Three-dimensional Immersed Finite Element Method For Second-order Elliptic Interface Problems

We call the jump phenomenon of solution on the discontinuous surface caused by thediscontinuous coefcients the interface problems and call the surface interface. Secondorder elliptic equations with discontinuous coefcients are often used to model interfaceproblems, for example, in material sciences and fluid dynamics when two or more mate-rials or fluids with diferent conductivities or densities or difusions due to conservationlaws. If the interface is smooth enough, then the solution of the interface is smooth, butdue to the jump of the coefcient across the interface, the global regularity is usually lowand has order of H^1+α（Ω）, 0≤α< 1 （see.[2,29,33]）. Because of the low global regularityand the irregular geometry of the interface, achieving accuracy is difcult with standardfinite element methods, unless the elements fit with the interface of general shape.At present, there are two classes of methods which can solve the interface problems,one important class of methods is the fitted finite element methods, which takes theadvantage of the fitted mesh to be aligned with the smooth interface; another class ofthe methods is unfitted or immersed interface method based on the interface-independentmesh, whose space can be established by using the jump condition. To our knowledge, thetheoretical analysis for IFE methods has been carried out only for 1-D and 2-D problems.The analysis for 3-D IFE methods is more complicated and much more important fromthe point of view of application. In this paper we present an immersed finite element method for the elliptic interfaceproblems in three-dimensional space as following:with the jump conditions on the interface:by constructing the nonconforming interface finite element space. we proved that thethree-dimensional immersed finite element space of the problem above can be uniquelydetermined by the values of vertexes on the elements and the interface jump condition.Then, we define the formulation of the problem and proved that the solution of it isuniquely. Optimal-order error estimates for the interpolation are derived by using themultipoint Taylor expansion technique. It shows that the immersed finite element spacehas an approximation capability similar to that of the standard linear finite element s-pace. At last, we derive the optimal error estimates by using trace space and the resultsof the interpolation error estimates.