New Two-level Finite Element Methods For The Incompressible Convective Brinkman-forchheimer Equations

The incompressible convective Brinkman-Forchheimer(B-F)equations are nonlinear reactive-convection-diffusion.In recent years,although there are many researches on the B-F equations,the finite element methods on the equations are still less.For solving this equations by standard finite element method,a large nonlinear system must be solved,which poses a great challenge to the existing computing resources.Hence,it is needful to construct and study algorithms with long-time stability,high efficiency and low consumption.The two-grid method is an efficient method for solving nonlinear partial differential equations,which can save a lot of calculation time.The variational multiscale method is to look at the space as an analytic scale space and projects the standard finite element space into an appropriate space as its large scale space.Combining the above methods,new high efficient and low-consumption algorithms for solving the B-F equations are constructed.The specific arrangements are as follows:In the first part,the two-level finite element methods with backtracking technique are utilized to solve the B-F equations.In the first,the basic idea of the methods is to solve a nonlinear system on a coarse grid.Then,a linear problem with two different methods on a fine grid are solved.Finally,a linear correction problem is settled on the coarse grid.At the same time,the error estimation is carried out.And then the numerical experiments are carried out to verify the correctness of the theoretical analysis.The optimal convergence order,the effectiveness and stability of our algorithms can be obtained.In the second part,the B-F equations are solved by the two-level variational multiscale method.Firstly,the corresponding numerical scheme is given.Then,the convergence is analyzed.Finally,the correctness of theoretical analysis,stability and effectiveness of the algorithm are verified by numerical experiments.