Linearized FE Approximations to a Nonlinear Gradient Flow

Abstract

We study fully discrete linearized Galerkin finite element approximations to a nonlinear gradient flow, applications of which can be found in many areas. Due to the strong nonlinearity of the equation, existing analyses for implicit schemes require certain restrictions on the time step and no analysis has been explored for linearized schemes. This paper focuses on the unconditionally optimal $L^2$ error estimate of a linearized scheme. The key to our analysis is an iterated sequence of time-discrete elliptic equations and a rigorous analysis of its solution. We prove the $W^{1,\infty}$ boundedness of the solution of the time-discrete system and the corresponding finite element solution, based on a more precise estimate of elliptic PDEs in $W^{2,2+\epsilon_1}$ and $H^{2+\epsilon_2}$ and a physical feature of the gradient-dependent diffusion coefficient. Numerical examples are provided to support our theoretical analysis.