# Combining the DPG method with finite elements

## TL;DR

This paper introduces a novel discretization scheme combining discontinuous Petrov-Galerkin and finite element methods for complex diffusion-advection-reaction problems, demonstrating its theoretical soundness and practical effectiveness.

## Contribution

It presents a new coupled DPG-FEM scheme with a heterogeneous variational formulation, including proof of well-posedness and convergence, for general diffusion-advection-reaction models.

## Key findings

- Proves well-posedness of the coupled scheme
- Establishes quasi-optimal convergence rates
- Numerical results confirm theoretical predictions

## Abstract

We propose and analyze a discretization scheme that combines the discontinuous Petrov-Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov-Galerkin) form with broken test space in one part, and of Bubnov-Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov-Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.