H1H1-Super-convergence of center finite difference method based on P1P1-element for the elliptic equation

In this paper, the center finite difference (CFD) method for the elliptic equation is deduced by the P1P1-element and the first-order discrete partial differential equation over the dual element K∗K∗ in the 1D or 2D domain. Next, the coefficient matrix of the CFD method is explicitly reduced and the H1H1-stability and convergence of the CFD solution uhuh is provided. Furthermore, the H1H1-super-convergence of uhuh to IhuIhu is obtained under the case of the almost-uniform mesh. Based on the H1H1-super-convergence of uhuh to IhuIhu, the optimal L2L2-error estimate of the numerical solution uhuh and the H1H1-super-convergence error estimate of the interpolation solution I2h2uh are derived respectively. Finally, some numerical tests are made to show the analytical results of the CFD method.