A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulat...A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.展开更多
In this paper,we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem.We establish optimal order error estimates for velocity and super convergence for pr...In this paper,we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem.We establish optimal order error estimates for velocity and super convergence for pressure.Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.展开更多
A modified weak Galerkin(MWG)finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuou...A modified weak Galerkin(MWG)finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuous polynomials on polytopal meshes.But its formulation is simple,symmetric,positive definite,and parameter independent,without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method.Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions.Error estimates in the L^(2)norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements.The numerical results are presented to confirm the theory of convergence.展开更多
In the realm of autoimmune and inflammatory diseases,the cyclic GMP-AMP synthase(cGAS)stimulator of interferon genes(STING)signaling pathway has been thoroughly investigated and established.Despite this,the clinical a...In the realm of autoimmune and inflammatory diseases,the cyclic GMP-AMP synthase(cGAS)stimulator of interferon genes(STING)signaling pathway has been thoroughly investigated and established.Despite this,the clinical approval of drugs targeting the cGAS-STING pathway has been limited.The Total glucosides of paeony(TGP)is highly anti-inflammatory and is commonly used in the treatment of rheumatoid arthritis(RA),emerged as a subject of our study.We found that the TGP markedly reduced the activation of the cGAS-STING signaling pathway,triggered by various cGAS-STING agonists,in mouse bone marrow-derived macrophages(BMDMs)and Tohoku Hospital Pediatrics-1(THP-1)cells.This inhibition was noted alongside the suppression of interferon regulatory factor 3(IRF3)phosphorylation and the expression of interferon-beta(IFN-β),C-X-C motif chemokine ligand 10(CXCL10),and inflammatory mediators such as tumor necrosis factor-alpha(TNF-α)and interleukin-6(IL-6).The mechanism of action appeared to involve the TGP’s attenuation of the STING-IRF3 interaction,without affecting STING oligomerization,thereby inhibiting the activation of downstream signaling pathways.In vivo,the TGP hindered the initiation of the cGAS-STING pathway by the STING agonist dimethylxanthenone-4-acetic acid(DMXAA)and exhibited promising therapeutic effects in a model of acute liver injury induced by lipopolysaccharide(LPS)and D-galactosamine(D-GalN).Our findings underscore the potential of the TGP as an effective inhibitor of the cGAS-STING pathway,offering a new treatment avenue for inflammatory and autoimmune diseases mediated by this pathway.展开更多
This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergen...This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergence features a rate that is two orders higher than the optimal-order error estimates in the usual energy and L^(2)norms.The extension from constant to variable coefficients for the modeling equations is highly non-trivial.The underlying technical analysis is based on a sequence of projections and decompositions.Numerical results confirm the superconvergence theory for second-order elliptic problems with variable coefficients.展开更多
In this paper,we introduce a stabilizer free weak Galerkin(SFWG)finite element method for second order elliptic problems on rectangular meshes.With a special weak Gradient space,an order two superconvergence for the S...In this paper,we introduce a stabilizer free weak Galerkin(SFWG)finite element method for second order elliptic problems on rectangular meshes.With a special weak Gradient space,an order two superconvergence for the SFWG finite element solution is obtained,in both L 2 and H1 norms.A local post-process lifts such a Pk weak Galerkin solution to an optimal order Pk+2 solution.The numerical results confirm the theory.展开更多
We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using H（div） conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for p...We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using H（div） conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the H（div） finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.展开更多
A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same tim...A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.展开更多
A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple form...A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple formulation.The main goal of this paper is to improve the above discontinuous Galerkinfinite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively.In addition,the method has been generalized in terms of approximation of the weak gradient.Error estimates of optimal order are established for the correspond-ing discontinuousfinite element approximation in both a discrete H1 norm and the L2 norm.Numerical results are presented to confirm the theory.展开更多
文摘A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.
基金supported in part by the National Science Foundation Grant DMS-1620016supported in parts by HKSAR grant Q81Q and JRI of The Hong Kong Polytechnic University.
文摘In this paper,we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem.We establish optimal order error estimates for velocity and super convergence for pressure.Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.
基金M.Cui was supported in part by the National Natural Science Foundation of China(Grant No.11571026)the Beijing Municipal Natural Science Foundation of China(Grant No.1192003)Xiu Ye was supported in part by the National Science Foundation Grant DMS-1620016.
文摘A modified weak Galerkin(MWG)finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuous polynomials on polytopal meshes.But its formulation is simple,symmetric,positive definite,and parameter independent,without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method.Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions.Error estimates in the L^(2)norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements.The numerical results are presented to confirm the theory of convergence.
基金This work was supported by the Natural Science Foundation of Beijing(No.7232321)the Cultivating and Improving the Service Ability of Traditional Chinese Medicine(No.2021ZY038)+1 种基金the Innovation Team and Talents Cultivation Program of National Administration of Traditional Chinese Medicine(No.ZYYCXTD-C-202005)the State Key Program of National Natural Science of China(Nos.81930110,82230118)。
文摘In the realm of autoimmune and inflammatory diseases,the cyclic GMP-AMP synthase(cGAS)stimulator of interferon genes(STING)signaling pathway has been thoroughly investigated and established.Despite this,the clinical approval of drugs targeting the cGAS-STING pathway has been limited.The Total glucosides of paeony(TGP)is highly anti-inflammatory and is commonly used in the treatment of rheumatoid arthritis(RA),emerged as a subject of our study.We found that the TGP markedly reduced the activation of the cGAS-STING signaling pathway,triggered by various cGAS-STING agonists,in mouse bone marrow-derived macrophages(BMDMs)and Tohoku Hospital Pediatrics-1(THP-1)cells.This inhibition was noted alongside the suppression of interferon regulatory factor 3(IRF3)phosphorylation and the expression of interferon-beta(IFN-β),C-X-C motif chemokine ligand 10(CXCL10),and inflammatory mediators such as tumor necrosis factor-alpha(TNF-α)and interleukin-6(IL-6).The mechanism of action appeared to involve the TGP’s attenuation of the STING-IRF3 interaction,without affecting STING oligomerization,thereby inhibiting the activation of downstream signaling pathways.In vivo,the TGP hindered the initiation of the cGAS-STING pathway by the STING agonist dimethylxanthenone-4-acetic acid(DMXAA)and exhibited promising therapeutic effects in a model of acute liver injury induced by lipopolysaccharide(LPS)and D-galactosamine(D-GalN).Our findings underscore the potential of the TGP as an effective inhibitor of the cGAS-STING pathway,offering a new treatment avenue for inflammatory and autoimmune diseases mediated by this pathway.
基金supported by U.S.National Science Foundation IR/D program while working at U.S.National Science Foundationsupported by U.S.National Science Foundation(Grant No.DMS-1620016)+1 种基金supported by Zhejiang Provincial Natural Science Foundation of China(Grant No.LY23A010005)National Natural Science Foundation of China(Grant No.12071184)。
文摘This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergence features a rate that is two orders higher than the optimal-order error estimates in the usual energy and L^(2)norms.The extension from constant to variable coefficients for the modeling equations is highly non-trivial.The underlying technical analysis is based on a sequence of projections and decompositions.Numerical results confirm the superconvergence theory for second-order elliptic problems with variable coefficients.
基金Xiu Ye was supported in part by National Science Foundation Grant DMS-1620016.
文摘In this paper,we introduce a stabilizer free weak Galerkin(SFWG)finite element method for second order elliptic problems on rectangular meshes.With a special weak Gradient space,an order two superconvergence for the SFWG finite element solution is obtained,in both L 2 and H1 norms.A local post-process lifts such a Pk weak Galerkin solution to an optimal order Pk+2 solution.The numerical results confirm the theory.
基金the NSF IR/D program,while working at the National Science FoundationThe research of Ye was supported in part by National Science Foundation Grant DMS-0612435
文摘We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using H（div） conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the H（div） finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.
基金supported in part by National Science Foundation Grant DMS-1115097supported in part by National Science Foundation Grants DMS-1016579 and DMS-1318898.
文摘A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.
基金supported in part by National Natural Science Foundation of China(NSFC No.11871038)supported in part by National Science Foundation Grant DMS-1620016.
文摘A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple formulation.The main goal of this paper is to improve the above discontinuous Galerkinfinite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively.In addition,the method has been generalized in terms of approximation of the weak gradient.Error estimates of optimal order are established for the correspond-ing discontinuousfinite element approximation in both a discrete H1 norm and the L2 norm.Numerical results are presented to confirm the theory.
