Theoretical analysis of the generalized finite difference method

Zheng, Zhi; Li, Xin

doi:10.1016/j.camwa.2022.06.017

Cited by 52 publications

(13 citation statements)

References 27 publications

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“…Many works have been devoted to the numerical solutions of semilinear elliptic problems such as finite element method (FEM) [2,3], finite difference method [4], finite volume element method [5] and discontinuous Galerkin method [6]. Recently, some collocation meshless (or meshfree) methods [7,8], Galerkintype meshless method [8] and generalized finite difference method [9,10] have been developed to solve the semilinear PDEs. Unlike mesh-based numerical methods, the shape functions used in the meshless methods [11][12][13][14] are linkage with nodes (or particles) scattered in the underlying computational domain, which reduces the dependence on the mesh.…”

Section: Introductionmentioning

confidence: 99%

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

Li¹

2024

AAMM

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A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in L 2 and H 1 norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance h does not appear explicitly in the parameter β introduced by the Nitsche method. The theoretical results are confirmed numerically.

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Section: Introductionmentioning

confidence: 99%

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

Li¹

2024

AAMM

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“…Since the GFDM is a meshless method that can arrange irregular nodes, it has unique advantages in dealing with problems with complex geometric boundaries. Many researchers have improved and perfected the GFDM through in-depth research over time [25][26][27], making its theoretical basis more mature and its application scope more extensive. The research of Hidayat [28] further demonstrates the effectiveness of the GFDM in solving two-dimensional elastic static problems.…”

Section: Introductionmentioning

confidence: 99%

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

Zhou,

Li,

Zhuang

et al. 2024

Mathematics

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This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems.

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“…The research on acoustic problems has aroused much attention in past decades. The methods for solving acoustic problems mainly include experiments, analytical methods (and semi-analytical methods) [1], and numerical methods [2][3][4][5]. Experiments usually require much expense and suitable experimental conditions.…”

Section: Introductionmentioning

confidence: 99%

A Coupled Overlapping Finite Element Method for Analyzing Underwater Acoustic Scattering Problems

Jiang,

Yu,

et al. 2023

JMSE

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It is found that the classic finite element method (FEM) requires much time for adequate meshes to acquire satisfactory numerical solutions, and is restricted to acoustic problems with low and middle frequencies. In this work, a coupled overlapping finite element method (OFEM) is employed by combining the overlapping finite element and the modified Dirichlet-to-Neumann (mDtN) boundary condition to solve underwater acoustic scattering problems. The main difference between the OFEM and the FEM lies in the construction of the local field approximation. In the OFEM, virtual nodes are utilized to form the partition of unity functions while no degree of freedom is assigned to these virtual nodes, which suppresses the linear dependence issue in other generalized finite element methods. Moreover, the user-defined enrichment functions can be flexibly utilized in the local field, and thus the numerical dispersions can be significantly mitigated. To truncate the infinite problem domain and satisfy the Sommerfeld radiation condition, an artificial boundary is constructed by incorporating the mDtN technique. Several numerical examples are studied and it is shown that the proposed method can greatly diminish the numerical error and is insensitive to distorted meshes, indicating that the proposed method is promising in predicting underwater acoustic scattering.

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Theoretical analysis of the generalized finite difference method

Cited by 52 publications

References 27 publications

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

A Coupled Overlapping Finite Element Method for Analyzing Underwater Acoustic Scattering Problems

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