Weak Galerkin finite element methods for a fourth order parabolic equation

Chai, Shimin; Zou, Yongkui; Zhou, Chenguang; Zhao, Wenju

doi:10.1002/num.22373

Cited by 13 publications

(4 citation statements)

References 23 publications

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“…Taking expectation on both sides of (64), together with the definitions (16) and (17), the first inequality of (62) is provided.…”

Section: Theorem 4 Under the Assumption Of Eorem 3 We Havementioning

confidence: 99%

“…en, the WG method is successfully applied to elliptic interface problems [11,12] and linear parabolic problems [13][14][15][16] and further developed for other applications, such as biharmonic problems [17][18][19][20][21][22], Cahn-Hilliard problems [23], and Stokes problems [24][25][26][27][28][29]. To ensure the method is highly flexible in element construction and mesh generation, the idea of a stabilization term is introduced in [30], which allows for arbitrary piecewise polynomial shape functions in deformed and honeycomb meshes.…”

Section: Introductionmentioning

confidence: 99%

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Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Zhou

Zou

Chai

et al. 2020

Mathematical Problems in Engineering

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This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

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“…Taking expectation on both sides of (64), together with the definitions (16) and (17), the first inequality of (62) is provided.…”

Section: Theorem 4 Under the Assumption Of Eorem 3 We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Zhou

Zou

Chai

et al. 2020

Mathematical Problems in Engineering

Self Cite

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“…The field of fluid mechanics includes numerous partial differential equations (PDEs) essential for studying fluid dynamics. Computational fluid dynamics (CFD) uses numerical analysis and data structures to analyze and solve fluid flow problems, incorporating various numerical solving methods such as the finite element method [ 1 ], finite volume method [ 2 ], and spectral methods [ 3 ]. Despite significant advancements over recent decades, these methods often face complexities due to mesh division in practical applications.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information

Liu,

Chen,

Chen

et al. 2024

Entropy

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Physics-informed neural networks (PINNs) have garnered widespread use for solving a variety of complex partial differential equations (PDEs). Nevertheless, when addressing certain specific problem types, traditional sampling algorithms still reveal deficiencies in efficiency and precision. In response, this paper builds upon the progress of adaptive sampling techniques, addressing the inadequacy of existing algorithms to fully leverage the spatial location information of sample points, and introduces an innovative adaptive sampling method. This approach incorporates the Dual Inverse Distance Weighting (DIDW) algorithm, embedding the spatial characteristics of sampling points within the probability sampling process. Furthermore, it introduces reward factors derived from reinforcement learning principles to dynamically refine the probability sampling formula. This strategy more effectively captures the essential characteristics of PDEs with each iteration. We utilize sparsely connected networks and have adjusted the sampling process, which has proven to effectively reduce the training time. In numerical experiments on fluid mechanics problems, such as the two-dimensional Burgers’ equation with sharp solutions, pipe flow, flow around a circular cylinder, lid-driven cavity flow, and Kovasznay flow, our proposed adaptive sampling algorithm markedly enhances accuracy over conventional PINN methods, validating the algorithm’s efficacy.

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“…In physics and engineering, many important physical models are described as partial differential equations (PDEs), such as Navier–Stokes equations [ 4 ] for fluid mechanics and Maxwell equations [ 5 ] for electromagnetic field theory. When solving partial differential equations using traditional numerical methods, such as the Finite Difference Method (FDM) [ 6 ], Finite Element Method (FEM) [ 7 ], Finite Volume Method (FVM) [ 8 ], Radial Basis Function Method (RBF) [ 9 ], etc., problems such as high computational costs and the curse of dimensionality are often encountered. Over the last few years, the use of machine learning to solve partial differential equations has also rapidly expanded [ 10 , 11 , 12 , 13 ].…”

Section: Introductionmentioning

confidence: 99%

A Second-Order Network Structure Based on Gradient-Enhanced Physics-Informed Neural Networks for Solving Parabolic Partial Differential Equations

Sun

Feng

2023

Entropy

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Physics-informed neural networks (PINNs) are effective for solving partial differential equations (PDEs). This method of embedding partial differential equations and their initial boundary conditions into the loss functions of neural networks has successfully solved forward and inverse PDE problems. In this study, we considered a parametric light wave equation, discretized it using the central difference, and, through this difference scheme, constructed a new neural network structure named the second-order neural network structure. Additionally, we used the adaptive activation function strategy and gradient-enhanced strategy to improve the performance of the neural network and used the deep mixed residual method (MIM) to reduce the high computational cost caused by the enhanced gradient. At the end of this paper, we give some numerical examples of nonlinear parabolic partial differential equations to verify the effectiveness of the method.

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Weak Galerkin finite element methods for a fourth order parabolic equation

Cited by 13 publications

References 23 publications

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information

A Second-Order Network Structure Based on Gradient-Enhanced Physics-Informed Neural Networks for Solving Parabolic Partial Differential Equations

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