Open Collections

Chu, Alex Shiu Lun

University of British Columbia

Closest point methods are a class of embedding methods that have been used to solve partial differential equations on surfaces with the closest point representation of the surface. Recently, several studies replaced the standard Cartesian grid methods in the original Closest point methods with radial basis function generated finite differences. This reduces the computational cost and allows scattered and unstructured grids as well as locally refined uniform grids. This thesis uses the polyharmonic spline function as the radial basis function in the combined method which is different from the usual choice of Gaussian or multiquadric to avoid the shape parameter. We first perform convergence tests of the combined method. In all cases, the radial basis function closest point method uses fewer points in the embedding space while achieving a similar accuracy and convergence rate as the original closest point method. We then focus on solving partial differential equation problems with irregular grids that match features of the surface or the solution. These include using more points near high curvature regions or using more points near fine scale solution features. This can reduce the computational cost compared to using a uniform fine grid over the entire surface. Lastly, we provide an adaptive version of the combined method that is able to solve partial differential equation problems on surfaces when either or both of the surface features and problem features are changing in time.

Text

2021-10-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/80067

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/