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<p>The main aim of this paper is to present a unified framework for the unconditional superconvergence analysis of an energy-stable 2-step backward differentiation formula (BDF2) fully discrete finite element method (FEM) for the strongly nonlinear Sobolev equation. Here, the energy dissipation is utilized to deal with the unique solvability of the discrete solution together with the Brouwer fixed-point theorem. Then, by leveraging the special properties of the finite element instead of the time-space splitting technique, along with an interpolation post-processing technique, the superclose and superconvergence results without any ratio restriction between the mesh size <i>h</i> and the time step <i>τ</i> are obtained rigorously. Additionally, the applicability of the proposed method to various well-known low-order membrane finite elements is discussed, and some numerical experiments about three kinds of finite elements are conducted to validate the theoretical results.</p>

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Unconditional superconvergence error estimates of a unified energy-stable BDF2-FEM for strongly nonlinear Sobolev equation

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • He Ma,
  • Dongyang Shi

摘要

The main aim of this paper is to present a unified framework for the unconditional superconvergence analysis of an energy-stable 2-step backward differentiation formula (BDF2) fully discrete finite element method (FEM) for the strongly nonlinear Sobolev equation. Here, the energy dissipation is utilized to deal with the unique solvability of the discrete solution together with the Brouwer fixed-point theorem. Then, by leveraging the special properties of the finite element instead of the time-space splitting technique, along with an interpolation post-processing technique, the superclose and superconvergence results without any ratio restriction between the mesh size h and the time step τ are obtained rigorously. Additionally, the applicability of the proposed method to various well-known low-order membrane finite elements is discussed, and some numerical experiments about three kinds of finite elements are conducted to validate the theoretical results.