Construction of DPG Fortin operators for second order problems

Abstract

The use of “ideal” optimal test functions in a Petrov–Galerkin scheme guarantees the discrete stability of the variational problem. However, in practice, the computation of the ideal optimal test functions is computationally intractable. In this paper, we study the effect of using approximate, “practical” test functions on the stability of the DPG (discontinuous Petrov–Galerkin) method and the change in stability between the “ideal” and “practical” cases is analyzed by constructing a Fortin operator.We highlight the construction of an “optimal” DPG Fortin operator for H1 and H(div) spaces; the continuity constant of the Fortin operator is a measure of the loss of stability between the ideal and practical DPG methods. We take a two-pronged approach: first, we develop a numerical procedure to estimate an upper bound on the continuity constant of the Fortin operator in terms of the inf–sup constant γh of an auxiliary problem. Second, we construct a sequence of approximate Fortin operators and exactly compute the continuity constants of the approximate operators, which provide a lower bound on the exact Fortin continuity constant.Our results shed light not only on the change in stability by using practical test functions, but also indicate how stability varies with the approximation order p and the enrichment order Δp. The latter has important ramifications when one wishes to pursue local hp-adaptivity.