Analysis of Discontinuous Bubble Immersed Finite Element Methods for Elliptic Interface Problems with Nonhomogeneous Interface Conditions

In this paper, we analyze the Lagrange and Crouzeix–Raviart type immersed finite element methods for elliptic interface problems with nonhomogeneous interface conditions. The solution of the method is represented as a sum of two functions: one is the so-called discontinuous bubble satisfying the interface conditions approximately, and the other is the immersed finite element solution of the elliptic interface problem with homogeneous interface conditions. The discontinuous bubble can be easily constructed, since it is a piecewise linear polynomial, supported only on the triangles intersecting with the interface, determined by the nodal values or edge averages on the triangles and the given interface conditions. We prove the optimal convergence under the piecewise \(H^2\) $H^2$ regularity assumption. Several numerical experiments are provided to confirm our theoretical results.