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This paper is for proving that the partially penalized immersed finite element (PPIFE) methods developed in [25] converge optimally under the standard piecewise H² regularity assump-tion for the exact solution. In energy norms, the error estimates given in this paper are better than those in [25] where a stronger piecewise H³ regularity was assumed. Furthermore, with the standard piecewise H² regularity assumption, this paper proves that these PPIFE methods also converge optimally in the L² norm which could not be proved in [25] because of the excessive H³ regularity requirement. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multi-layer porous wall model for coronary drug-eluting stents that leads to an interface problem whose coefficients have multiple discontinuous points, and an imperfect contact interface jump condition is imposed at the first discontinuous point where the stent meets the artery. The existence and uniqueness of the solution to the related weak problem are established. A linear and a quadratic immersed finite element (IFE) methods are developed for solving this interface problem. Error estimation is carried out to show that the proposed IFE methods converge optimally. Numerical examples are presented to demonstrate features of these IFE methods. [ABSTRACT FROM AUTHOR]

This article presents an error analysis of a Hermite cubic immersed finite element (IFE) method for solving interface problems of the differential equation modeling a Euler-Bernoulli beam made up of multiple materials together with suitable jump conditions at material interfaces. The analysis consists of three essential groups. The first group is about IFE functions including bounds for the IFE shape functions and inverse inequalities. The second group is about error bounds for IFE interpolation derived with a multi-point Taylor expansion technique. The last group, and perhaps the most important group, is for proving the optimal convergence of the IFE solution generated by the usual Galerkin scheme based on the Hermite cubic IFE space considered in this article. [ABSTRACT FROM AUTHOR]

This paper is to develop immersed finite element (IFE) functions for solving second order elliptic boundary value problems with discontinuous coefficients and non-homogeneous jump conditions. These IFE functions can be formed on meshes independent of interface. Numerical examples demonstrate that these IFE functions have the usual approximation capability expected from polynomials employed. The related IFE methods based on the Galerkin formulation can be considered as natural extensions of those IFE methods in the literature developed for homogeneous jump conditions, and they can optimally solve the interface problems with a nonhomogeneous flux jump condition. [ABSTRACT FROM AUTHOR]

We present a least squares framework for constructing p-th degree immersed finite element (IFE) spaces for typical second-order elliptic interface problems. This least squares formulation enforces interface jump conditions including extended ones already proposed in the literature, and it guarantees the existence of p-th IFE shape functions on interface elements. The uniqueness of the proposed p-th degree IFE shape functions is also discussed. Computational results are presented to demonstrate the approximation capabilities of the proposed p-th IFE spaces as well as other features. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]