arXiv subjects

Peimeng Yin

Publications and source records attributed to Peimeng Yin.

At least 19 recordsLinked to original sources

arXiv2026

A Convex Splitting Spectral Method for the Phase Field Crystal Equation: Energy Stability, Computational Stability Maps, and Three-Dimensional GPU Simulations

We present an efficient Fourier spectral method based on the convex splitting framework of Eyre~\cite{eyre1998unconditionally} for the phase field crystal (PFC) equation. The proposed first-order scheme is unconditionally energy stable and conserves mass to machine precision. An energy stability theorem is established using a truncated potential argument, and a semi-analytical neutral stability curve is derived from a dominant-mode energy balance, providing a closed-form characterization of the practical stability boundary. The classical sufficient condition $a \geq 2$ is shown to be conservative: a computational stability map obtained from 40,000 GPU-accelerated PFC simulations reveals that energy-stable solutions persist for values of $a$ significantly below this threshold. Crucially, accuracy analysis demonstrates that smaller values of $a$ within the stable region consistently yield lower $L^2$ errors. This high-fidelity regime is rigorously verified through an extended asymptotic stress test consisting of a long-time three-dimensional simulation on a $256^3$ grid up to $T_f = 10,000$, successfully executing $10^6$ continuous temporal increments deep within the relaxed stability regime, well below the classical convex splitting limit ($a < 2$), while preserving strict monotonic energy dissipation and machine-precision mass conservation. Finally, two-dimensional and three-dimensional simulations at resolutions up to $512^3$ are performed on a single consumer GPU, demonstrating the scalability of the proposed framework for resolving complex phase-field dynamics without requiring HPC infrastructure. The code is made publicly available on GitHub.

math.NA↗

arXiv2026

A linear, fully decoupled, and unconditionally energy-stable SAV-FEM for the Cahn--Hilliard--Navier--Stokes model

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.

math.NA↗

arXiv2026

Unconditional energy stable hybrid IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations

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.

math.NA↗

arXiv2026

A fully decoupled and structure-preserving relaxation Crank--Nicolson finite element method for Gross--Pitaevskii--Poisson model

We propose a fully decoupled, structure-preserving relaxation Crank--Nicolson finite element method (FEM) for the coupled Gross--Pitaevskii--Poisson (GPP) system modeling ultracold plasmas. By introducing suitable auxiliary variables to reformulate the nonlinear interaction and charge density terms, the original system is recast into an equivalent form that enables a linear, fully decoupled numerical scheme. The proposed method preserves key physical invariants, including the mass of each component and a modified discrete energy, at the fully discrete level. We establish the well-posedness and uniqueness of the scheme and rigorously derive optimal error estimates, achieving second-order accuracy in time and optimal $(k+1)$-th order convergence in space for $P^k$ finite element approximations. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the method in preserving conservation properties and accurately capturing complex dynamical behaviors of the coupled GPP system.

math.NA↗

arXiv2026

Towards $C^0$ finite element methods for fourth-order elliptic equation. Part I: general boundary conditions

This paper is part of a series developing $C^0$ finite element methods for fourth-order elliptic equations on polygonal domains. Here, we investigate how boundary conditions influence the design of effective $C^0$ schemes, specifically focusing on equations without lower-order terms, namely the biharmonic equation. We propose a modified mixed formulation that decomposes the problem into a system of Poisson equations, where the number of equations depends on both the largest interior angle and the boundary conditions on its two adjacent sides. In contrast to the naive mixed formulation, which involves only two Poisson problems, the proposed approach guarantees convergence to the true solution for arbitrary polygonal domains and general boundary conditions, including Navier, Neumann, and mixed boundary conditions. $C^0$ finite element algorithms are developed, rigorous error estimates are established, and numerical experiments are presented to demonstrate the well-posedness and effectiveness of the proposed method.

math.NA↗

arXiv2025

Weights initialization of neural networks for function approximation

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.

cs.LG↗

arXiv2025

A $C^0$ finite element algorithm for the sixth order problem with simply supported boundary conditions

In this paper, we investigate a sixth order elliptic equation with the simply supported boundary conditions in a polygonal domain. We propose a new method that decouples the sixth order problem into a system of second order equations. Unlike the direct decomposition, which yields three Poisson problems but is restricted to polygonal domains with the largest interior angle no more than $π/{2}$, we rigorously analyze and construct extra Poisson problems to confine the solution into the same function space as that of the original sixth order problem. Consequently, the proposed method can be applied to general polygonal domains. In turn, we also present a $C^0$ finite element algorithm to discretize the new resulting system and establish optimal error estimates for the numerical solution on quasi-uniform meshes. Finally, numerical experiments are performed to validate the theoretical findings.

math.NA↗

arXiv2025

A structure-preserving relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation

In this paper, we propose a mass- and modified energy-conservative relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schrödinger equation and the Poisson equation into their equivalent expressions, constructing a system equivalent to the original Schrödinger-Poisson equation. Our proposed scheme, derived from this equivalent system, is implemented linearly, avoiding the need for iterative techniques to solve the nonlinear equation. Additionally, it is executed sequentially, eliminating the need to solve a coupled large linear system. We in turn rigorously derive the optimal error estimates for the proposed scheme, demonstrating second order accuracy in time and $(k+1)$th order accuracy in space when employing polynomials of degree up to $k$. Numerical experiments validate the accuracy and effectiveness of our method and emphasize its conservation properties over long-time simulations.

math.NA↗

arXiv2025

A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems

In earlier work [H. Liu and Z. Wang, J. Comput. Phys., 328(2017)], an arbitrary high-order conservative and energy-dissipative direct discontinuous Galerkin (DDG) scheme was developed. Although this scheme enforced solution positivity using cell averages as reference values, it lacked a theoretical guarantee for the positivity of those cell averages. In this study, we develop a novel arbitrary high-order DDG method with rigorously proven positivity-preserving properties. Specifically, the positivity of the cell averages is ensured through a modified numerical flux in combination with forward Euler time discretization. To achieve point-wise positivity of ion concentrations, we introduce a hybrid algorithm that integrates a positivity-preserving limiter. The proposed method is further extended to higher-dimensional problems with rectangular meshes. Numerical results confirm the scheme's high-order accuracy, guaranteed positivity preservation, and consistent discrete energy dissipation.

math.NA↗

arXiv2025

Neural network-enhanced $hr$-adaptive finite element algorithm for parabolic equations

In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element solution from the previous step in the updated mesh. The interpolation dependent on the new mesh must be recomputed at each adaptive iteration, resulting in high computational costs. The new approach addresses this challenge by introducing a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the minimizer of the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZE and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement of the conventional methods and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of singularities. The code is made publicly available on GitHub.

math.NA↗

arXiv2025

A second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation

In this paper, we propose a novel second-order dynamical low-rank mass-lumped finite element method for solving the Allen-Cahn (AC) equation, a semilinear parabolic partial differential equation. The matrix differential equation of the semi-discrete mass-lumped finite element scheme is decomposed into linear and nonlinear components using the second-order Strang splitting method. The linear component is solved analytically within a low-rank manifold, while the nonlinear component is discretized using a second-order augmented basis update & Galerkin (BUG) integrator, in which the $S$-step matrix equation is solved by the explicit 2-stage strong stability-preserving Runge-Kutta method. The algorithm has lower computational complexity than the full-rank mass-lump finite element method. The dynamical low-rank finite element solution is shown to conserve mass up to a truncation tolerance for the conservative Allen-Cahn equation. Meanwhile, the modified energy is dissipative up to a high-order error and is hence stable. Numerical experiments validate the theoretical results. Symmetry-preserving tests highlight the robustness of the proposed method for long-time simulations and demonstrate its superior performance compared to existing methods.

math.NA↗

arXiv2024

A posteriori error estimators for fourth order elliptic problems with concentrated loads

In this paper, we study two residual-based a posteriori error estimators for the $C^0$ interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples.

math.NA↗

arXiv2024

Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation

In this paper, we present several unconditionally energy-stable invariant energy quadratization (IEQ) finite element methods (FEMs) with linear, first- and second-order accuracy for solving both the Cahn-Hilliard equation and the Allen-Cahn equation. For time discretization, we compare three distinct IEQ-FEM schemes that position the intermediate function introduced by the IEQ approach in different function spaces: finite element space, continuous function space, or a combination of these spaces. Rigorous proofs establishing the existence and uniqueness of the numerical solution, along with analyses of energy dissipation for both equations and mass conservation for the Cahn-Hilliard equation, are provided. The proposed schemes' accuracy, efficiency, and solution properties are demonstrated through numerical experiments.

math.NA↗

arXiv2023

A semi-implicit dynamical low-rank discontinuous Galerkin method for space homogeneous kinetic equations. Part I: emission and absorption

Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. In this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the DG scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the classical DG solution space. Similar to the classical DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings.

math.NA↗

arXiv2023

Recovery type a posteriori error estimation of an adaptive finite element method for Cahn--Hilliard equation

In this paper, we derive a novel recovery type a posteriori error estimation of the Crank-Nicolson finite element method for the Cahn--Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator.

math.NA↗

arXiv2022

An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead of regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem with zero source term and nonzero flux jumps on line fractures. The transmission problem is defined in the same domain as the original problem excluding on line fractures, and the solution is therefore shown to be more regular. The estimator relies on meshes conforming to the line fractures and its edge jump residual essentially uses the flux jumps of the transmission problem on line fractures. The error estimator is proven to be both reliable and efficient, an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests show that quasi-optimal convergence rates are achieved even for high order approximations and the adaptive meshes are only locally refined at singular points.

math.NA↗

arXiv2022

A $C^0$ finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain

In this paper, we study the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

math.NA↗

arXiv2021

Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

math.NA↗