Bivariate Splines of Various Degrees for Numerical Solution of Partial Differential Equations

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these splines is presented. Upon this form, the multivariate spline method for numerical solution of partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston in [The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations, in Wavelets and Splines, G. Chen and M. J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 24-76] is generalized to obtain a new spline method. It is observed that, combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest, the new spline method of bivariate splines of various degrees is able to solve linear PDEs very effectively and efficiently.

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