Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

Abstract

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H2‐norm by the temporal error. Superconvergence results of order O(h2 + τ3) in H1‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Virtual element method for nonlinear Sobolev equation on polygonal meshes

“…There are also some non-standard finite element methods based on the nonlinear Sobolev equations have been developed such as the two-grid finite element method [11], mixed finite element method [12], split least-squares mixed finite element method [13] and low order characteristicnonconforming finite element method [14]. The superconvergence result of order O(h 2 +τ 3 ) in H 1 -norm was displayed in [15], which developed a linearized three-step backward differential formula Galerkin finite element method. In addition, fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations were introduced in [16] and the interior penalty discontinuous Galerkin method was applied in [17].…”

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

“…For example, BDF2-finite element method (FEM) and BDF3-FEM were made use of in References [1,10], respectively to obtain optimal error estimates unconditionally through splitting technique [14,[20][21][22][23]28]. Further, in References [25,27], the authors studied unconditional superconvergent error estimates with BDF2-FEM for nonlinear parabolic equation and for Sobolev equation, respectively. BDF2-FEM was utilized to (1.1) in Reference [26], energy stability is testified under the condition that 𝜏 < 1 and unconditional superconvergent error estimates was deduced.…”