Biquadratic element discrete duality finite volume method for solving elliptic equations on quadrilateral mesh

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H1-norm, with a 2nd-order convergence, and in the L2-norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L∞-norm on several distorted meshes.