@article{1aa8f0ee711944808a6a6dea7d317127,

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

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.",

keywords = "Cord`es condition, Discontinuous coefficients, Finite element methods, Non-divergence form, Polyhedral meshes, Weak Galerkin, Weak Hessian operator",

author = "Chunmei Wang and Junping Wang",

note = "Funding Information: Received by the editor October 21, 2015 and, in revised form, September 16, 2016. 2010 Mathematics Subject Classification. Primary 65N30, 65N12, 35J15, 35D35. Key words and phrases. Weak Galerkin, finite element methods, non-divergence form, weak Hessian operator, discontinuous coefficients, Cord{\`e}s condition, polyhedral meshes. The research of the first author was partially supported by National Science Foundation Awards #DMS-1522586 and #DMS-1648171. The research of the second author was supported by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Funding Information: We would like to express our gratitude to an anonymous referee for bringing the identities (2.5) to our attention. This has led to a sharp estimate for the Cord{\`e}s constant. The research of the second author was supported by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation Publisher Copyright: {\textcopyright} 2017 American Mathematical Society.",

year = "2018",

doi = "10.1090/mcom/3220",

language = "English",

volume = "87",

pages = "515--545",

journal = "Mathematics of Computation",

issn = "0025-5718",

number = "310",

}