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

Wang, Chunmei; Wang, Junping

doi:10.1090/mcom/3220

A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form
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by Chunmei Wang and Junping Wang PDF

Math. Comp. 87 (2018), 515-545 Request permission

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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