HIGHER DEGREE IMMERSED FINITE ELEMENT METHODS FOR SECOND-ORDER ELLIPTIC INTERFACE PROBLEMS

Abstract. We present higher degree immersed finite element (IFE) spaces that can be used to solve two dimensional second order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. The interpolation errors in the proposed piecewise pth degree spaces yield optimal O(hp+1) and O(hp) convergence rates in the L2 and broken H1 norms, respectively, under mesh refinement. A partially penalized method is developed which also converges optimally with the proposed higher degree IFE spaces. While this penalty is not needed when either linear or bilinear IFE space is used, a numerical example is presented to show that it is necessary when a higher degree IFE space is used. Key words. Immersed finite element, immersed interface, interface problems, Cartesian mesh method, structured mesh, higher degree finite elemen