Goal-oriented error estimation for the automatic variationally stable FE method for convection-dominated diffusion problems

Abstract

We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart–Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Demkowicz and Gopalakrishnan, 2010; Carstensen et al., 2014; Demkowicz and Gopalakrishnan, 2011a; Demkowicz and Gopalakrishnan, 2011b; Demkowicz and Gopalakrishnan, 2012), leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart–Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element-by-element fashion.To establish the error estimators we present two approaches: (i) following the classical approach of Becker and Rannacher (Becker and Rannacher, 2001), i.e., the dual solution is sought in the (broken) test space, and (ii) introducing an alternative approach in which we seek C0, or Raviart–Thomas, AVS-FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h-adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.