16 citations · 33 across the 16 of their papers we have counts for
17 papers · 1 filter
Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous Discretizations
Shun Zhang
Weakly imposed boundary conditions in first-order system least-squares methods go back to Jespersen (1977), but two questions have remained open for decades: the right weights, esp…
Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations
Yuxiang Liang, Shun Zhang
In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may…
Several Proofs of Coerciveness of First-Order System Least-Squares Methods for General Second-Order Elliptic PDEs
Shun Zhang
In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal…
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…
Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations
Qunjie Liu, Shun Zhang
In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous…
Battling Gibbs Phenomenon: On Finite Element Approximations of Discontinuous Solutions of PDEs
Shun Zhang
In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions…
Adaptive First-Order System Least-Squares Finite Element Methods for Second Order Elliptic Equations in Non-Divergence Form
Weifeng Qiu, Shun Zhang
This paper studies adaptive first-order least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical f…
Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Approximations
Shun Zhang
For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the Raviart-Thomas and Brezzi-…
Primal-Dual Reduced Basis Methods for Convex Minimization Variational Problems: Robust True Solution A Posteriori Error Certification and Adaptive Greedy Algorithms
Shun Zhang
In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develo…
Adaptive Least-Squares Finite Element Methods for Linear Transport Equations Based on an H(div) Flux Reformulation
Qunjie Liu, Shun Zhang
In this paper, we study the least-squares finite element methods (LSFEM) for the linear hyperbolic transport equations. The linear transport equation naturally allows discontinuous…
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…
An Efficient Implementation of Brezzi-Douglas-Marini (BDM) Mixed Finite Element Method in MATLAB
Shun Zhang
In this paper, a MATLAB package bdm_mfem for a linear Brezzi-Douglas- Marini (BDM) mixed finite element method is provided for the numerical solution of elliptic diffusion problems…
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…
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…