Superconvergent DPG methods for second order elliptic problems
This work provides theoretical advancements for numerical analysts and computational scientists working with DPG methods for solving partial differential equations, particularly in understanding superconvergence and optimal L2 approximation.
This paper investigates DPG methods for second-order elliptic problems, achieving superconvergence for the scalar field variable either by increasing polynomial degree or local postprocessing. It provides a uniform analysis showing that only the quasi-optimal test norm yields higher convergence rates in the presence of convection, and offers the first theoretical explanation for the method's ability to deliver the best L2 approximation of the scalar field variable.
We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general second order elliptic problems. Under some assumptions on the regularity of solutions of the model problem and its adjoint, superconvergence for the scalar field variable is achieved by either increasing the polynomial degree in the corresponding approximation space by one or by a local postprocessing. We provide a uniform analysis that allows to treat different test norms. Particularly, we show that in the presence of convection only the quasi-optimal test norm leads to higher convergence rates, whereas other norms considered do not. Moreover, we also prove that our DPG method delivers the best $L^2$ approximation of the scalar field variable up to higher order terms, which is the first theoretical explanation of an observation made previously by different authors. Numerical studies that support our theoretical findings are presented.