A novel class of Hessian recovery-based numerical methods for solving biharmonic equations and their applications in phase field modeling

In this paper, we introduce unified Hessian recovery-based $C^0$ finite element methods (HRB–FEM) and finite volume methods (HRB–FVM) for 2D biharmonic equations. Within the framework of Petrov–Galerkin methods, we propose a novel $H^3-H^1$ formulation. Initially, we employ the Hessian recovery operator to discretize the Laplacian operator, subsequently integrating it into both the standard $C^0$ Lagrange finite element framework and finite volume framework. Through tailored treatments of Neumann-type boundary conditions aimed at reducing computational overhead, we extend our Hessian recovery-based FEM to address phase field equations. Numerical experiments confirm optimal order of convergence under $L^2$ and $H^1$ norms, demonstrating rates of $O\left(h^{k+1}\right)$ and $O\left(h^k\right)$ respectively for both proposed methods. Furthermore, a series of benchmark tests highlight the robustness of our approach and its ability to faithfully capture the physical characteristics during prolonged simulations of phase field equations.