A high order HDG method for curved-interface problems via approximations from straight triangulations

TL;DR

This paper extends high-order HDG methods to curved-interface problems using straight triangulations, demonstrating that interpolating boundaries with linear segments achieves optimal convergence.

Contribution

It generalizes existing techniques to elliptic problems with complex interfaces, providing a practical approach for high-order accuracy on curved domains.

Findings

Optimal high-order convergence achieved with linear boundary interpolation.

Distance between computational and exact boundary is $O(h^2)$.

Numerical results confirm effectiveness of the proposed method.

Abstract

We generalize the technique of [Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, pp. A497--A519 (2012)] to elliptic problems with mixed boundary conditions and elliptic interface problems involving a non-polygonal interface. We study first the treatment of the Neumann boundary data since it is crucial to understand the applicability of the technique to curved interfaces. We provide numerical results showing that, in order to obtain optimal high order convergence, it is desirable to construct the computational domain by interpolating the boundary/interface using piecewise linear segments. In this case the distance of the computational domain to the exact boundary is only $O(h^2)$.