A Nonstandard Mixed Finite Element Method

When using the standard mixed finite element method to solve second-order elliptic problems,we generally introduce the flow variable=-?as the intermediate vari-able,and use its equivalent form^-1=-?in the actual calculation.However,many practical problems such as porous media diffusion,nonlinear radiative heat conduction problems,etc.,where the diffusion?or heat conduction?coefficientis often degenerated,that is equal to or approaches to zero at some points or in some areas.Obviously,it is no longer feasible to directly use standard mixed finite element methods to solve such problems.In this paper,the second order partial differential equation is taken as the model,and a new mixed finite element method is constructed by introducing the intermediate variable,which is different from the standard mixed finite element method.This method uses the piecewise polynomial function to approximate the original scalar function and the introduced vector-valued function.In the theoretical analysis,this paper constructs a weighted Sobolev space related to the diffusion coefficient,and the proof for posi-tive definiteness of bilinear form solution space is weakened to its subspace by using the Orthogonal decomposition theorem;Then we use the Babu?ska-Brezzi?i.e.-?con-dition of the continuous problem to derive the-condition of the discrete problem;Combining the positive definiteness of the bilinear form with the discrete-condition to obtain the existence and uniqueness of the solution;With the help of the projection operator of the solution space of the continuous problem to the space of the mixed finite element solution,it is obtained that the original and intermediate variables in the con-structed non-standard mixed finite element solution respectively have second-order and first-order convergence rates depending on²-norm.Through a large number of numerical experiments,the validity of the constructed non-standard mixed finite element method and the correctness of the theoretical analysis are verified.Although this non-standard mixed finite element method is constructed,which is based on linear non-degenerate elliptic problems,it is believed that this method can be further extended to degenerate non-linear problems.