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This paper presents a new numerical method and analysis for solving second-order elliptic interface problems. The method uses a modified nonconforming rotated

Interface problems arise in many applications, such as mechanical analysis in material sciences or fluid dynamics, where two distinct materials or fluids with different conductivities or densities encounter on an interface [

In recent years, the convergence analysis of the finite element method (FEM) for the interface problems has been discussed in many publications. For instance, Babuška [

Immersed methods are effective for solving interface problems, which contain immersed finite difference method (IFDM) and immersed finite element method (IFEM). LeVeque and Li [

However, the works mentioned above are mainly contributions to the conforming FEM. In fact, nonconforming finite element method (NFEM) has some advantages compared with the conforming ones. For example, for the Crouzeix–Raviart-type nonconforming finite element, since the unknowns of the elements are associated with the element edges or faces, each degree of freedom belongs to at most two elements, so using the nonconforming elements facilitates the exchange of information across each subdomain and provides spectral radius estimates for the iterative domain decomposition operator, see [

The remainder of the paper is organized as follows. Firstly, a nonconforming modified rotated

Let

Sketch of the domain

In (

The corresponding variational form of (

Because of the low global regularity of the exact solution in (

As the usual Sobolev spaces, when

In this section, local nonconforming

Assume that

It is well known that the standard

So, by direct calculation, the corresponding finite element interpolation function can be expressed as

Let

There holds the following interpolation error estimate for any given

Consider the discrete variational form of (

Now, some reasonable restrictions on the quadrilateral subdivision

The edges meet the interface at no more than two points

Each edge is passed through at most once except passed at two vertices

It is easy to check that

To describe the local IFE space on an interface element

Note that each piecewise polynomial in

Given a reference interface element, the piecewise function defined by (

For any function

Under this affine mapping

So, we only need to prove that the desired result holds on the reference element.

The values

If we choose

In this section, the convergence analysis and error estimates of the IFEM will be carried out for the elliptic interface problem. In order to do this, the following two important lemmas are proven as follows.

Let

If

Firstly, by the trace theorem, it can be derived that

Secondly, let

Thus, it can be derived that

Let

Noticing that

Applying Lemma

Let

By Strong’s second lemma, it can be obtained that

From now on,

In order to prove (

From [

The error estimate order in Theorem

In order to conquer the asymmetry of the basis function space, one can first choose

In this section, two numerical experiments will be carried out for an elliptic interface problem.

(see [

Errors in

Order | Order | |||
---|---|---|---|---|

0.372938117902 | ||||

1.835491491 | 0.941876033 | |||

1.817310966 | 0.965547404 | |||

0.003679917736 | 1.910475983 | 0.970145853 | ||

0.000962451092 | 1.934888380 | 0.025235556981 | 1.007837138 | |

0.000234073307 | 2.039752808 | 0.012610758212 | 1.000802910 | |

0.000062682198 | 1.900832744 | 0.006318603068 | 0.996977475 |

Errors in

Order | Order | |||
---|---|---|---|---|

1.013510946696 | ||||

0.268072065256 | 1.935145013 | 0.532621122372 | 0.928180123 | |

0.073483750865 | 1.867123716 | 0.275850326414 | 0.949223956 | |

1.917946532 | 0.145081776373 | 0.927019373 | ||

0.005066576895 | 1.940395475 | 0.073089894557 | 0.989122457 | |

0.001387910935 | 1.868096366 | 0.037931441458 | 0.946277755 | |

0.000327592545 | 2.082940559 | 0.019698456819 | 0.945311584 |

Convergence rates in

In this example, the domain is chosen as

Obviously,

Figure

From Tables

Convergence rates in

Errors in

Order | Order | |||
---|---|---|---|---|

0.178346555746 | ||||

1.728749267 | 0.094748546340 | 0.912507640 | ||

0.027942781629 | 1.830309436 | 0.048211021354 | 0.974740816 | |

0.007621493172 | 1.874330067 | 0.958975695 | ||

0.001861050498 | 2.033956470 | 0.012082578254 | 1.037458961 | |

0.000491029936 | 1.922234315 | 0.005817501298 | 1.054456806 | |

0.000117408662 | 2.064272133 | 0.002914705261 | 0.997049624 |

Errors in

Order | Order | |||
---|---|---|---|---|

0.524504640793 | ||||

0.136643751524 | 1.808803586 | 0.274358019661 | 0.934895790 | |

0.058418679138 | 1.225917846 | 0.146031238772 | 0.909782726 | |

0.018747166324 | 1.639757193 | 0.075697236338 | 0.947964487 | |

0.004139677913 | 2.179082117 | 0.038901806739 | 0.960403469 | |

0.001182704768 | 1.807428537 | 0.019973384828 | 0.961758318 | |

0.000325298041 | 1.862255947 | 0.009713801702 | 1.039970902 |

This paper discusses a modified nonconforming rotated

We should point out that this method is suitable for parabolic-type or hyperbolic-type interface problems by using a suitable full discretization scheme. However, the method cannot be applied to other very popular nonconforming FEs, such as

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant nos. 11501527, 11301392, and 71601119).

_{1}-nonconforming triangular finite element method for elliptic and parabolic interface problems

_{1}element on non-tensor product anisotropic meshes