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

@article{Guzmn2015OnTA, 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={2015}, 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…

## 30 Citations

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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.

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A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter and a new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements.

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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…

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- 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.

### Adaptive discontinuous Galerkin methods for elliptic interface problems

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An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semipermeable membranes is considered and upper and lower bounds of the error in the respective dG-energy norm are proven.

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

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- 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.

## 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…

### Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions

- MathematicsSIAM J. Numer. Anal.
- 2007

A class of new finite- element methods, called immersed-interface finite-element methods, is developed to solve elliptic interface problems with nonhomogeneous jump conditions to provide fast simulation of interface dynamics that does not require remeshing.

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

- Computer ScienceJ. Comput. Phys.
- 2010

### Convergence proof of the velocity field for a stokes flow immersed boundary method

- Mathematics
- 2008

The immersed boundary (IB) method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures. It is characterized by the use of a uniform Cartesian…

### 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…