arXiv subjects

Zhiqiang Cai

Publications and source records attributed to Zhiqiang Cai.

42 records · Page 3Linked to original sources

arXiv2016

Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented.

math.NA↗

arXiv2016

An Empirical Study on Academic Commentary and Its Implications on Reading and Writing

The relationship between reading and writing (RRW) is one of the major themes in learning science. One of its obstacles is that it is difficult to define or measure the latent background knowledge of the individual. However, in an academic research setting, scholars are required to explicitly list their background knowledge in the citation sections of their manuscripts. This unique opportunity was taken advantage of to observe RRW, especially in the published academic commentary scenario. RRW was visualized under a proposed topic process model by using a state of the art version of latent Dirichlet allocation (LDA). The empirical study showed that the academic commentary is modulated both by its target paper and the author's background knowledge. Although this conclusion was obtained in a unique environment, we suggest its implications can also shed light on other similar interesting areas, such as dialog and conversation, group discussion, and social media.

cs.CY↗

arXiv2015

Finite Element Methods for Interface Problems: Robust and Local Optimal A Priori Error Estimates

For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to local regularity of the solution. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.

math.NA↗

arXiv2015

A Recovery-Based A Posteriori Error Estimator for H(curl) Interface Problems

This paper introduces a new recovery-based a posteriori error estimator for the lowest order Nedelec finite element approximation to the H(curl) interface problem. The error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results. Under certain assumptions, it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients.

math.NA↗

arXiv2014

Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bounds

math.NA↗

arXiv2014

Recovery-Based Error Estimators for Diffusion Problems: Explicit Formulas

We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e., requiring solutions of global problems with mass matrices). In this paper, we develop fully explicit recovery-based error estimators for lower order conforming, mixed, and non- conforming finite element approximations to diffusion problems with full coefficient tensor. When the diffusion coefficient is piecewise constant scalar and its distribution is local quasi-monotone, it is shown theoretically that the estimators developed in this paper are robust with respect to the size of jumps. Numerical experiments are also performed to support the theoretical results.

math.NA↗