Optimal $L^2$-error estimates for expanded mixed finite element methods of semilinear Sobolev equations

摘要

In this paper we derive a priori $L^iy (L^2)$ error estimates for expanded mixed finite element formulations of semilinear Sobolev equations. This formulation expands the standard mixed formulation in the sense that three variables, the scalar unknown, the gradient and the flux are explicitly treated. Based on this method we construct finite element semidiscrete approximations and fully discrete approximations of the semilinear Sobolev equations. We prove the existence of semidiscrete approximations of $u$, $-a u$ and $-a u - a u_t$ and obtain the optimal order error estimates in the $L^iy (L^2)$ norm. And also we construct the fully discrete approximations and analyze the optimal convergence of the approximations in $ell^iy(L^2)$ norm. Finally we also provide the computational results.