C0 finite element approximations of linear elliptic equations in non-divergence form and Hamilton–Jacobi–Bellman equations with Cordes coefficients

Abstract

This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C0 on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex. The main novelty of the non-standard finite element methods is to introduce an interior stabilization term to argument the PDE-induced variational form of the linear elliptic equations in non-divergence form or the HJB equations. As a distinctive feature of the proposed methods, no stabilization parameter is involved in the variational forms. As a consequence, the coercivity constant (resp. monotonicity constant) for the linear elliptic equations in non-divergence form (resp. the HJB equations) at discrete level is exactly the same as that from PDE theory. The quasi-optimal order error estimates as well as the convergence of the semismooth Newton method are established. Numerical experiments are provided to validate the convergence theory and to illustrate the accuracy and computational efficiency of the proposed methods.

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Acknowledgements

The work of Shuonan Wu is supported in part by the National Natural Science Foundation of China Grant No. 11901016 and the startup grant from Peking University. The author would like to express his gratitude to Guangwei Gao and Prof. Jun Hu in Peking University for their helpful discussions, and to the anonymous referees for their valuable comments.

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Wu, S. C0 finite element approximations of linear elliptic equations in non-divergence form and Hamilton–Jacobi–Bellman equations with Cordes coefficients. Calcolo 58, 9 (2021). https://doi.org/10.1007/s10092-021-00400-1

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Keywords

  • Elliptic PDEs in non-divergence form
  • Hamilton–Jacobi–Bellman equations
  • Cordes condition
  • C 0 (non-Lagrange) finite element methods

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 65N15
  • 35J12
  • 35D35