A New Approach to Optimal Error Analysis of Linearized Semi-implicit Galerkin Methods for a Large Class of Nonlinear Parabolic Systems

Many physical phenomena can be described by time-dependent parabolic partial differential equations. Due to the nature of the evolution, a stable numerical approximation to the time integration in nonlinear time-dependent PDEs is always a key issue in developing efficient numerical methods and their theoretical analysis. However, a time stepsize condition was often required in previous analysis for many numerical methods. The time-step restriction may result in extremely time-consuming in solving various physical problems. The aim of this project is to develop a new approach to analyze several popular numerical methods for a large class of nonlinear parabolic PDEs and establish unconditionally optimal error estimates. Our approach may provide a new understanding on the commonly-used schemes and clear up the misgivings for the time stepsize restriction in practical computations.