Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

TL;DR

This paper introduces a shearlet system on bounded domains that provides superior approximation rates for functions with smooth jumps along curves, demonstrating its potential for PDE discretization and solving elliptic PDEs efficiently.

Contribution

The paper develops a new shearlet system on bounded domains that achieves better approximation rates for Sobolev spaces than traditional wavelet systems.

Findings

Shearlet system yields improved approximation rates for functions with jump discontinuities.

Adaptive shearlet-based algorithm effectively solves elliptic PDEs.

Analysis shows favorable computational complexity and convergence properties.

Abstract

In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(\Omega)$ for some $s\in \mathbb{N}$, $\Omega \subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(\Omega)$ norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties.