Linearization of the finite element method for gradient flows by Newton’s method

Abstract

Abstract The implicit Euler scheme for nonlinear partial differential equations of gradient flows is linearized by Newton’s method, discretized in space by the finite element method. With two Newton iterations at each time level, almost optimal order convergence of the numerical solutions is established in both the $L^q(\varOmega )$ and $W^{1,q}(\varOmega )$ norms. The proof is based on techniques utilizing the resolvent estimate of elliptic operators on $L^q(\varOmega )$ and the maximal $L^p$-regularity of fully discrete finite element solutions on $W^{-1,q}(\varOmega )$.