Adaptive Node Refinement Collocation Method for Partial Differential Equations

Muñoz, Antonio; González–Casanova, Pedro; Gómez, Gustavo Rodríguez

doi:10.1109/enc.2006.4

Cited by 6 publications

(5 citation statements)

References 25 publications

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“…Applying the proposed adaptive algorithm resulted in N=1281 nodes and RMS=6.936e-5 after three steps of adaptation, while Libre et al [25] Achieved lower accuracy using a wavelet scheme with N=1823 nodes and after four steps of adaptation. Munoz-Gomez et al [23] Achieved this accuracy after 12 iterations and with 2646 points The numerical results with uniform and adaptive distributions are shown in Fig. 3.…”

Section: Numerical Resultsmentioning

confidence: 78%

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A new adaptive-node refinement algorithm based on multiquadric method for the nearly singular problems

koushki

jabbari

ahmadinia

2020

Preprint

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In this paper, a new adaptive-node refinement algorithm for multiquadric (MQ) method applied to nearly singular PDEs, has been presented. Besides, for the first time, the concept of the gradient has been employed in the refinement index. Regions with high gradients are identified based on the average value of the proposed function solution. The solution is approximated using a first-order derivative of interpolation with MQ. In the framework of the adaptive algorithm, the average of the proposed function was used as an indicator that determines where the point distribution can be refined and nodes can be added or removed based on this indicator. Different applications of the proposed adaptive algorithm are investigated through numerical examples. It has been revealed that the proposed algorithm is able to identify the singularities both in the domain and near boundaries and the numerical results of the MQ method confirm the accuracy and efficiency of the algorithm. The main advantage of this algorithm is that in the first steps, the regions with high gradients are identified correctly and the convergence speed of the algorithm increases continuously.

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Section: Numerical Resultsmentioning

confidence: 78%

“…The error indicator was used to identify locations that require higher accuracy and Behrens et al [22] applied this algorithm to linear evolutionary PDEs and obtained accurate results. Gomez et al [23] presented a RBF dynamic domain decomposition algorithm. This algorithm was applied in a large PDEs with high gradient.…”

Section: Introductionmentioning

confidence: 99%

A new adaptive-node refinement algorithm based on multiquadric method for the nearly singular problems

koushki

jabbari

ahmadinia

2020

Preprint

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“…This can be accom plished in two ways: by means of the clas si cal Carte sian h-re fine ment scheme or an adap tive node re finement scheme (ANR). We se lect the sec ond ap proach be cause it has been showed to be com pu ta tional ef ficient (Muñoz-Gómez et al, 2006;Driscoll et al, 2006).…”

Section: Expe Ri Mental Error Analysis For Implicit Methodsmentioning

confidence: 99%

Exponential Convergence of Multiquadric Collocation Method: a Numerical Study

Muñoz-Gómez

González–Casanova

Rodríguez-Gómez

2009

IIT

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Recent numer ical studies have proved that multiquadric collo ca tion methods can achieve expo nen tial rate of conver gence for elliptic prob lems. Although some investi ga tions has been performed for time dependent prob lems, the influ ence of the shape param eter of the multiquadric kernel on the conver gence rate of these schemes has not been studied. In this article, we inves ti gate this issue and the influence of the Péclet number on the rate of conver gence for a convec tion diffu sion problem by using both an explicit and implicit multiquadric collo ca tion tech niques. We found that for low to moderate Péclet number an expo nen tial rate of convergence can be attained. In addi tion, we found that increasing the value of the Péclet number produces a value reduc tion of the coef fi cient that deter mines the expo nential rate of conver gence. More over, we numer i cally showed that the optimal value of the shape param eter decreases monotonically when the diffusive coefficient is reduced.Keywords: Radial basis func tions, multiquadric, convec tion-diffu sion, partial differ en tial equa tion. ResumenExpe ri men tos nu mé ri cos re cien tes so bre los mé to dos de co lo ca ción con mu lit cuá dri cos han de mos tra do que és tos pue den al can zar ra zo nes de con ver gen cia ex po nen cial en pro ble mas de ti po elíp ti cos. Si bien, al gu nas in ves ti ga cio nes se han rea li za do pa ra pro ble mas de pendien tes del tiem po, la in fluen cia del pa rá me tro c del nú cleo mul ti cuá dri co en la ra zón de con ver gen cia de éstos es que mas no ha si do es tu dia da. En la pre sen te in ves ti ga ción se anali za es te tó pi co y la in fluen cia del nú me ro de Pé clet en la ra zón de con ver gen cia pa ra un pro ble ma con vec ti vo di fu si vo, con si de ran do un es que ma de dis cre ti za ción im plí ci to y ex pli ci to con téc ni cas de co lo ca ción con mu lit cuá dri cos. De mos tra mos nu mé ri ca men te que para va lo res ba jos a mo de ra dos del coe fi cien te de Pé clet se ob tie ne una ra zón de con ver gen cia expo nen cial. Ade más, en con tra mos que al au men tar el nú me ro de Pé clet ori gi na una re duc ción en va lor del coe fi cien te que de ter mi na la ra zón de con ver gen cia ex po nen cial. Adi cio nal men te, de ter mi na mos que el va lor óp ti mo del parámetro c decrece monótonicamente cuando el coeficiente difusivo es disminuido.Des cip to res: Fun cio nes de ba se ra dial, mul ti cuá dri co, con vec ción-di fu sión, ecua ción di feren cial parcial.

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“…For example, RBF-specific preconditioners [18] and adaptive selection of data centres [19]. However, the only reliable method currently available is domain decomposition [19], [20], [21], [22], [23].…”

Section: Introductionmentioning

confidence: 99%

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

Thewlis

Stevens²,

Power

et al. 2023

Computer Methods and Programs in Biomedicine

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Adaptive Node Refinement Collocation Method for Partial Differential Equations

Abstract: In this work, by using the local node refinement technique purposed in [1,2], and a quad-tree type algorithm [3,4]

Cited by 6 publications

References 25 publications

A new adaptive-node refinement algorithm based on multiquadric method for the nearly singular problems

A new adaptive-node refinement algorithm based on multiquadric method for the nearly singular problems

Exponential Convergence of Multiquadric Collocation Method: a Numerical Study

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

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