Date of Final Oral Examination (Defense)
Type of Culminating Activity
Master of Science in Mathematics
Grady B. Wright, Ph.D.
Varun Shankar, Ph.D.
Jodi Mead, Ph.D.
Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary configurations of points, and do not introduce artificial singularities, as surface parameterizations are known to do. In this thesis, we develop a new RBF-FD method that uses projections on the tangent plane to approximate the Laplace-Beltrami operator (surface Laplacian). We then compare this method to two other previously developed RBF-FD methods: the Projected Gradient method and the Hermite RBF-FD method, on a set of benchmark problems posed on the sphere and torus. We also provide guidelines for choosing the various parameters involved in the methods.
Shaw, Sage Byron, "Radial Basis Function Finite Difference Approximations of the Laplace-Beltrami Operator" (2019). Boise State University Theses and Dissertations. 1587.