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

TL;DR

This paper introduces a novel spectral/hp hull method that reduces degrees of freedom by directly using convex hulls for complex geometries, enabling high-accuracy discretization of conservation laws in fluid flow.

Contribution

It proposes a new approach to construct basis functions on convex hulls, including a closed-form approximation of Fekete points, and demonstrates its effectiveness for compressible fluid flow simulations.

Findings

Achieves high-order accuracy satisfying Weierstrass theorem.

Demonstrates stability and accuracy on benchmark fluid flow problems.

Reduces degrees of freedom compared to traditional methods.

Abstract

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.