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### Session S20 - Applied Math and Computational Methods and Analysis across the Americas

Friday, July 16, 19:30 ~ 20:00 UTC-3

## Stabilizing radial basis functions techniques for a local boundary integral method

### Luciano Ponzellini Marinelli

#### Universidad Nacional de Rosario, CIFASIS-CONICET-UNR, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak0c5573f930241c6be3117e5c5b54b1e9').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy0c5573f930241c6be3117e5c5b54b1e9 = 'l&#117;c&#105;&#97;n&#111;p&#111;nz&#101;ll&#105;n&#105;' + '&#64;'; addy0c5573f930241c6be3117e5c5b54b1e9 = addy0c5573f930241c6be3117e5c5b54b1e9 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text0c5573f930241c6be3117e5c5b54b1e9 = 'l&#117;c&#105;&#97;n&#111;p&#111;nz&#101;ll&#105;n&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak0c5573f930241c6be3117e5c5b54b1e9').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy0c5573f930241c6be3117e5c5b54b1e9 + '\'>'+addy_text0c5573f930241c6be3117e5c5b54b1e9+'<\/a>';

Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions.

One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter ($\varepsilon$) which controls the flatness of the function. It is observed that best accuracy is often achieved when $\varepsilon$ tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned.

A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit like the RBF-QR method and the RBF-GA method.

We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations.

Numerical results for small shape parameter that stabilize the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.

Joint work with Nahuel Caruso (Universidad Nacional de Rosario, CIFASIS-CONICET-UNR, Argentina) and Margarita Portapila (Universidad Nacional de Rosario, CIFASIS-CONICET-UNR, Argentina).