A Bivariate Spline Method for Second Order Elliptic Equations in Non-Divergence Form

TL;DR

This paper introduces a bivariate spline method for numerically solving second order elliptic PDEs in non-divergence form, establishing theoretical properties and demonstrating effectiveness through computational results.

Contribution

It develops a novel spline-based numerical approach for non-divergence elliptic equations, with proven stability and approximation properties.

Findings

Method is effective for various spline degrees

Theoretical stability and convergence are established

Computational results confirm efficiency

Abstract

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.