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

Authors: Chunmei Wang and Junping Wang
Journal: Math. Comp. 87 (2018), 515-545
MSC (2010): Primary 65N30, 65N12, 35J15, 35D35
Published electronically: June 27, 2017
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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 $ 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.

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Additional Information

Chunmei Wang
Affiliation: Department of Mathematics, Texas State University, San Marcos, Texas 78666-4684

Junping Wang
Affiliation: Division of Mathematical Sciences, National Science Foundation, Arlington, Virginia 22230

Keywords: Weak Galerkin, finite element methods, non-divergence form, weak Hessian operator, discontinuous coefficients, Cord\`es condition, polyhedral meshes
Received by editor(s): October 21, 2015
Received by editor(s) in revised form: September 16, 2016
Published electronically: June 27, 2017
Additional Notes: 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
Article copyright: © Copyright 2017 American Mathematical Society

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