Finite Element Analysis Of The Second Order Parabolic Partial Differential Equations And Variational Inequality Problem With Obstacle Displacement

In this paper, the finite element methods of two kind second order parabolic partial differential equations and the variational inequality problem with obstacle displacement and the convergence and super-convergence are investigated for different conditions.Firstly, a kind of parabolic type integro-differential equation is discussed with the bilinear finite element. By use of novel technique of interpolation and Ritz projection combination and the interpolation post-processing method, the super-close and super-convergence results in Hl norm with order O?h²? can be obtained under lower regularity requirements of the exact solution, which cannot be derived by Ritz projection or interpolation alone in the previous literature. Moreover, the comparison of the different methods and results is also made.Secondly, the famous low order nonconforming EQ₁^rot finite element is applied to a class of nonlinear reaction-diffusion equations. On one hand, a priori bound of the approximation solution is proved for semi-discrete scheme through Lyapunov functional.Meanwhile, with the help of two special properties of EQ₁^rot element: ??? the consistency error can reach order O?h²?, one order higher than its interpolation error 0?h? ,when the exact solution belongs to H3??? ; ??? the associated interpolation operator is identical to its Ritz projection operator, the super-close estimate of order O?h²? in the broken H¹norm without the traditional assumption on the numerical solution in L_???? . On the other hand, the linearized backward Euler and Crank-Nicolson fully discrete schemes are developed, and the super-close estimates of order O?h² +?? and O?h² +?2? are obtained respectively by use of a new splitting technique in dealing with the consistency error terms. Furthermore, the corresponding global super-convergent results are derived by interpolation post-processing approach. Besides, a numerical example is provided to confirm the theoretical analysis.Finally, the convergence and super-convergence analysis of the second order variational inequality problem with displacement obstacle is investigated with the low order nonconforming constrained rotated Q1 element( CNQ₁^rot ) and EQ₁^rot element,respectively. On the one hand, a useful lemma?see Lemma 4.1? of CNQ₁^rot element is proved for quadrilateral meshes, and the convergence analysis is presented and optimal order error estimate in H¹ norm is obtained. On the other hand, for EQ₁^rot element the super-convergent result in H¹ norm is deduced through some sharp estimates for rectangular meshes. At the same time, numerical results are provided for these two elements to verify the theoretical analysis. It is worthy to be mentioned is that this result has never been reported in the existing literature.