# Adaptive Discontinuous Galerkin Finite Elements for Advective Allen-Cahn Equation

## TL;DR

This paper develops an adaptive interior penalty discontinuous Galerkin method to solve the advective Allen-Cahn equation with complex velocity fields, improving accuracy in capturing sharp layers.

## Contribution

It introduces a residual-based a posteriori error estimator tailored for non-divergence-free velocity fields in the adaptive DG framework.

## Key findings

- Effective resolution of sharp layers in numerical examples
- High accuracy achieved with adaptive method
- Error estimator accounts for non-divergence-free velocities

## Abstract

We apply a space adaptive interior penalty discontinuous Galerkin method for solving advective Allen-Cahn equation with expanding and contracting velocity fields. The advective Allen-Cahn equation is first discretized in time and the resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a residual-based a posteriori error estimator. The a posteriori error estimator contains additional terms due to the non-divergence-free velocity field. Numerical examples demonstrate the effectiveness and accuracy of the adaptive approach by resolving the sharp layers accurately.