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Chunmei Wang

Publications and source records attributed to Chunmei Wang.

86 records · Page 5Linked to original sources
arXiv2018

Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions

This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of ${\cal O}(h^r)$, $1.5\leq r \leq 2$, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is ${\cal O}(h)$ for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.

math.NA↗
arXiv2018

Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems

The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak $L^2$ topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.

math.NA↗
arXiv2017

Superconvergence of Ritz-Galerkin Finite Element Approximations for Second Order Elliptic Problems

In this paper, the author derives an $O(h^4)$-superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor $A$ and the usual shape functions on each element, called $A$-equilateral assumption in this paper. Several examples are presented for the coefficient tensor $A$ and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

math.NA↗
arXiv2017

A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations

This paper presents a primal-dual weak Galerkin (PD-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. The method is based on a variational form where all the derivatives are applied to the test functions so that no regularity is necessary for the exact solution of the model equation. The numerical scheme is designed by using locally constructed weak second order partial derivatives and the weak gradient commonly used in the weak Galerkin context. Optimal order of convergence is derived for the resulting numerical solutions. Numerical results are reported to demonstrate the performance of the numerical scheme.

math.NA↗
arXiv2017

A Bivariate Spline Method for Second Order Elliptic Equations in Non-Divergence Form

A bivariate spline method is developed to numerically solve second order elliptic partial differential equations (PDE) in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. A plenty of computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.

math.NA↗
arXiv2015

A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography

In this paper, a new and efficient numerical algorithm by using weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography(FT). Fluorescence tomography is an emerging, in vivo non-invasive 3-D imaging technique which reconstructs images that characterize the distribution of molecules that are tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

math.NA↗
arXiv2015

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Optimal order error estimates are derived for the finite element approximations in a discrete $H^2$-norm, as well as the usual $H^1$- and $L^2$-norms. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains.

math.NA↗
arXiv2015

A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation

This paper presents an arbitrary order locking-free numerical scheme for linear elasticity on general polygonal/polyhedral partitions by using weak Galerkin (WG) finite element methods. Like other WG methods, the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element. Such local problems are derived from weak formulations of the corresponding differential operators through integration by parts. Locking-free error estimates of optimal order are derived in a discrete $H^1$-norm and the usual $L^2$-norm for the approximate displacement when the exact solution is smooth. Numerical results are presented to demonstrate the efficiency, accuracy, and the locking-free property of the weak Galerkin finite element method.

math.NA↗
arXiv2015

Discretization of div-curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions

In this paper, the authors devise a new discretization scheme for div-curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in the algorithm development: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is devised and analyzed. The novelty of the technique lies in the discretization of the divergence operator applied to vector fields with heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.

math.NA↗
arXiv2014

An embryo of protocell membrane: The capsule of graphene oxide

Many signs indicate that the graphene could widely occur on the early Earth. Here, we report a new theory that graphene might be an embryo of protocell membrane, and found several evidences. Firstly, the graphene oxide and phospholipid-graphene oxide composite would curl into capsules in strongly acidic saturated solution of Pb(NO3)2 at low temperature, providing a protective space for biochemical reactions. Secondly, L-animi acids exhibit higher reactivity than D-animi acids for graphene oxides in favor of the formation of left-handed proteins. Thirdly, monolayer graphene with nanopores prepared by unfocused 84Kr25+ has high selectivity for permeation of the monovalent metal ions (Rb+ > K+ > Cs+ > Na+ > Li+), but does not allow Cl- through, which could be attributed to the ion exchange of oxygen-containing groups on the rim of nanopores. It is similar to K+ channels, which would cause efflux of some ions from capsule of graphene oxides with the decrease of pH in the primitive ocean, creating a suitable inner condition for the origin of life. Consequently, the strongly acidic, high salinity and strong radiation as well as temperature changes in the early Earth, regarded as negative factors, would be indispensable for the origin of protocell. In short, graphene bred life, but digested gradually by the evolution.

q-bio.PE↗
arXiv2014

A Hybridized Weak Galerkin Finite Element Method for the Biharmonic Equation

This paper presents a hybridized formulation for the weak Galerkin finite element method for the biharmonic equation. The hybridized weak Galerkin scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier is verified to provide a numerical approximation for certain derivatives of the exact solution. An optimal order error estimate is established for the numerical approximations arising from the hybridized weak Galerkin finite element method. The paper also derives a computational algorithm (Schur complement) by eliminating all the unknown variables on each element, yielding a significantly reduced system of linear equations for unknowns on the boundary of each element.

math.NA↗
arXiv2013

An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes

This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual $H^2$-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

math.NA↗
arXiv2009

Influence of uniaxial tensile stress on the mechanical and piezoelectric properties of short-period ferroelectric superlattice

Tetragonal ferroelectric/ferroelectric BaTiO3/PbTiO3 superlattice under uniaxial tensile stress along the c axis is investigated from first principles. We show that the calculated ideal tensile strength is 6.85 GPa and that the superlattice under the loading of uniaxial tensile stress becomes soft along the nonpolar axes. We also find that the appropriately applied uniaxial tensile stress can significantly enhance the piezoelectricity for the superlattice, with piezoelectric coefficient d33 increasing from the ground state value by a factor of about 8, reaching 678.42 pC/N. The underlying mechanism for the enhancement of piezoelectricity is discussed.

cond-mat.mes-hall↗