A new analysis of isoparametric bilinear finite volume element scheme based on trapezoidal rule for anisotropic diffusion problems

We studied the coercivity and error estimate of a modified isoparametric bilinear finite volume element scheme for anisotropic diffusion problems on quadrilateral meshes, where the scheme is obtained by employing the trapezoidal rule to approximate the line integrals in classical $Q_1$-finite volume element method. By an element analysis approach, we propose a new sufficient condition to ensure the coercivity result of the scheme, which is better than the existing results in [Q. Hong and J. Wu, Adv. Comput. Math., 44 (2018), 897-922]. Under $h^2$-uniform quadrilateral mesh assumption, we prove the superconvergence $|u_I-u_h{|}_1=\mathcal{O}(h^2)$, where $u_I$ is the isoparametric bilinear interpolation of exact solution $u$, and $u_h$ is the finite volume element solution. As a result, an optimal $L^2$ error estimate of $u_h$ is obtained. Some numerical experiments were carried out to verify the theoretical findings.