A New Immersed Interface FEM for Elliptic Problems with Discontinuous Coefficients and Nonlinear Local Own Source

)). ( ( )] [( ξ β ξ u g u x x = = (4) Problems of this type arise when we consider a diffusion equation with nonlinear localized chemical reactions. As a result of the reactions the derivatives are discontinuous across the interfaces (local sites of reactions). Some 2D problems with jump conditions, that depend on the solution on the interface are considered by J. Kandilarov and L. Vulkov [2,3,4] using finite difference schemes. The main goal of this work is the application of the Finite Element Immersed Interface Method (FEIIM) to the proposed elliptic problem and theoretical validation of its implementation. The organization of the paper is as follows. In Section 2 the weak formulation of the problem is given. Next, in section 3 the theoretical analysis for the linear source case is done. In Section 4 the nonlinear case is discussed. Finally, numerical experiments that confirm second order of the method are presented.