The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains

TL;DR

The paper introduces the Projection Extension method, a spectrally accurate technique for solving PDEs on complex domains without domain decomposition, leveraging extension procedures and least squares boundary condition enforcement.

Contribution

It presents a novel extension-based spectral method that achieves exponential convergence on complex geometries without domain partitioning.

Findings

Rapidly yields exponentially convergent solutions

Applicable to elliptic, parabolic, and fluid flow problems

Stable and generalizable to higher dimensions

Abstract

An essential ingredient of a spectral method is the choice of suitable bases for test and trial spaces. On complex domains, these bases are harder to devise, necessitating the use of domain partitioning techniques such as the spectral element method. In this study, we introduce the Projection Extension (PE) method, an approach that yields spectrally accurate solutions to various problems on complex geometries without requiring domain decomposition. This technique builds on the insights used by extension methodologies such as the immersed boundary smooth extension and Smooth Forcing Extension (SFE) methods that are designed to improve the order of accuracy of the immersed boundary method. In particular, it couples an accurate extension procedure, that functions on arbitrary domains regardless of connectedness or regularity, with a least squares minimization of the boundary conditions. The resulting procedure is stable under iterative application and straightforward to generalize to higher dimensions. Moreover, it rapidly and robustly yields exponentially convergent solutions to a number of challenging test problems, including elliptic, parabolic, Newtonian fluid flow, and viscoelastic problems.