# On the accuracy of finite element approximations to a class of interface problems

@article{Guzmn2016OnTA, title={On the accuracy of finite element approximations to a class of interface problems}, author={Johnny Guzm{\'a}n and Manuel A. S{\'a}nchez and Marcus Sarkis}, journal={Math. Comput.}, year={2016}, volume={85}, pages={2071-2098} }

The jump is defined as [∇u · n] = ∇u · n + ∇u ·n where u = u|Ω± and n is the unit outward pointing normal to Ω (see figure 1). Also, we denote [u] = u − u. Many numerical methods have been developed for problem (1.1). Perhaps the most notable ones are the finite difference method of Peskin [18] (i.e., immersed boundary method) and the method of LeVeque and Li [11] (i.e., the immersed interface method ; see also the method of Mayo [14, 15, 16]) .The method of LeVeque and Li [11] was developed…

## 28 Citations

### Analysis of the immersed boundary method for a finite element Stokes problem

- MathematicsNumerical Methods for Partial Differential Equations
- 2018

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force…

### Finite Element Methods for Interface Problems Using Unfitted Meshes: Design and Analysis

- Mathematics, Computer Science
- 2016

Novel numerical methods are proposed for the model interface problems, proving optimal convergence of the errors and robust error estimates with respect to the curvature of the interface and the physical parameters of the problems, under discretizations non-fitted with the interface.

### A high-order source removal finite element method for a class of elliptic interface problems

- Computer ScienceApplied Numerical Mathematics
- 2018

### Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems

- Computer ScienceJ. Comput. Appl. Math.
- 2020

### Monotone Finite Volume Schemes for Diffusion Equation with Imperfect Interface on Distorted Meshes

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2018

In this paper, we prove the solution of diffusion equation with imperfect interface is positivity-preserving and a monotone finite volume method is presented to obtain the nonnegative solution on…

### A group of immersed finite-element spaces for elliptic interface problems

- Mathematics
- 2019

We present a unified framework for developing and analysing immersed finite element (IFE) spaces for solving typical elliptic interface problems with interface independent meshes. This framework…

### A Nonconforming Immersed Finite Element Method for Elliptic Interface Problems

- MathematicsJ. Sci. Comput.
- 2019

A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient and error estimates in energy and L2-norms are proved to be better than O(h\sqrt{|\log h|}) and O( h2|logh|), respectively, where the jump discontinuity factors reflectJump discontinuity.

### Finite Element Methods For Interface Problems On Local Anisotropic Fitting Mixed Meshes

- Computer ScienceArXiv
- 2020

A simple and efficient interface-fitted mesh generation algorithm which can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface and a new finite element method is proposed for second order elliptic interface problems based on the resulting mesh.

### A new parameter free partially penalized immersed finite element and the optimal convergence analysis

- Mathematics, Computer ScienceNumerische Mathematik
- 2022

The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature and a new trace inequality which is necessary to prove the optimal convergence of immersed finiteelement methods is established on interface elements.

### An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid

- Computer Science
- 2018

This paper designs and implements a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane and forms a time discretization that yields an unconditionally energy stable scheme.

## References

SHOWING 1-10 OF 39 REFERENCES

### ON THE ACCURACY OF FINITE DIFFERENCE METHODS FOR ELLIPTIC PROBLEMS WITH INTERFACES

- Mathematics
- 2006

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque…

### A numerical method for solving variable coefficient elliptic equation with interfaces

- Mathematics
- 2005

### A new multiscale finite element method for high-contrast elliptic interface problems

- Computer ScienceMath. Comput.
- 2010

We introduce a new multiscale finite element method which is
able to accurately capture solutions of elliptic interface problems with high
contrast coefficients by using only coarse quasiuniform…

### A second order virtual node method for elliptic problems with interfaces and irregular domains

- Computer ScienceJ. Comput. Phys.
- 2010

### A weak formulation for solving elliptic interface problems without body fitted grid

- Computer ScienceJ. Comput. Phys.
- 2013

### Numerical Approximation of Large Contrast Problems with the Unfitted Nitsche Method

- Mathematics
- 2011

The application of Nitsche’s method to set up a robust approximation of interface conditions in the framework of the finite element method is studied, which occurs when the computational mesh does not conform with the interface between subproblems.

### Properties of Discrete Delta Functions and Local Convergence of the Immersed Boundary Method

- MathematicsSIAM J. Numer. Anal.
- 2012

This work shows how a recently introduced property of discrete delta functions---the smoothing order---is important in the determination of local convergence rates.

### Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions

- Mathematics
- 2011

Abstract. 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…

### The convergence of the bilinear and linear immersed finite element solutions to interface problems

- Mathematics, Computer Science
- 2012

It has been shown that the bilinear and linear IFE solutions converge to the exact solution under the usual assumptions about the meshes and regularity.