A Modified $P_1$ -- Immersed Finite Element Method

Abstract

In recent years, the immersed finite element methods (IFEM) introduced in [20], [21] to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface to cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence.In this paper, we propose an improved version of the P 1 based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional P 1 finite element method.We prove H 1 and L 2 error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme.

Discontinuous Bubble Immersed Finite Element Method for Poisson-Boltzmann Equation

“…The main advantage of IFEM is that IFEM use structured grids so that the data structure of discretized system is simpler than those of fitted grids and it is easy to develop efficient solvers. The convergence analysis for the early version of IFEM is given in [6,23,24] for the homogeneous jump cases. IFEM has been applied to other problems such as multiphase flows in porous media [19], Robin type elliptic problems [21] and elasticity problem [18,22,25].…”

“…IFEM based on Crouzeix-Raviart (CR) P 1 -nonconforming space [7] was introduced in [24]. Also, IFEM was enriched by consistency terms of DG [23,28]. Since it can use uniform grids, we can apply the geometric multigrid solver.…”

A Modified Immersed Finite Volume Element Method for Elliptic Interface Problems

“…In Figure 8, we plot the L ∞ error of MIFVE and IFVE method in the log-log scale versus the mesh size h. Figure 9 shows the MIFVE solutions with N = 64 for these two contrasting coefficients. [21]. The exact solution is chosen as…”

“…To overcome the difficulties, the partially penalized immersed finite element (PPIFE) methods are proposed in the literature [1,[19][20][21]29]. The penalization terms are added to IFE schemes to deal with the negative impacts caused by the discontinuity in the IFE functions.…”

A Consistent Immersed Finite Element Method for the Interface Elasticity Problems

“…Since the estimates of other terms are given in [10], it suffices to show that the term ∑ ∈E ∫ ℎ|{ (u − ℎ u) ⋅ n}| 2 is bounded by ‖u − ℎ u‖ 1,ℎ . It can be proved by splitting (u − ℎ u) ⋅ n into normal and tangential component; see [18] for details. The conclusion will follow from Proposition 2.…”