## Preprint 2020-10

## Rommel Bustinza, Jonathan Munguia:

### A mixed hybrid high-order formulation for linear interior transmission elliptic problems

### Abstract:

In this paper, we analyse a linear transmission elliptic problem in a bounded domain, applying the already known Hybrid High-Order (HHO) method. This approach gives approximation of unknowns in the interior volume of each element and on the faces of its boundary, in the following sense: obtaining L2-projection on the space of polynomials of total degree at most k on the mesh elements and faces. Thus, we obtain a non-conforming discrete formulation, which is well posed, and after a condensation process, we can reduced it to another scheme defined on the skeleton induced by the mesh. This allows us to obtain a more compact system and reduce significantly the number of unknowns. We point out that we need to introduce an auxiliary unknown in order to deal with the non homogeneous transmission conditions, that will act as Lagrange multiplier. We prove that the method is optimally convergent in the energy norm, as well as in the L2-norm for the potential and a weighted L2-norm for the Lagrange multiplier, for smooth enough solutions. Finally, we include some numerical experiments that validate our theoretical results, even in situations not covered by the current analysis.