An Immersed Interface Finite Element Method For Anisotropic Flow Models In Porous Media

We consider second order elliptic interface equations with discontinuous difusion-coefcients, which describes many physical models in material sciences and fluiddynamics when two or more distinct materials or fluids with diferent densities ordifusivity are involved. We call the jump phenomenon of solution on the discon-tinuous surface caused by the discontinuous coefcients the interface problems andcall the discontinuous surface the interface. The solution of these interface problemsmust satisfy interface jump conditions due to conservation laws. If the interface issmooth enough, then the solution of the interface problem is also smooth in individ-ual regions where the coefcient is smooth, but due to the jump of the coefcientacross the interface, the global regularity is usually low, and the solution usually be-longs to H1+α() for some0≤α <1. Standard finite element methods are difcultto achieve high accuracy.Various numerical experiments have recognized that the immersed interfacefinite element(IIFE) method is very efective for solving elliptic interface problems.There are two classes for IIFE method, one important class of the method is theone based on Lagrange-nodal-polynomials proposed by [14] and developed in [3,5,6,10,11,17](also see the references cited therein for the literature of the immersedmethod), especially, the theoretical analysis of error estimates was obtained in [3];another subclass of the IIFE method was developed in [8] based on Crouzeix-Raviart-type polynomials.The interface problems in the above studies are all assumed that the difusioncoefcients are piecewise scale functions, which describe the isotropic flow models. In this paper, we extend the difusion coefcient to piecewise definite-positive matrixdescribing a kind of anisotropic flow models in porous media. We divide the paperinto two parts. As for the analysis of the two classes IIFE methods in [3] and [8],we prove the possibility of the generalizations, point out the problems in the errorestimates and improve the estimates respectively.In the first part, we apply the IIFE method based on Lagrange-nodal-polynomialsto solve the anisotropic flow models. At first, with the help of three counter-examples, we derive the condition of the generalization. Based on the condition,we construct the IIFE space. Due to the problems of error estimates in [14], wepropose partially penalty IIFE method, add two penalty terms to weaken the non-conforming in the common edge of two adjacent interface elements. After the proofof the consistence and solvability of the procedure, we prove the IIFE solution pos-sesses the optimal-order error estimates in the energy norm and L2norm.In the second part, we apply the IIFE method based on Crouzeix-Raviart-type polynomials to solve the anisotropic flow models. Firstly, we prove that thegeneralization is workable for any definite-positive difusion matrix and we do notneed to add any condition like the first part. Then we construct the IIFE procedure,which is used to solve two numerical examples. And the numerical experiments showthat the finite element solution possesses the optimal-order error estimates in bothH1norm and L2norm. As for theoretical analysis of error estimates, consideringthe problems existed in the error estimates in [8], we can also apply the partiallypenalty idea to discuss.