Stability and error estimation of θ-difference finite element method with C-Bezier basis

Partial differential equations (PDEs) can be solved numerically by finite element method (FEM). theta-difference FEM scheme carries out spatial discretization with FEM and temporal discretization with theta-difference scheme which is the generalization of forward Euler scheme, backward Euler scheme and Crank-Nicolson (CN) scheme. In this paper, C-Bezier basis is used to construct finite dimensional subspace of FEM and full-discrete scheme of parabolic equations is developed with theta-difference FEM scheme. In addition, the stability and error estimation of theta-difference FEM scheme are analyzed. While 0 < theta < 1/2, theta-difference FEM scheme is conditionally stable, and 1/2 <= theta <= 1, this scheme is unconditionally stable. Furthermore, some numerical examples are given to verify the effectiveness of theta-difference FEM scheme. It's worth noting that the numerical precision of C-Bezier basis is improved 3-5 orders of magnitude comparing with classical Lagrange basis and it also has higher numerical accuracy than CN finite difference method(FDM) and backward FDM.