A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

Abstract

We continue our theoretical and numerical study on the Discontinuous Petrov-Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: = 10 −11 for 1D and = 10 −7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only.

Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

“…By (9), c * Bc = 0 if and only if T w = 0, which hold if and only if c is the zero vector. As a consequence, we can use powerful iterative solvers for positive definite systems (like the conjugate gradient method) to obtain the DPG solution, even though the original Helmholtz operator is indefinite.…”

“…We begin by summarizing the DPG framework developed in [7,8,9,25] in § 2.1. We then apply it to the Helmholtz setting.…”

“…The guiding principle in designing Petrov-Galerkin methods is that one chooses trial spaces for good approximation properties, but one designs test spaces to obtain good stability properties. We have exemplified this principle in our earlier works [7,8,9,25]. In [25], we considered the one-dimensional Helmholtz equation and obtained (1) with C(ω) independent of ω.…”

Adaptivity and variational stabilization for convection-diffusion equations

“…While these developments ultimately require dealing with classical H −1 -inner products, we encounter here additional difficulties due to the singularly perturbed nature of the underlying problem (1.1). Moreover, after completion of this paper we became aware of recent related work reported in [10,11]. It centers on the notion of "optimal test spaces" for Petrov-Galerkin discretizations which is closely related to the functional analytic framework developed below in Section 2.…”

A toolbox for a class of discontinuous Petrov-Galerkin methods using trilinos.

“…Recently, L. Demkowicz and J. Gopalakrishnan have proposed a new class of discontinuous Petrov-Galerkin (DPG) methods [4,5,6,9,3], which compute test functions that are adapted to the problem of interest to produce stable discretization schemes. An important choice that must be made in the application of the method is the definition of the inner product on the test space.…”