During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented.

A novel method for detection of lines in a digital image used in line scale calibration process is described. The line images are recorded by a moving microscope and a CCD camera. This method is based on the center of the line instead of the edges of the line. Line detection is one of the most important problems in the line scale calibration. This method uses a Gabor filter for each row in the digital image. Likewise, based on robust statistics, some outlier points due to imperfections in the mark on the scale are ignored.

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.