The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

Wang, Jufeng; Hao, Su; Cheng, Yumin

doi:10.1155/2014/641592

Cited by 6 publications

(7 citation statements)

References 77 publications

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“…Then similar to the finite element method, it follows that Vρ(Ω)⊂H01. Then using the error estimates of the IMLS method (Sun et al , 2015; Wang et al , 2014) and Céa’s lemma we have proven that (Wang et al , 2014): …”

Section: Stability and Convergencementioning

confidence: 89%

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An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Wang

Sun

2019

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Purpose This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions. Design/methodology/approach In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly. Findings Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate. Originality/value This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.

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Section: Stability and Convergencementioning

confidence: 89%

“…The basis functions are chosen to be the complete polynomial functions of order m. Then by using the Aubin–Nitsche method, the error estimates in the L 2 norm can be obtained as (Wang et al , 2014): …”

Section: Stability and Convergencementioning

confidence: 99%

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Wang

Sun

2019

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“…where u h (x(s)) and q h (x(s)) are the approximation functions obtained by the IMLS method. en, from the error estimates of the IMLS method [28,44,45], there exist…”

Section: Error Estimations Of the Ibefm For Two-dimensional Potential Problemsmentioning

confidence: 99%

“…When the Gaussian point and the source point do not coincide, it can be seen from equations ( 6) and ( 7) that the basic solutions are bounded. Also, from the errors of the IMLS method [28,45], there exists constant C independent of ρ such that…”

Section: Error Estimations Of the Ibefm For Two-dimensional Potential Problemsmentioning

confidence: 99%

Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

Wang

Sun

2020

Mathematical Problems in Engineering

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The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

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“…It has been widely used in many engineering fields. For example, it is known that it formed many meshless methods to solve mathematical and physical problems where traditional calculation methods are not applicable [9][10][11], such as the element-free Galerkin method [12], the meshless local Petrov-Galerkin [13], and the boundary element free method [14]. In recent years, many scholars have studied and enhanced the moving least squares method [15,16].…”

Section: Introductionmentioning

confidence: 99%

A trimmed moving total least-squares method for curve and surface fitting

Tang

et al. 2020

Meas. Sci. Technol.

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The Moving Least Squares (MLS) method has been developed for fitting of the measurement data contaminated with errors. The local approximants of the MLS method only take the random errors of the dependent variable into account, whereas the independent variables of measurement data always contain errors. To consider the influence of errors of dependent and independent variables, the Moving Total Least Squares (MTLS) offers a better choice. However, both MLS and MTLS method are sensitive to outliers, which greatly affects the fitting accuracy and robustness. This paper presents an improved method-Trimmed Moving Total Least Squares (TrMTLS) method, in which Total Least Squares (TLS) method with truncation procedure is adopted to determine the local coefficients in the influence domain. This method can deal with outliers and random errors of all variables without setting the threshold or adding small weights subjectively. The numerical simulation and measurement experiments results indicate that the proposed algorithm has better fitting accuracy and robustness compared with the MTLS and MLS method.

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The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

Cited by 6 publications

References 77 publications

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

A trimmed moving total least-squares method for curve and surface fitting

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