Polynomial Preserving Recovery Method On Elliptic Interface Problem Based On The Interface Relaxation Method

The elliptical interface problem often occurs in fluid mechanics and materials science where the environment is composed of multiple substances.The main difficulty in the numerical calculation of the ellipse interface problem is that the discontinuity of the parameters on the interface(such as the dielectric constant)causes the overall solution to lack global regularity.The research on the ellipse interface is of great significance in both nature and engineering practice.For Poisson problem and elliptic interface problem,this thesis conducts research on gradient reconstruction algorithm based on Nitsche stabilization and interface relaxation method and polynomial preserving reconstruction(PPR).The content of this article is mainly divided into the following three parts:(1)Realize the gradient reconstruction algorithm,and compare the calculation performance of SPR and PPR.In particular,we expand the PPR(polynomial preserving reconstruction algorithm)under the finite element framework,and construct the unit decomposition expansion layer based on the finite element mesh and the linear basis functions on it,making this type of method easy Extend to high-order finite element.Then we realized the performance comparison of the SPR and PPR methods under the two-dimensional calculation example.(2)On the basis of the Nitsche stabilization finite element format,we introduce the PPR post-processing algorithm based on the Nitsche stabilization finite element method,and use the basic idea of the Nitsche stabilization method to incorporate the mandatory boundary conditions into the variational form.(3)The Nitsche interface relaxation finite element format for general elliptic interface problems is given,and on this basis,the PPR post-processing algorithm for elliptic interface problems based on the Nitsche interface relaxation finite element format is established.Numerical experiments show that the Nitsche interface relaxation finite element method has obvious advantages in the accuracy of the numerical solution of the elliptic interface problem,while the accuracy of the gradient approximation of the numerical solution is worse.The problem of insufficient precision of numerical gradient is solved by designing a PPR post-processing algorithm based on the Nitsche interface relaxation finite element format.