Peimeng Yin

Jinting Yang, Nianyu Yi, Peimeng Yin

In this paper, we develop a linear, fully decoupled, and unconditionally energy-stable fully discrete finite element scheme for the Cahn--Hilliard--Navier--Stokes (CHNS) system by employing the scalar auxiliary variable (SAV) approach. Unlike existing SAV-based formulations that typically introduce multiple auxiliary variables or additional techniques to handle different nonlinearities, we introduce only one scalar auxiliary variable together with a novel update of the auxiliary variable to reformulate all nonlinear terms arising from the Cahn--Hilliard and Navier--Stokes equations, yielding an equivalent reformulation of the original CHNS system. An implicit--explicit (IMEX) Euler scheme is applied for temporal discretization, where the linear terms are treated implicitly and the nonlinear terms explicitly, while a finite element method is adopted for spatial discretization. The resulting fully discrete scheme can be efficiently decomposed into two linear subproblems and one scalar quadratic algebraic equation, which significantly simplifies the implementation. Furthermore, we prove that the proposed scheme satisfies an unconditional discrete energy dissipation law and establish its stability with respect to several relevant norms. Optimal-order $L^2$ error estimates are also derived for the fully discrete finite element approximation. Finally, a series of numerical experiments are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

Hailiang Liu, James Ralston, Peimeng Yin

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.

Yaoyao Chen, Dongqian Li, Yin Yang et al.

We investigate two unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) [Chen et al. Numerical Algorithms, DOI: 10.1007/s11075-024-01910-z, 2024] for solving the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. \textcolor{black}{The auxiliary energy function introduced by the IEQ approach, modeling the square root of the nonlinear part of the energy, does not belong to the finite element space used for the spatial discretization.} These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed method.

Xinwen Hu, Yunqing Huang, Nianyu Yi et al.

Neural network-based function approximation plays a pivotal role in the advancement of scientific computing and machine learning. Yet, training such models faces several challenges: (i) each target function often requires training a new model from scratch; (ii) performance is highly sensitive to architectural and hyperparameter choices; and (iii) models frequently generalize poorly beyond the training domain. To overcome these challenges, we propose a reusable initialization framework based on basis function pretraining. In this approach, basis neural networks are first trained to approximate families of polynomials on a reference domain. Their learned parameters are then used to initialize networks for more complex target functions. To enhance adaptability across arbitrary domains, we further introduce a domain mapping mechanism that transforms inputs into the reference domain, thereby preserving structural correspondence with the pretrained models. Extensive numerical experiments in one- and two-dimensional settings demonstrate substantial improvements in training efficiency, generalization, and model transferability, highlighting the promise of initialization-based strategies for scalable and modular neural function approximation. The full code is made publicly available on Gitee.