L2-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations

We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal [superscript][cursive l]∞[/superscript] ([superscript]L2[/superscript] ) error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.