### 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 H^{2}-norm, as well as the usual H^{1}- and L^{2}-norms. The convergence theory is based on the assumption that the solution of the model problem is H^{2}-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 language | English |
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Pages (from-to) | 515-545 |

Number of pages | 31 |

Journal | Mathematics of Computation |

Volume | 87 |

Issue number | 310 |

DOIs | |

State | Published - 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