Journal
MATHEMATICS OF COMPUTATION
Volume 87, Issue 310, Pages 515-545Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/mcom/3220
Keywords
Weak Galerkin; finite element methods; non-divergence form; weak Hessian operator; discontinuous coefficients; Cordes condition; polyhedral meshes
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Funding
- National Science Foundation [DMS-1522586, DMS-1648171]
- NSF IR/D program
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1648171] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1849483] Funding Source: National Science Foundation
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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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