Polynomial Preserving Recovery Finite Element Method For The Stokes-Darcy System

Polynomial Preserving Recovery(PPR)is a method of reconstructing continuous approx-imate gradients through post-processing technology.Error estimation using PPR is a recon-structive error estimation,which is widely used in adaptive in the finite element method.In this thesis,a polynomial gradient-preserving reconstruction method for the finite element of the Stokes-Darcy equation is studied,and the PPR method is applied to the adaptive finite element method to solve the Stokes-Darcy equation.Firstly,the main idea of the PPR method is introduced.Secondly,the Stokes-Darcy e-quation and its finite element discretization are introduced.Furthermore,based on the finite element numerical results of the Stokes-Darcy equation,construct the PPR gradient reconstruc-tion operator,PPR gradient reconstruction posterior error estimates for the finite element of the Stokes-Darcy equation are given.Then,according to the properties of the PPR gradient reconstruction operator and com-bined with the finite element super-convergence theory,the PPR super-convergence analysis under uniform uniform grid and lightly structured grid is given,which verifies the PPR gradient reconstruction under different grid conditions satisfy the superconvergence property.Finally,some numerical example are given to verify the polynomial gradient-preserving reconstruction of the finite element method of the Stokes-Darcy equation.Example 1 gives the posterior error estimation under the~2norm of the reconstructed numerical solution gradient of the Stokes-Darcy equation under a uniform uniform grid.Example 2 applies the PPR method to the adaptive solution of the Stokes-Darcy equation,an adaptive finite element algorithm for the Stokes-Darcy equation is proposed to verify the reliability and validity of the posterior error es-timator based on PPR gradient reconstruction.By reconstructing the gradient approximation at the nodes of the finite element mesh,it is verified that the reconstructed gradient superconverges to the exact gradient,and the posterior error estimator based on the PPR gradient reconstruction given in this thesis is asymptotically accurate,and it can be used for the adaptive refinement of finite element meshes.Numerical experiments verify that the use of PPR for error estimation is effective and feasible.