinder, airfoil, vortex convection and compressible vortex
shedding from a triangle.
