Abstract
This chapter reviews the fundamentals of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. The main idea admits three different interpretations: a Petrov–Galerkin method with (optimal) test functions that realize the supremum in the inf–sup condition; a minimum‐residual method with residual measured in a dual norm; and a mixed formulation where one solves simultaneously for the Riesz representation of the residual. The methodology can be applied to any well‐posed variational problem, but it is especially effective in context of discontinuous (broken) test spaces. We discuss how one can “break” test functions in any variational formulation, and use the convection‐dominated diffusion model problem to illustrate challenges related to the choice of an appropriate test norm.
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