Error analysis of the meshless finite point method

Li, Xin; Dong, Haiyun

doi:10.1016/j.amc.2020.125326

Cited by 21 publications

(15 citation statements)

References 27 publications

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“…• The error estimate in Equation It should be mentioned that the condition number of matrices K and N depends on h. So as h → 0, the term cond(KN −1 ) could increase in such a way that the cond(KN −1 )h m − l + 1 goes to infinity. However, this problem solved very recently when Li and Dong [14] proposed the discrete error estimate of the finite point method which is independent of the condition number of the coefficient matrix.…”

Section: Comparing Error Estimate (38) With Existing Resultsmentioning

confidence: 99%

“…The finite point method (FPM) is a specific meshless scheme in which the MLS approximation is combined with the collocation approach to solve the strong form of the PDE (PIDE) under consideration. This method has been applied successfully to approximate solutions of time‐independent [13, 14] and also time‐dependent PDEs [9]. The method is also applied to the numerical solution of mathematical models arising from engineering [15, 16] and also valuation of American options under the Black–Sholes model [17, 18].…”

Section: Introductionmentioning

confidence: 99%

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Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

Shirzadi

Dehghan

Bastani

2020

Numerical Methods Partial

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In this study, we derive optimal uniform error bounds for moving least-squares (MLS) mesh-free point collocation (also called finite point method) when applied to solve second-order elliptic partial integro-differential equations (PIDEs). In the special case of elliptic partial differential equations (PDEs), we show that our estimate improves the results of Cheng and Cheng (Appl. Numer. Math. 58 (2008), no. 6, 884-898) both in terms of the used error norm (here the uniform norm and there the discrete vector norm) and the obtained order of convergence. We then present optimal convergence rate estimates for second-order elliptic PIDEs. We proceed by some numerical experiments dealing with elliptic PDEs that confirm the obtained theoretical results. The article concludes with numerical approximation of the linear parabolic PIDE arising from European option pricing problem under Merton's and Kou's jump-diffusion models. The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the MLS collocation scheme is an efficient and reliable numerical method to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.

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Section: Comparing Error Estimate (38) With Existing Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

Shirzadi

Dehghan

Bastani

2020

Numerical Methods Partial

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“…Since the CVMLS shape function Φ j (x) lacks the delta function property [18,20] , i.e., Φ j (x I ) = δ jI , we have u I = u(x I ) and q I = q(x I ). Thus, as in other meshless methods [24][25][26][27][28] , the boundary conditions in the present CVBEFM must be dealt with carefully. For problems in potential theory and linear elasticity, the boundary conditions in the boundary node method can be satisfied by coupling discretized BIEs in terms of approximate nodal variables, together with equations relating these approximations to their exact values through the MLS approximation [18] .…”

Section: Discretization For Mixed Boundary Conditionsmentioning

confidence: 99%

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

Chen

2020

Appl. Math. Mech.-Engl. Ed.

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The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method (CVBEFM) for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers. To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative, a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures. To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables, a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure. The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers, even for extremely large wavenumbers such as κ = 10 000.

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“…In this paper, we adopt the multiple-scale technique [28][29] , which is a convenient and effective pre-conditioning technique to reduce the condition number of the resultant matrix of the MPS. Certainly, there also exist other superior methods to deal with the above ill-conditioned problem, such as the finite point method [30] and the localized method of fundamental solutions [31] . Besides, due to the utilization of the concept of localization [31] , the condition number can be markedly decreased, and the error can be analyzed theoretically.…”

Section: Introductionmentioning

confidence: 99%

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Feng

Yan

Lin

et al. 2020

Appl. Math. Mech.-Engl. Ed.

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Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

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Error analysis of the meshless finite point method

Cited by 21 publications

References 27 publications

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

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