Robust DPG test spaces and Fortin operators
— the H¹ and H(div) cases
Thomas Führer and Norbert Heuer
SIAM J. Numer. Anal. 62 (2), 718–748, 2024.
At the fully discrete setting, stability of the discontinuous
Petrov–Galerkin (DPG) method with optimal test functions
requires local test spaces that ensure the existence of Fortin operators.
We construct such operators for H¹ and H(div) on simplices
in any space dimension and arbitrary polynomial degree.
The resulting test spaces are smaller than previously analyzed cases.
For parameter-dependent norms, we achieve uniform boundedness by the inclusion
of face bubble functions that are polynomials on faces and decay
exponentially in the interior.
As an example, we consider a canonical DPG setting for reaction-dominated
diffusion. Our test spaces guarantee uniform stability and quasi-optimal
convergence of the scheme.
We present numerical experiments that illustrate the loss of stability and
error control by the residual for small diffusion coefficient when using
standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.