An optimal adaptive wavelet method for First Order System Least Squares

TL;DR

This paper introduces an optimal adaptive wavelet method for solving well-posed first order systems derived from second order PDEs, including elliptic and Navier-Stokes equations, with proven computational efficiency.

Contribution

It reformulates second order PDEs as first order least squares systems and develops an adaptive wavelet solver with optimal complexity.

Findings

Achieves optimal computational complexity for the systems

Successfully applies to elliptic PDEs with inhomogeneous boundary conditions

Extends to stationary Navier-Stokes equations

Abstract

In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.