An Immersed Finite Element Method For Parabolic Interface Problems

Many actual seepage flow process, such as in material sciences and fluid dy-namics when two or more distinct materials or fluids with different densities or diffusivity are involved, can be described by the interface problems. To establish its accurate and efficient numerical simulation and numerical analysis is of significant importance for revealing the movement mechanism of the actual seepage and the guidance of scientific and engineering practice. This article aims to put forward to the corresponding finite element simulation for a class of second order parabolic interface problems, and establish a strict numerical analysis theory. The outline of the paper is as follows:1. The partially penalty immersed interface finite element method based on the linear Lagrange interpolationIn this part, we use the linear Lagrange interpolation to construct the im-mersed interface finite element space. With the condition that the diffusion coeffi-solvability of a function by its values at three vertex of a triangle in the finite element space. Further, we weaken the restriction used in [12] in the con-struction of the space to and prove that a function in the finite element space can be determined uniquely by the value of three vertex of the triangle element. Based on the condition, we establish the partially penalty symmetric, nonsymmetric and incomplete immersed interface finite element(PIFE) formulation for the second order parabolic interface problems. Using the numerical analysis technique such as the elliptic projection and Gronwall’s inequality, we prove the solvability, stability, the optimal error-order estimates in the energy norm and L2norm.2. The partially penalty immersed interface finite element method based on the rotated Q1finite elementIn this part, we use the rotated Q1finite element to construct the immersed interface finite element space. With the condition that the diffusion coefficient we prove that a function in the finite element space can be determined uniquely by the values of four midpoint of the rectangle element’s edges. Based on the condi-tion, we establish the partially penalty symmetric, nonsymmetric and incomplete immersed interface bilinear finite element(PIFE) formulation for the second order parabolic interface problems. Using the numerical analysis technique such as the elliptic projection and Gronwall’s inequation, we prove the solvability, stability and the suboptimal error estimates in the energy norm.