Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods

Abstract

In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contrast.Key words. immersed finite element methods, a posteriori error estimation, interface problems, residual-based. AMS subject classifications. 35R05, 65N15, 65N301. Introduction. Interface problems arise widely in the multi-physics and multimaterial applications in the fluid mechanics and material science. The governing partial differential equations (PDEs) for interface problems are usually characterized with discontinuous coefficients that represent different material properties. The solutions to the interface problems often involve kinks, singularities, discontinuity, and other non-smooth behaviors. It is therefore challenging to obtain accurate numerical approximations for interface problems. Moreover, the complexity of the interface geometry may add an extra layer of difficulty to the numerical approximation.In general, there are two classes of numerical methods for solving interface problems. The first class of methods use interface-fitted meshes, i.e., the meshes are tailored to fit the interface, see the left plot in Figure 1.1. Methods of this type include classical finite element methods [14], discontinuous Galerkin methods [4] and recently developed weak Galerkin methods [30], to name a few. The second class of methods use unfitted meshes which are independent of the interface, as illustrated in the right plot in Figure 1.1. In the past few decades, many numerical methods based on unfitted meshes have been developed. In the finite difference framework, since the pioneering work of immersed boundary method [31] by Peskin, many numerical methods of finite difference type have been developed such as immersed interface method [20,23], matched interface and boundary method [35]. In finite element framework, there are quite a few numerical methods developed, for instance, the general finite element method [5], unfitted finite element method [17], multi-scale finite element method [19], extended finite element method [29], and immersed finite element method (IFEM) [22,24]. A great advantage for unfitted numerical methods is that they can circumvent (re)meshing procedure which can be very expensive especially for time dependent problems with complex interface geometry or for shape optimization processes that require repeated updates of the mesh.The IFEM was first developed in [22] for a one-dimensional elliptic interface problem and then extended to higher-order approximations [1,2,10,11] and to higher- *

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

“…This algorithm constitutes an important building block of this paper. We refer to [38,37,25] for the hp a posteriori error analysis on conforming finite element meshes and the recent work on a posteriori error analysis for immersed finite element methods [29] and the cut finite element method [17].…”

An a posteriori error estimator for isogeometric analysis on trimmed geometries

“…Therefore, our work is strongly related to the immersed or unfitted mesh methods. In particular, a reliable and efficient residual-based a posteriori error estimator is studied in [28] for partially penalized linear immersed finite element method applied to elliptic interface problems. In the context of the cut finite element method, an implicit a posteriori estimator of the energy error due to numerical approximation is introduced in [29] for any polynomial degree, but its efficiency is only demonstrated numerically.…”

“…In that paper, to avoid the dependence on the location of domain-mesh intersection, the efficiency for the term of ghost penalty is shown globally. Both [28] and [30] deal with a linear finite element basis, while in the very recent work [31], a reliable and efficient hp-residual type error estimator is given in the case of high-order unfitted finite element for interface problems in the framework of the local discontinuous Galerkin method.…”

A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

“…In the past decades, several unfitted-mesh methods have been developed for solving Stokes interface problems, such as CutFEM [15], Nitsche's FEM [36], XFEM [9], fictitious domain FEM [31,34], to name only a few. The immersed finite element method (IFEM) [24,26,18,11,14,30] is a class of unfitted-mesh finite element methods for solving interface problems. The main idea of IFEM is to incorporate the interface jump conditions in the construction of IFE basis functions.…”