In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution u and the flux σ are approximated using finite element sp...In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution u and the flux σ are approximated using finite element spaces consisting of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection, superconvergent H1-error estimates of both the primary solution approximation uh and the flux approximation σh are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of O(hr+2) for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order r are employed with optimal error estimate of O(hr+1).展开更多
Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger...Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1)in the H1norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1+ τ2) in the H1norm can be obtained in the Crank-Nicolson fully discrete scheme.展开更多
针对一维带有不连续系数和奇异源项的椭圆型方程,采用匹配界面和边界(MIB,matched interface and boundary)方法进行求解.该方法对微分方程和跳跃条件的离散是分别进行的,通过在界面附近构造虚拟点达到提高差分格式整体精度的目的,文中...针对一维带有不连续系数和奇异源项的椭圆型方程,采用匹配界面和边界(MIB,matched interface and boundary)方法进行求解.该方法对微分方程和跳跃条件的离散是分别进行的,通过在界面附近构造虚拟点达到提高差分格式整体精度的目的,文中对Neumann边界也给出了处理办法.通过数值算例对文中构造的差分方法进行了验证,并与文献中的浸入界面方法进行了对比,数值结果证明了方法的有效性和可行性.展开更多
基金Supported by National Science Foundation of China and the Foundation of China State Education Commission and the Special Funds for Major State Basic Research Projects.
文摘In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution u and the flux σ are approximated using finite element spaces consisting of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection, superconvergent H1-error estimates of both the primary solution approximation uh and the flux approximation σh are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of O(hr+2) for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order r are employed with optimal error estimate of O(hr+1).
基金Project supported by the National Natural Science Foundation of China(No.11671157)
文摘Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1)in the H1norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1+ τ2) in the H1norm can be obtained in the Crank-Nicolson fully discrete scheme.
文摘针对一维带有不连续系数和奇异源项的椭圆型方程,采用匹配界面和边界(MIB,matched interface and boundary)方法进行求解.该方法对微分方程和跳跃条件的离散是分别进行的,通过在界面附近构造虚拟点达到提高差分格式整体精度的目的,文中对Neumann边界也给出了处理办法.通过数值算例对文中构造的差分方法进行了验证,并与文献中的浸入界面方法进行了对比,数值结果证明了方法的有效性和可行性.
