Mixed Element Method For Fourth-Order Semilinear Elliptic Equation And G-Navier-Stokes Equation

This paper mainly studies the following two problems.In the first part,we propose a hybrid finite element method for a class of fourth-order Semilinear Equations.Firstly,we prove the existence and uniqueness of the solution of the mixed problem by constructing an auxiliary problem.In addition,an iterative scheme for solving nonlinear terms is given,and the convergence of the problem is proved under some assumptions.In addition,in order to improve the computing efficiency,we establish a two-grid algorithm to obtain the same optimal error estimation,but reduce the CPU running time.Finally,some numerical examples are given to compare the convergence speed and running time of traditional grid algorithm and two-grid algorithm.In the second part,we discuss a class of evolution equations,and construct a semi discrete and fully discrete scheme for the g-Navier-Stokes equation with weight function g.the semi discrete scheme only uses the finite difference scheme for time,while the fully discrete scheme disperses the finite element in space on this basis,and deduces the optimal error estimation.Using the continuity and discrete BB compatibility conditions,the convergence order O(h + τ)of the numerical solution with weight function g is obtained.