A Bivariate Spline Method for Second Order Elliptic Equations in Non-Divergence Form
For researchers in numerical PDEs, this provides a new method for a class of elliptic equations, but the contribution appears incremental as it adapts existing spline techniques with LBB analysis.
The paper develops a bivariate spline method for solving second-order elliptic PDEs in non-divergence form, establishing existence, uniqueness, and stability via the LBB condition, with computational results demonstrating effectiveness across various spline degrees.
A bivariate spline method is developed to numerically solve second order elliptic partial differential equations (PDE) in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. A plenty of computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.