Skip to main content

Jinru Chen

Followers

5

Following

3

Co-authors

2

Public Views

Interests

Uploads

Papers by Jinru Chen

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

arXiv (Cornell University), Aug 6, 2021

In this paper, an important discovery has been found for nonconforming immersed finite element (I... more In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H 1 -and L 2 -norms are proved and confirmed with numerical experiments.

Contrast analysis between the trajectory of the planetary system and the periodicity of solar activity

Annales Geophysicae, May 17, 2017

The relationship between the periodic movement of the planetary system and its influence on solar... more The relationship between the periodic movement of the planetary system and its influence on solar activity is currently a serious topic in research. The kinematic index of the planet juncture index has been developed to find the track and variation of the Sun around the centroid of the solar system and the periodicity of solar activity. In the present study, the kinematic index of the planetary system's heliocentric longitude, developed based on the orbital elements of planets in the solar system, and it is used to investigate the periodic movement of the planetary system. The kinematic index of the planetary system's heliocentric longitude and that of the planet juncture index are simulated and analyzed. The numerical simulation of the two kinematic indexes shows orderly orbits and disorderly orbits of 49.9 and 129.6 years, respectively. Two orderly orbits or two disorderly orbits show a period change rule of 179.5 years. The contrast analysis between the periodic movement of the planetary system and the periodicity of solar activity shows that the two phenomena exhibit a period change rule of 179.5 years. Moreover, orderly orbits correspond to high periods of solar activity and disorderly orbits correspond to low periods of solar activity. Therefore, the relative movement of the planetary system affects solar activity to some extent. The relationship provides a basis for discussing the movement of the planetary system and solar activity.

A two-level additive Schwarz preconditioner for the Nitsche extended finite element approximation of elliptic interface problems

Computers & mathematics with applications, Oct 1, 2023

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

ESAIM, Jul 1, 2023

In this paper, an important discovery has been found for nonconforming immersed finite element (I... more In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both 1-and 2-norms are proved and confirmed with numerical experiments.

Unfitted finite element methods for the heat conduction in composite media with contact resistance

Numerical Methods for Partial Differential Equations, Oct 5, 2016

This article is concerned with the heat conduction problem in composite media. In practical appli... more This article is concerned with the heat conduction problem in composite media. In practical applications, the composite materials often do not contact well and there exist gaps between the contacting materials. This leads to the thermal contact resistance effect which results in a discontinuity of the temperature across the interface. In this article, an unfitted finite element method is proposed to solve the problem. Different from the traditional finite element method, the proposed method uses structured meshes that allow the interface to cut through. To avoid integrating on curved domains and interfaces, the interface is approximated by a broken line/plane corresponding to the triangulation. In addition, a ghost‐penalty is added to recover the condition number of the stiffness matrix to O ( h − 2 ) with a hidden constant independent of the mesh‐interface geometry. A rigorous analysis is provided. Finally, numerical tests are presented to verify the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 354–380, 2017

A Posteriori Error Estimates of a Weakly Over-Penalized Symmetric Interior Penalty Method for Elliptic Eigenvalue Problems

East Asian Journal on Applied Mathematics, Nov 1, 2015

A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue... more A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.

An extended mixed finite element method for elliptic interface problems

Computers & mathematics with applications, May 1, 2022

In this paper, we propose an extended mixed finite element method for elliptic interface problems... more In this paper, we propose an extended mixed finite element method for elliptic interface problems. By adding some stabilization terms, we present a mixed approximation form based on Brezzi-Douglas-Marini element space and the piecewise constant function space, and show that the discrete inf-sup constant is independent of how the interface intersects the triangulation. Furthermore, we derive that the optimal convergence holds independent of the location of the interface relative to the mesh. Finally, some numerical examples are presented to verify our theoretical results.

An additive Schwarz preconditioner for the mortar-type rotated FEM for elliptic problems with discontinuous coefficients

Applied Numerical Mathematics, Jul 1, 2009

In this paper, we propose an additive Schwarz preconditioner for the mortar-type rotated Q 1 fini... more In this paper, we propose an additive Schwarz preconditioner for the mortar-type rotated Q 1 finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. The work here is an extension of the research presented in [L. Marcinkowski, Additive Schwarz method for mortar discretization of elliptic problems with P 1 non-conforming elements, BIT 45 (2005) 375-394]. Our analysis is valid for rectangular or L-shaped domains, which are partitioned by rectangular subdomains and meshes. We have shown that our proposed method has a quasioptimal convergence behavior, i.e., the condition number of the preconditioned problem is O((1 + log(H/h)) 2), which is independent of the jump in the coefficient. Numerical experiments presented in this paper have confirmed our theoretical analysis.

Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems

Numerical Methods for Partial Differential Equations, Mar 22, 2017

In this article, we apply a modified weak Galerkin method to solve variational inequality of the ... more In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.

A Conforming Virtual Element Method Based on Unfitted Meshes for the Elliptic Interface Problem

Journal of Scientific Computing, May 28, 2023

Cascadic Multigrid Method for Mortar-Type Rotated Q1 Element

In this paper, the cascadic multigrid method for mortar-type rotated Q1 element is discussed. It ... more

A Cartesian grid nonconforming immersed finite element method for planar elasticity interface problems

Computers & Mathematics with Applications, 2017

In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian m... more In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian meshes is developed for solving planar elasticity interface problems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover, the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimentally, extensive numerical examples are given to show that the approximation orders in L 2 norm and semi-H 1 norm are optimal under various Lamé parameters settings and different interface geometry configurations.

26 Multigrid for the Mortar-type Nonconforming Element Method for Nonsymmetric and Indefinite Problems

A Priori and a Posteriori Error Estimates for H(div)-Elliptic Problem with Interior Penalty Method

Communications in Computational Physics, 2013

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(di... more In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

Science China Mathematics, 2012

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element app... more In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L 2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

Robust multigrid method for the planar linear elasticity problems

Numerische Mathematik, 2009

A Unified Mortar Condition for Nonconforming Finite Elements

Journal of Scientific Computing, 2014

A continuously interpolated mortar condition is proposed for 2D and 3D P k nonconforming finite e... more A continuously interpolated mortar condition is proposed for 2D and 3D P k nonconforming finite elements on nonmatching grids. The resulting finite element method is an optimal order one in solving elliptic equations. Numerical tests on the 2D P 1 , 2D P 2 and 3D P 1 nonconforming finite elements are provided.

Partition of unity method on nonmatching grids for the Stokes problem

Journal of Numerical Mathematics, 2005

We consider the Stokes Problem on a plane polygonal domain Ω ⊂ R 2. We propose a finite element m... more We consider the Stokes Problem on a plane polygonal domain Ω ⊂ R 2. We propose a finite element method for overlapping or nonmatching grids for the Stokes Problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.

A Local Multilevel Preconditioners for the Adaptive Mortar Finite Element Method

Journal of Computational Mathematics, 2013

In this paper, we propose a local multilevel preconditioner for the mortar finite element approxi... more In this paper, we propose a local multilevel preconditioner for the mortar finite element approximations of the elliptic problems. With some mesh assumptions on the interface, we prove that the condition number of the preconditioned systems is independent of the large jump of the coefficients but depends on the mesh levels around the cross points. Some numerical experiments are presented to confirm our theoretical results.

An Unfitted Discontinuous Galerkin Method for Elliptic Interface Problems

Journal of Applied Mathematics, 2014

An unfitted discontinuous Galerkin method is proposed for the elliptic interface problems. Based ... more An unfitted discontinuous Galerkin method is proposed for the elliptic interface problems. Based on a variant of the local discontinuous Galerkin method, we obtain the optimal convergence for the exact solutionuin the energy norm and its fluxpin theL2norm. These results are the same as those in the case of elliptic problems without interface. Finally, some numerical experiments are presented to verify our theoretical results.