Primal HDG Methods for Elliptic Problems on Curved Meshes

Resumo

This work addresses stability and locking of three classes of hybrid methods for elliptic problems on
straight and curved meshes in 2d. We consider here modifications of the hybrid methods presented in [1] and [2],
and compare them with the hybrid high order method discussed in [3]. For straight meshes, the approximation
order for polynomial spaces over edges can either be equal or one order smaller than the approximation order for
polynomial spaces in the interior. However, having one order smaller for edges is actually mandatory for obtaining
locking-free estimates from our modification of the primal method introduced in [1] on straight meshes (except
for triangles, where the approximation spaces are divergence-free). The other two hybrid methods are locking-free
whenever they are stable. In the case of curved meshes, we verify that all three methods are unstable if one chooses
approximation orders for the edges one order smaller than the interior. However, we also verify numerically that
all three methods are locking-free in curved meshes if these polynomial orders are equal.