In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuou...In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method,though they have essentially different bilinear forms.More precisely,we prove its reliability and efficiency for the actual error measured in the standard DG norm.We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution.Numerical results are presented to verify the theoretical analysis.展开更多
In this paper,based on the Lagrange Multiplier approach in time and the Fourierspectral scheme for space,we propose efficient numerical algorithms to solve the phase field crystal equation.The numerical schemes are u...In this paper,based on the Lagrange Multiplier approach in time and the Fourierspectral scheme for space,we propose efficient numerical algorithms to solve the phase field crystal equation.The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential.The unconditional energy stability of the three semi-discrete schemes is proven.Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.展开更多
基金The first author was supported by Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2018A030307024 and 2020A1515011032)by National Natural Science Foundation of China(Grant No.11526097)+2 种基金The second author was supported by National Natural Science Foundation of China(Grant Nos.11871272 and 11871281)The third author was supported by National Natural Science Foundation of China(Grant No.11701197)The fourth author was supported by Guangdong Basic and Applied Basic Research Foundation(Grant No.2018A0303100016).
文摘In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method,though they have essentially different bilinear forms.More precisely,we prove its reliability and efficiency for the actual error measured in the standard DG norm.We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution.Numerical results are presented to verify the theoretical analysis.
基金The work of Q.Zhuang is supported by the National Natural Science Foundation of China(No.11771083)The research of S.Zhai is supported in part by the Natural Science Foundation of China(No.11701196)+3 种基金the Natural Science Foundation of Fujian Province(No.2020J01074)The work of Z.Weng is supported in part by the Natural Science Foundation of China(No.11701197)Supported by the Fundamental Research Funds for the Central Universities(No.ZQN-702)the Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education(Xiangtan University)(No.2020ICIP03).
文摘In this paper,based on the Lagrange Multiplier approach in time and the Fourierspectral scheme for space,we propose efficient numerical algorithms to solve the phase field crystal equation.The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential.The unconditional energy stability of the three semi-discrete schemes is proven.Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.
