Goal-oriented adaptive mesh refinement for non-symmetric functional settings
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In this article, a Petrov-Galerkin duality theory is developed. This theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of steady boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. The primal problem is simply the system of linear equations coming from a standard DPG method. The dual problem is a similar system of equations, coming from a new method which is dual to DPG; having the same coefficient matrix as the DPG method but a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poisson's boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of goal-oriented adaptive mesh refinement with DPG methods for quantities of interest with either interior or boundary terms.
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