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In conventional spectral/finite element methods, thetriangulation/quadrilateralization of the domain produces many interior edgeswhich require additional DOF. What if we could directly use the original hullwithout going to triangulation/quadrilateralization? There are three majordifficulties with this type of approach that are addressed: 1- How can a convexhull tessellation be obtained for complex geometries encountered in practicalengineering applications? 2- How can basis functions and quadrature points bedefined on these hulls? 3- How can this type of grid and corresponding basisfunctions be used in practice to yield accurate discretization of nonlinearconservation laws? Geometrical approaches to tackle the first challenge arediscussed . To solve the second challenge, for the first time in theliterature, a closed form relation is proposed to approximate Fekete points (aNondeterministic Polynomial (NP) problem) on a general convex/concavepolyhedral. The approximate points are used to generate basis functions usingthe SVD of the Vandermonde matrix. The type of basis functions derived includeLagrange basis, orthogonal and orthonormal hull basis and radial basis. It isshown that the hull basis is the best choice to enforce minimum DOF whilemaintaining a small Lebesgue constant when very high p-refinement is done. Theproposed hull basis is rigorously proven to achieve arbitrary order of accuracyby satisfying Weierstrass approximation theorem in $\mathbb{R}^d$. The thirdchallenge is met using a standard Discontinuous Galerkin (DG) and DiscontinuousLeast-Squares (DLS) spectral hull formulations. The accuracy and stability ofthe formulation is demonstrated for the linearized acoustics andtwo-dimensional compressible Euler equations on some benchmark problemsincluding a cylinder, airfoil, vortex convection and compressible vortexshedding from a triangle.

Spectral/hp Hull: A Degree of Freedom Reducing Discontinuous Spectral  Element Method for Conservation Laws with Application to Compressible Fluid  Flow