Two Dimensional Fourier Continuation for Domains with Corners

TL;DR

This paper introduces a fast 2D Fourier Continuation method for smooth function extension on domains with corners, achieving high accuracy and efficiency, demonstrated on complex Poisson problems.

Contribution

The paper develops a novel 2D-FC algorithm with O(N log N) complexity for domains with corners, enabling high-precision solutions for large-scale Poisson problems.

Findings

Achieves near machine precision accuracy on complex domains.

Operates with O(N log N) computational complexity.

Successfully applied to a 4-million degree of freedom Poisson problem.

Abstract

This paper presents a fast "two-dimensional Fourier Continuation" (2D-FC) method for the construction of biperiodic extensions of smooth, non-periodic functions defined over general two-dimensional (2D) domains, including domains with corners. The algorithm operates with an O(N log N) computational cost, for an N-point discretization grid, and it achieves a user-prescribed d-th order of accuracy. The methodology can be generalized to non-smooth domains of arbitrary dimensionality, but such extensions are not considered in the present work. The usefulness and performance of the 2D-FC method are demonstrated through applications to the Poisson problem posed on bounded 2D domains with corners. One illustrative application concerns a Poisson problem on a non-smooth drop-shaped domain with a highly oscillatory forcing term; employing a discretization containing approximately four million degrees of freedom, the method produces the Poisson solution with accuracies of the order of machine precision in computing times on the order of one second on a single core of a present-day computer.