### Spectral Hulls: a Degree of Freedom Reducing hp-Strategy in Space/Time

#### Arash Ghasemi, August 2016

Abstract:
Reducing the degrees of freedom (DOF) of modern finite element methods is investigated using a systematic hp-process. The elements are first agglomerated (h-coarsening) to form convex/concave hulls and then the polynomial degree of the hull basis, is increased (p-refinement). Compared to the conventional continuous/discontinuous FEM, this mechanism yields more accurate solutions with smaller DOF. This methodology is validated throughout the dissertation using various methods including Fourier-Chebyshev collocation, Continuous Galerkin (CG), Discontinuous Galerkin (DG) and Discontinuous Least-Squares (DLS) on structured and/or arbitrary unstructured grids. The feasibility of such procedure is first investigated in time only by letting the spatial discretization to be fixed to an arbitrary spectral/finite element discretization. In this scenario, lower order time steps (elements) are agglomerated into a space-time hull. A general system of Volterra integral equation is then developed which is simultaneously applicable to $\partial^v /\partial t^v$ time dependency of the PDE. The reduction in DOF is demonstrated by validating a one-dimensional periodic convection test case and two-dimensional scattering from engineering geometries. Motivated by these results, the ideas are then generalized to space. This requires special grid generation and general polyhedral basis functions, called spectral hull basis, which are addressed in detail. In particular, a new set of basis functions are derived based on the SVD of the Vandermonde matrix which are proven to have small Lebesgue constant. Various theoretical results are presented including the derivation of a closed form relation for the Lebesgue constant on a polyhedron, derivation of a closed form relation for approximate Fekete points on a polyhedron and a new proof of Weierstrass approximation theorem in a polyhedral subset of d-dimensional space. One application of the proposed hull basis is to reduce the DOF of discontinuous FEM such that it can compete in practice with CG. The accuracy and efficiency of spectral hulls are demonstrated in a linear acoustics test case and a two-dimensional compressible vortex shedding problem.

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