Immersed Finite Element Spaces Research Articles

Linear and bilinear immersed finite elements for planar elasticity interface problems

This article is to discuss the linear (which was proposed in [18,19]) and bilinear immersed finite element (IFE) methods for solving planar elasticity interface problems with structured Cartesian meshes. Basic features of linear and bilinear IFE functions, including the unisolvent property, will be discussed. While both methods have comparable accuracy, the bilinear IFE method requires less time for assembling its algebraic system. Our analysis further indicates that the bilinear IFE functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear IFE functions. Numerical examples are provided to demonstrate that both linear and bilinear IFE spaces have the optimal approximation capability, and that numerical solutions produced by a Galerkin method with these IFE functions for elasticity interface problem also converge optimally in both L2 and semi-H1 norms.

Approximation capability of a bilinear immersed finite element space

AbstractThis article discusses a bilinear immersed finite element (IFE) space for solving second‐order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems

We carry out error estimation of a class of immersed finite element (IFE) methods for elliptic interface problems with both perfect and imperfect interface jump conditions. A key feature of these methods is that their partitions can be independent of the location of the interface. These quadratic IFE spaces reduce to the standard quadratic finite element space when the interface is not in the interior of any element. More importantly, we demonstrate that these IFE spaces have the optimal (slightly lower order in one case) approximation capability expected from a finite element space using quadratic polynomials.

Quadratic immersed finite element spaces and their approximation capabilities

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces.

An immersed finite element space and its approximation capability

AbstractThis article discusses an immersed finite element (IFE) space introduced for solving a second‐order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body‐fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004