A new quadrilateral finite element based on quadratic spline

Abstract

This study presents a novel quadrilateral finite element constructed from quadratic cardinal B-splines to solve two-dimensional partial differential equations. The formulation is developed through a tensor-product approach, first defined in one dimension and then generalized to two dimensions. Its effectiveness is examined using two representative test cases: an elliptic Dirichlet problem and a convection--diffusion equation with constant coefficients, both discretized on quadrilateral meshes. The proposed element demonstrates superior smoothness, enhanced local approximation accuracy, and improved numerical stability compared with traditional biquadratic Lagrange elements.