p-th degree immersed finite element for boundary value problems with discontinuous coefficients

In this manuscript we present a p-th degree immersed finite element method for solving boundary value problems with discontinuous coefficients. In this method, interface jump conditions are employed in the finite element basis functions, and the mesh does not have to be aligned with coefficient discontinuity. We show that under h refinement the immersed finite element solution converges to the true solution at the optimal O(hp+1) and O(hp) rates in the L2 and H1 norms, respectively. Furthermore, we show that the immersed finite element solution converges exponentially fast under p refinement and hp refinement. Numerical examples are provided to illustrate features of this immersed finite element method