Haijun Wu, Professor, Nanjing University

Peijun Li, Professor

Adaptive Finite Element Method for a Nonlinear Helmholtz Equation with High Wave Number

14:00-15:00 April 9 (Tuesday) , N533

A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for the NLH problem, a priori stability and error estimates are proved for the FEM on shape regular meshes including the case of locally refined meshes. Then a posteriori upper and lower bounds using a new residual-type error estimator, which is equivalent to the standard one, are derived for the FE solutions to the NLH problem. These a posteriori estimates have confirmed a significant fact that is also valid for the NLH problem, namely the residual-type estimator seriously underestimates the error of the FE solution in the preasymptotic regime, which was first observed by Babuˇska et al. [Int J Numer Methods Eng 40 (1997)] for a one-dimensional linear problem. Based on the new a posteriori error estimator, both the convergence and the quasi-optimality of the resulting adaptive finite element algorithm are proved the first time for the NLH problem, when the initial mesh size lying in the preasymptotic regime. Finally, numerical examples are presented to validate the theoretical findings and demonstrate that applying the continuous interior penalty (CIP) technique with appropriate penalty parameters can reduce the pollution errors efficiently. In particular, the nonlinear phenomenon of optical bistability with Gaussian incident waves is successfully simulated by the adaptive CIPFEM.

报告人介绍：武海军，南京大学数学系教授、博导。研究领域包括高波数散射问题有限元方法、界面问题的非拟合界面罚有限元方法、等等。获得2012年江苏省数学杰出成就奖，2015年获国家杰出青年科学基金资助。现任江苏省数学会秘书长。