Numerical analysis of a space-time continuous Galerkin method for the nonlinear Sobolev equation

Abstract

In this work, we shall use space-time continuous Galerkin (STCG) method to analyze the nonlinear Sobolev equation. This method is different from the conventional finite element methods, that is, it not only uses finite element to discrete spatial variables but also uses finite element to discrete temporal variable. Therefore it is easier to obtain high order accuracy in time and space than the conventional finite element methods and has the good stability. Also, it does not need time step size and space mesh parameter to satisfy the stability conditions. In this paper, we show a detailed analysis including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. Especially, the analysis method given here can be similarly applied to the more general nonlinear problems. At last, the numerical examples reveal that the iterative method is more efficient than the linearized method for the nonlinear problems with STCG method. Also, the numerical experiments show that the STCG method is more efficient than the space-time discontinuous Galerkin (STDG) method in the practical calculations.