## Multivariate Splines for Numerical Solution of PDE

- April 25, 2017
- 1:30 p.m.
- LeConte 317R

## Abstract

Multivariate splines are smooth piecewise polynomial functions defined on a triangulation or polygonal partition in 2D and a tetrahedral/hexahdral partition (in 3D). They are extremely convenient for numerical solution of partial differential equations.

I first explain how to use splines of arbitrary degree, arbitrary smoothness over arbitrary triangulation to numerically solve linear and nonlinear PDEs. In this approach, no macro-elements are needed to construct; no numerical quadratures are needed if the right-hand side of the PDE or the coefficients of PDE is piecewise continuous. Divergence free conditions can be coded as side constraints in terms of coefficient vectors of spline functions. Our spline method enables to use the stream function formulation for Navior-Stokes equations and potential function formulation for Maxwell equations.

Several PDE including a biharmonic equation, a general second order of PDE in non-divergence form and reaction-diffusion time dependent partial differential equations will be demonstrated to show the flexibility of using splines of variable degrees and variable smoothness over a given triangulation. Adaptive triangulation based approaches will be shown.

Finally I will explain how to use polygonal splines for numerical solution of PDE. Polygonal splines are defined over any collection of polygons. They can be more efficient than spline functions over triangulations, using less of degrees of freedom to achieve more accurate solution.