In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal ...Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.展开更多
The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approx...The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.展开更多
Error estimation for the differential quadrature (DQ)with various sets of grid spacing is presented. A general formula is given to compute the weighting coefficients directly. It is found that the maximum error of the...Error estimation for the differential quadrature (DQ)with various sets of grid spacing is presented. A general formula is given to compute the weighting coefficients directly. It is found that the maximum error of the Do method with roots of Chebyshev polynomials including two end POints (-1 and + 1 ) is the smallest among the several sets of grid spacing investigated herein.展开更多
A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal der...A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.展开更多
It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rle in hybrid modeling and computations.Being constructed on various kinds of hybrid grids,that is,time sc...It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rle in hybrid modeling and computations.Being constructed on various kinds of hybrid grids,that is,time scales,dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important nat- ural processes with hard-to-predict singularities,such as the epidemic growth with un- predictable jump sizes and option market changes with high uncertainties,as com- pared with conventional derivatives.In this article,we shall review the novel new concepts,explore delicate relations between the most frequently used second-order dy- namic derivatives and conventional derivatives.We shall investigate necessary condi- tions for guaranteeing the consistency between the two derivatives.We will show that such a consistency may never exist in general.This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hy- brid grids.Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications.Numerical experiments will also be given.展开更多
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
基金Subsidized by NSFC(11571274 and 11171269)the Ph.D.Programs Foundation of Ministry of Education of China(20110201110027)
文摘Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
文摘The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.
文摘Error estimation for the differential quadrature (DQ)with various sets of grid spacing is presented. A general formula is given to compute the weighting coefficients directly. It is found that the maximum error of the Do method with roots of Chebyshev polynomials including two end POints (-1 and + 1 ) is the smallest among the several sets of grid spacing investigated herein.
基金supported by The Key Project of Natural Science Foundation of China G10531080National Basic Research Program of China No.2005CB321702Natural Science Foundation of China G10771178.
文摘A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.
文摘It has been evident that the theory and methods of dynamic derivatives are playing an increasingly important rle in hybrid modeling and computations.Being constructed on various kinds of hybrid grids,that is,time scales,dynamic derivatives offer superior accuracy and flexibility in approximating mathematically important nat- ural processes with hard-to-predict singularities,such as the epidemic growth with un- predictable jump sizes and option market changes with high uncertainties,as com- pared with conventional derivatives.In this article,we shall review the novel new concepts,explore delicate relations between the most frequently used second-order dy- namic derivatives and conventional derivatives.We shall investigate necessary condi- tions for guaranteeing the consistency between the two derivatives.We will show that such a consistency may never exist in general.This implies that the dynamic derivatives provide entirely different new tools for sensitive modeling and approximations on hy- brid grids.Rigorous error analysis will be given via asymptotic expansions for further modeling and computational applications.Numerical experiments will also be given.
