9870178 Lai This project concerns investigation of multivariate spline functions. Bivariate and trivariate splines of smoothness r and degree d, will be studied more carefully to enhance their computational efficiency which is essential for applications in computer aided geometric design (CAGD) and numerical solutions of partial differential equations (PDE). The PI will identify the best spline spaces for these applications in the bivariate setting, comparing the dimension of all spline spaces of a fixed smoothness, the number of triangles of their underlying triangulations and their approximation power. The PI will implement these best spline spaces when r=1 and r=2 for typical applications in CAGD and numerical solution of PDE's, e.g., scattered data fitting, filling polygonal holes, numerical solutions of linear and nonlinear biharmonic equations. The PI will study the construction of compactly supported orthonormal wavelets under H1 norm using bivariate splines so that the numerical solution of the standard elliptic equations can be solved without inverting the linear systems. Furthermore, the PI will also study the trivariate spline spaces and identify the best one for the application in numerical solutions of 3D partial differential equations, in particular 3D Navier-Stokes equations. Once the best spline space for r=1 is identified, the PI will implement it for numerical solution of PDE's in the trivariate setting. When computing with these splines, multi-level and domain decomposition methods will be employed to improve the performance of the computation.
