Two-grid mixed finite element analysis of semi-linear second order hyperbolic problem

Abstract

A novel two-grid symmetric mixed finite element analysis is considered for semi-linear second order hyperbolic problem. To overcome the saddle-point problem resulted by the traditional mixed element methods, a new symmetric and positive definite mixed procedure is first introduced to solve semi-linear hyperbolic problem. Then the a priori error estimates both in $L^2$ and $L^p$-norm senses are derived. Meanwhile, the two-grid technique proposed by Xu is applied to improve the resulting nonlinear numerical algorithm. Theoretical analysis is considered and the corresponding error estimate is derived under the relation $h=O\left(H^2\right)$. Finally, numerical examples are provided to test theoretical results and the efficiency of the proposed two-grid mixed element method.