A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth -order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

A fourth-order unfitted characteristic finite element method is proposed for solving free-boundary problems of time-dependent partial differential equations. The method utilizes a fourth-order forward flow map and a fourth-order backward flow map to track the domain and discretize the PDE, providing a framework for designing high-order numerical methods. Extensive numerical experiments confirm the effectiveness and accuracy of the method.