Researches On An Efficient Finite Element Method For Nonlinear Coupled Bacterial Equation

In this paper,the nonlinear biological bacteria model?parabolic coupled e-quation?is studied.From the points of Galerkin finite element method and H¹-Galerkin mixed finite element method view,the superclose and superconvergence results of two-grid algorithm for different fully discrete schemes are obtained,respe-ctively,and some numerical examples are given to verify the theoretical analysis.First,the backward Euler and Crank-Nicolson fully discrete schemes for the model are established by use of the bilinear finite element.Based on the fixed point theorem,the existence and uniqueness of solutions for two fully discrete schemes are given,respectively.And with the help of the known high accuracy re-sults,the combination technique of the interpolation and Ritz projection and dis-crete derivative transfer trick,the superclose results O?h²+??and O?h²+?²?are obtained for the above two schemes,respectively.After that,in order to improve the efficiency of calculation,the two-grid algorithms are constructed for both two schemes.And by use of the above technologies and results,the superclose result-s O?h²+H⁴+??and O?h²+H⁴+?²?for corresponding algorithms are obtained.At the same time,by the interpolation postprocessing approach,the satisfying results of global superconvergence are obtained.Then,by constructing auxiliary variables ???=?u,???=?v and by use of the element pair Q₁₁/Q_1,0 × Q_0,1,the simidiscrete and backward Euler fully discrete schemes of H¹-Galerkin mixed finite element for the model are established.And by existence and uniqueness theorem of solutions of differential equation and fixed point theorem,the existence and uniqueness of solutions for above schemes are given,respectively.At the same time,with the help of the known high accuracy results,the superclose results O?h²+??are obtained for original variables u,v in H¹ norm and auxiliary variables p,qin L2 norm,respectively.After that,the two-grid algorithm is also established,and the corresponding superclose and super-convergence results for original and auxiliary variables are obtained,respectively.Finally,numerical examples of the above schemes are given,and it can be seen that numerical results are consistent with the theoretical analysis.Apart from these,the examples show that two-grid algorithms have high computing efficiency,and the CPU cost required are only half of Galerkin finite element methods for the examples given in this paper.