A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form

Chunmei Wang, Junping Wang

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Abstract

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Error estimates of optimal order are derived for the corresponding finite element approximations in a discrete H2-norm, as well as the usual H1- and L2-norms. The convergence theory is based on the assumption that the solution of the model problem is H2-regular, and that the coefficient tensor in the PDE is piecewise continuous and uniformly positive definite in the domain. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains, which not only confirm the developed convergence theory but also a superconvergence result.

Original languageEnglish
Pages (from-to)515-545
Number of pages31
JournalMathematics of Computation
Volume87
Issue number310
DOIs
StatePublished - Jan 1 2018

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Keywords

  • Cord`es condition
  • Discontinuous coefficients
  • Finite element methods
  • Non-divergence form
  • Polyhedral meshes
  • Weak Galerkin
  • Weak Hessian operator

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