115 citations · 220 across the 19 of their papers we have counts for
19 papers
Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian
Zhaopeng Hao, Zhiqiang Cai, Zhongqiang Zhang
We study the numerical evaluation of the integral fractional Laplacian and its application in solving fractional diffusion equations. We derive a pseudo-spectral formula for the in…
Adaptive Least-Squares Methods for Convection-Dominated Diffusion-Reaction Problems
Zhiqiang Cai, Binghe Chen, Jing Yang
This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system…
Hybrid A Posteriori Error Estimators for Conforming Finite Element Approximations to Stationary Convection-Diffusion-Reaction equations
Difeng Cai, Zhiqiang Cai
We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an expli…
Adaptive Two-Layer ReLU Neural Network: II. Ritz Approximation to Elliptic PDEs
Min Liu, Zhiqiang Cai
In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adap…
Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation
Min Liu, Zhiqiang Cai, Jingshuang Chen
In this paper, we introduce adaptive neuron enhancement (ANE) method for the best least-squares approximation using two-layer ReLU neural networks (NNs). For a given function f(x),…
Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation
Zhiqiang Cai, Jingshuang Chen, Min Liu
This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equi…
Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law
Zhiqiang Cai, Jingshuang Chen, Min Liu
We introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperfor…
Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements
Cuiyu He, Zhiqiang Cai, Shun Zhang
The well-known Prager-Synge identity is valid in and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this pape…
Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
Zhiqiang Cai, Jingshuang Chen, Min Liu +1
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to ap…
A finite element method for Dirichlet boundary control of elliptic partial differential equations
Shaohong Du, Zhiqiang Cai
This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjo…
A Discretization-Accurate Stopping Criterion for Iterative Solvers for Finite Element Approximation
Zhiqiang Cai, Shuhao Cao, Robert D. Falgout
This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element appro…
Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements
Zhiqiang Cai, Cuiyu He, Shun Zhang
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 o…
Finite Element Methods for Interface Problems: Robust and Local Optimal A Priori Error Estimates
Zhiqiang Cai, Shun Zhang
For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and disconti…
Finite Element Methods for Interface Problems: Robust Residual-Based A Posteriori Error Estimates
Zhiqiang Cai, Cuiyu He, Shun Zhang
For elliptic interface problems, this paper studies residual-based a posteriori error estimations for various finite element approximations. For the conforming and the Raviart-Thom…
Robust A Posteriori Error Estimation for Finite Element Approximation to H(curl) Problem
Zhiqiang Cai, Shuhao Cao, Rob Falgout
In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H(curl) problem with inhomogeneous media and with the right-…
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements
Zhiqiang Cai, Cuiyu He, Shun Zhang
In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The…
A Recovery-Based A Posteriori Error Estimator for H(curl) Interface Problems
Zhiqiang Cai, Shuhao Cao
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 estima…
Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations
Zhiqiang Cai, Rob Falgout, Shun Zhang
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 efficie…
Recovery-Based Error Estimators for Diffusion Problems: Explicit Formulas
Zhiqiang Cai, Shun Zhang
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 rec…