Spectral approach to D-bar problems

TL;DR

This paper introduces a spectral numerical method for solving D-bar problems with real analytic, rapidly decreasing potentials, achieving spectral convergence and applying Fourier and Krylov techniques.

Contribution

It presents the first spectral convergence numerical approach for D-bar problems, combining integral formulation, analytical regularization, and Krylov solvers.

Findings

Spectral convergence achieved for D-bar problems

Method successfully applied to Davey-Stewartson II equations

Provides accurate numerical solutions for PDE testing

Abstract

We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.