Higher-order immersed finite element spaces for second-order elliptic interface problems with quadratic interface

Abstract

In this manuscript, we present quadratic immersed finite element (IFE) spaces to be used with the interior penalty IFE method proposed in Adjerid (Int. J. Numer. Anal. Model., 2013, accepted) to solve interface problems with a quadratic interface. Quadratic IFE spaces for interface problems with quadratic interfaces are developed using an affine mapping between the reference and the physical elements. Two different approaches for imposing the interface jump conditions are proposed: (i) a weak form of jump conditions using Legendre polynomials and (ii) a pointwise form by imposing the conditions at some particular points. We give a procedure to construct IFE shape functions, investigate the optimal approximation capability of the proposed IFE spaces, and present numerical results showing optimal convergence.