New Mixed Element Schemes For Parabolic Equation And Elliptic Eigenvalue Problem

In this article, a new mixed variational formulation to the parabolic equation is givenbased on the less regularity of fux in practical problems. Since in the new mixed vari-ational formulation, the pressure belongs to the square integrable space instead of theclassical divergence space, the choices of fnite element pairs become simple and easy.Based on this new formulation, we give its corresponding stable conforming fnite elementapproximation for the P0-P1pair by using Crank-Nicolson time-discretization scheme.Moreover, we derive optimal error estimates in H1-norm and L2-norm for the approxi-mation of pressure p and in L2-norm for the approximation of velocity u. Finally, somenumerical experiments verify the efectiveness and theoretical results of this method.Besides, this method can be extended to utilize the lowest equal order fnite elementpair by adding some simple stabilization terms. In this article, we present a new stabilizedmixed fnite element method based on the less regularity of fux (velocity) for the two-dimensional parabolic equation in practice. The method combines the Crank-Nicolsonscheme with a stabilized mixed fnite element method which is based on local Gauss inte-gral technique by using the lowest equal-order pair for the velocity and pressure (i.e., theP1P1pair). In contrast to other stabilized methods, the new stabilization technique isfree of stabilization parameters, does not require any calculation of high-order derivativesor edge-based data structures, and can be implemented at the element level. For the sta-bilized fnite element solution, the optimal error estimates have been obtained. Finally,the numerical results demonstrate the better stability and convergence of the stabilizedfnite element method for the parabolic equations.Last,we present a new mixed variational formulation to the elliptic eigenvalue problemin this paper. Moreover, it's proven that eigenvalue optimal error is derived for P0P1in some sense. Finally, we give numerical experiments to verify the efectiveness andadvantage of this method.