Imposing jump conditions on nonconforming interfaces for the Correction Function Method: a least squares approach

TL;DR

This paper presents a new least squares-based technique for imposing jump conditions on nonconforming interfaces within the Correction Function Method, enabling high-order accurate solutions to Poisson's equation in 2-D and 3-D.

Contribution

It introduces a novel formulation of the least squares approach that simplifies integral computations over interfaces with implicit representations.

Findings

Achieves fourth-order accuracy in 2-D and 3-D Poisson problems.

Effectively handles complex interfaces with implicit level set representations.

Demonstrates practical implementation of the method with numerical examples.

Abstract

We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the Correction Function Method (CFM). The CFM offers a general framework to solve Poisson's equation in the presence of discontinuities to high order of accuracy, while using a compact discretization stencil. A key concept behind the CFM is enforcing the jump conditions in a least squares sense. This concept requires computing integrals over sections of the interface, which is a challenge in 3-D when only an implicit representation of the interface is available (e.g., the zero contour of a level set function). The technique introduced here is based on a new formulation of the least squares procedure that relies only on integrals over domains that are amenable to simple quadrature after local coordinate transformations. We incorporate this technique into a fourth order accurate implementation of the CFM, and show examples of solutions to Poisson's equation computed in 2-D and 3-D.