Fully Discrete Finite Element Method Based On Second Order Temporal Schemes For Darcy-Brinkman Equation

The temperature gradient exists when the fluid is heated inhomogeneous and the concentration gradient exists when the composition of the fluid is not uniform.Two diffusion mechanisms(thermal diffusion and concentration diffusion)will lead to double-diffusion convection.Darcy-Brinkman equation can describe some double-diffusive phenomena and come from the combined heat and mass transfer in saturated porous medium.DarcyBrinkman equation plays an important role in many industrial and technological applications which is an important equation that can describe metallurgy,electrochemistry,grain storage,groundwater pollutant transportation and so on.In Darcy-Brinkman equation,there is not only velocity field,but also temperature field and concentration field.It is a multi-physical field coupling problem.Based on the finite element method,this paper presents some efficient numerical algorithms for the Darcy-Brinkman model,which includes the defect-deferred correction finite element method and the time filter method.Firstly,we give a defect-deferred correction finite element method for solving the Darcy-Brinkman equation.The presented method combines the defect-correction method with the deferred correction strategy and involves two steps.The first step(defect step)is to solve a Darcy-Brinkman equation with artificial viscosity term.Then in the second step,a Darcy-Brinkman equation based on the deferred correction approach is solved.This method combines the advantages of the defect correction and deferred correction methods,which can alleviate the difficulty in solving the small viscosity coefficient and improve the accuracy of the time from the first order to the second order.Secondly,we construct the time filter method of the Darcy-Brinkman equation.This method is based on adding a simple time filter to the stable backward Euler numerical format,which increases the accuracy of the backward Euler format from the first order to the second order without increasing the complexity of the calculation.In addition,we give the stability and convergence analysis of the two algorithms,and verify the effectiveness of the algorithms through some numerical examples.