A Posteriori Error Control for DPG Methods

A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a posteriori error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. The presented abstract framework for a posteriori error analysis applies to known DPG discretizations of Laplace and Lame equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete space...

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