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Analysis of HDG methods for Stokes flow. (English) Zbl 1410.76164

Summary: We analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree \( k\) for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of \( k+1\) in \( L^2\) for any \( k \geq 0\). Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div )-conforming, and converges with order \( k+2\) for \( k\geq 1\) and with order 1 for \( k=0\). Numerical experiments are presented which validate the theoretical results.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
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