A DGFEM for Nondivergence Form Elliptic Equations with Cordes Coefficients on Curved Domains

TL;DR

This paper extends a discontinuous Galerkin finite element method for nondivergence form elliptic equations with Cordes coefficients to Lipschitz domains with curved boundaries, broadening its applicability beyond convex polytopal domains.

Contribution

It introduces a novel analysis framework that accommodates curved domain boundaries, enhancing the method’s flexibility for practical applications.

Findings

Method successfully applied to curved domains

Maintains convergence and stability properties

Broadens the class of feasible computational domains

Abstract

In "I. Smears, E. S\"{u}li, \emph{Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cord\'{e}s coefficients. SIAM J. Numer Anal., 51(4):2088-2106, 2013}" the authors designed and analysed a discontinuous Galerkin finite element method for the approximation of solutions to elliptic partial differential equations in nondivergence form. The results were proven, based on the assumption that the computational domain was convex and \emph{polytopal}. In this paper, we extend this framework, allowing for Lipschitz continuous domains with piecewise curved boundaries.