Advancements in the Ultraspherical Spectral Method

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The numerical solution of real-world fluid flow and airfoil problems needs an accurate, flexible, and fully-adaptive spectral element method. The so-called ultraspherical spectral method, with its sparsity and regularity preserving discretizations, is promising to overcome many of the traditional computational barriers. This research project will exploit and investigate the remarkable properties of the ultraspherical spectral method with the aim of producing a high quality and industrial-strength spectral element solver for partial differential equations. One key feature will be its robustness to pinching boundary features, typical with airfoils, that will alleviate the current tremendous burden on mesh generation algorithms. The project will radically alter the perception of spectral methods in the computational mathematics and engineering communities by extensively demonstrating that, when done carefully, they can be a flexible, general, and powerful numerical tool.Today's pseudospectral methods deliver both convenience and spectrally accurate discretizations for the solution of differential equations. However, they lead to dense discretizations, numerical instability, and a severe limitation to simple geometries. The novel ultraspherical spectral method is an alternative that retains the same accuracy and convenience, but leads to almost banded well-conditioned discretizations that faithfully preserves the regularity of the underlying differential operator while also being amenable to specialized fast linear algebra routines. Based on this new spectral method, the PI will derive a new mathematically-grounded fully-adaptive spectral element method for meshed geometries. Key novel computational features will include: (1) A high accuracy on mesh elements that is independent of the aspect ratio; (2) True hp-adaptivity that allows for essentially arbitrarily large element degree p and small average mesh element size h (without concern of ill-conditioning); and (3) The flexibility to solve a wide range of differential equations with general boundary constraints; and (4) Local refinement and mesh coarsening for the resolution of corner singularities. This new spectral element method will be applied to challenging partial differential equations for the state-of-the-art numerical simulation of advection-dominated fluid flow problems.