Weifeng Qiu
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Weifeng Qiu has a total of 69 co-authors. Here are the top co-authors by number of shared publications:
- Huangxin Chen
School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, China
10 shared publications - Ke Shi
Department of Mathematics and Statistics, Old Dominion University Norfolk, VA 23529, USA#TAB
9 shared publications - Buyang Li
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong
5 shared publications - Jay Gopalakrishnan
Portland State University PO Box 751 97207 Portland OR USA
5 shared publications - Bernardo Cockburn
School of Mathematics, University of Minnesota, Minneapolis, USA
4 shared publications - Huadong Gao
School of Mathematics and Statistics Huazhong University of Science and Technology 430074 Wuhan P.R. China
4 shared publications
Latest Publications
- Open Access
Optimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier--Stokes flow without surface tension
Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(Ω))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.Open AccessOptimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier--Stokes flow without surface tension
Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(Ω))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results. - Open Access
- 14 November 2024
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 101 (3) , 1-24
Error Analysis with Polynomial Dependence on $\varepsilon ^{-1}$ in SAV Methods for the Cahn-Hilliard EquationThe optimal error estimate that depends only on the polynomial degree of $$ \varepsilon ^{-1}$$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is converting the structure of the SAV time-stepping scheme back to a form compatible with the original format of the Cahn-Hilliard equation, which makes it feasible to use spectral estimates to handle the nonlinear term. Based on the transformation of the SAV numerical scheme, the optimal error estimate for the temporal semi-discrete scheme which depends only on the low polynomial order of $$\varepsilon ^{-1}$$ instead of the exponential order, is derived by using mathematical induction, spectral arguments, and the superconvergence properties of some nonlinear terms. Numerical examples are provided to illustrate the discrete energy decay property and validate our theoretical convergence analysis.- 14 November 2024
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 101 (3) , 1-24
Open AccessError Analysis with Polynomial Dependence on $\varepsilon ^{-1}$ in SAV Methods for the Cahn-Hilliard EquationThe optimal error estimate that depends only on the polynomial degree of $$ \varepsilon ^{-1}$$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is converting the structure of the SAV time-stepping scheme back to a form compatible with the original format of the Cahn-Hilliard equation, which makes it feasible to use spectral estimates to handle the nonlinear term. Based on the transformation of the SAV numerical scheme, the optimal error estimate for the temporal semi-discrete scheme which depends only on the low polynomial order of $$\varepsilon ^{-1}$$ instead of the exponential order, is derived by using mathematical induction, spectral arguments, and the superconvergence properties of some nonlinear terms. Numerical examples are provided to illustrate the discrete energy decay property and validate our theoretical convergence analysis. - Open Access
- 17 June 2024
- Published by Springer Nature in Numerische Mathematik
- Vol. 156 (4) , 1639-1678
$H(\textrm{div})$-conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equationsIn this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and $$H(\textrm{div};\Omega )$$ -conforming space of order $$k+1$$ , respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete $$H^1$$ -stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the $$L^2$$ projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete $$H^1$$ -norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete $$H^1$$ -norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.- 17 June 2024
- Published by Springer Nature in Numerische Mathematik
- Vol. 156 (4) , 1639-1678
Open Access$H(\textrm{div})$-conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equationsIn this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and $$H(\textrm{div};\Omega )$$ -conforming space of order $$k+1$$ , respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete $$H^1$$ -stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the $$L^2$$ projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete $$H^1$$ -norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete $$H^1$$ -norm is achieved. Several numerical experiments are carried out to confirm the proposed theories. - 2 May 2024
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 99 (3) , 1-30
Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs
- 2 May 2024
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 99 (3) , 1-30
Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs
- 22 November 2023
- Published by Springer Nature in Advances in Computational Mathematics
- Vol. 49 (6) , 1-20
An unconditionally stable and $L^2$ optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations- 22 November 2023
- Published by Springer Nature in Advances in Computational Mathematics
- Vol. 49 (6) , 1-20
An unconditionally stable and $L^2$ optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations- 2 August 2023
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 93 (345) , 1-34
Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra
The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than π \pi , which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard H 2 H^2 elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by H 2 H^2 and W 1 , p W^{1,p} estimates, respectively. As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz’s argument due to the nonconformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the W 1 , ∞ W^{1,\infty } regularity estimate of the Poisson equation uniformly with respect to the perturbation.
- 2 August 2023
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 93 (345) , 1-34
Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra
The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than π \pi , which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard H 2 H^2 elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by H 2 H^2 and W 1 , p W^{1,p} estimates, respectively. As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz’s argument due to the nonconformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the W 1 , ∞ W^{1,\infty } regularity estimate of the Poisson equation uniformly with respect to the perturbation.
- 18 April 2023
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 95 (3) , 1-22
New Analysis of Mixed Finite Element Methods for Incompressible Magnetohydrodynamics
- 18 April 2023
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 95 (3) , 1-22
New Analysis of Mixed Finite Element Methods for Incompressible Magnetohydrodynamics
- Open Access
Spectral analysis of a mixed method for linear elasticity
The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an $O(h^2)$ initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete $H^1$-stability of numerical displacement, while only an $O(h)$ approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments.Open AccessSpectral analysis of a mixed method for linear elasticity
The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an $O(h^2)$ initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete $H^1$-stability of numerical displacement, while only an $O(h)$ approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments. - Open Access
New analysis of Mixed finite element methods for incompressible Magnetohydrodynamics
The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to Navier-Stokes equations and the N\'ed\'elec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order N\'ed\'elec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order N\'ed\'elec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented. Numerical results confirm our theoretical analysis.
Open AccessNew analysis of Mixed finite element methods for incompressible Magnetohydrodynamics
The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to Navier-Stokes equations and the N\'ed\'elec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order N\'ed\'elec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order N\'ed\'elec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented. Numerical results confirm our theoretical analysis.
- Open Access
Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra
The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than $π$, which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard $H^2$ elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by $H^2$ and $W^{1,p}$ estimates, respectively. As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz's argument due to the non-conformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the $W^{1,\infty}$ regularity estimate of the Poisson equation uniformly with respect to the perturbation.Open AccessWeak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra
The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than $π$, which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard $H^2$ elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by $H^2$ and $W^{1,p}$ estimates, respectively. As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz's argument due to the non-conformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the $W^{1,\infty}$ regularity estimate of the Poisson equation uniformly with respect to the perturbation.