A $C^1$-conforming arbitrary-order two-dimensional virtual element method for the fourth-order phase-field equation

TL;DR

This paper introduces a high-order, $C^1$-conforming virtual element method for solving the complex fourth-order phase-field equation in two dimensions, ensuring accuracy and convergence through novel function space design.

Contribution

The work develops a new arbitrary-order virtual element method with $C^1$ regularity for high-order phase-field equations, including detailed construction and convergence analysis.

Findings

The method achieves high-order accuracy in numerical tests.

Error estimates confirm the convergence of the scheme.

Numerical experiments demonstrate the method's effectiveness.

Abstract

We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.