Error estimates of EDG-HDG methods for the Stokes equations with Dirac measures. (English) Zbl 07698894
Summary: In this paper, we analyze the hybridized, embedded-hybridized and embedded discontinuous Galerkin methods for the Stokes equations with Dirac measures. The velocity, the velocity traces and the pressure traces are approximated by polynomials of degree \(k\ge 1\), and the pressure is discretized by polynomials of degree \(k-1\). An attractive property, named divergence-free, is satisfied by the discrete velocity field. Moreover, the discrete velocity fields derived by hybridized and embedded-hybridized discontinuous Galerkin methods are \(H(\mathrm{div})\)-conforming. Using duality argument and Oswald interpolation, a priori and a posteriori error estimates are obtained for the velocity in \(L^2\)-norm. In addition, a posteriori error estimates for the velocity in \(W^{1,q}\)-seminorm and the pressure in \(L^q\)-norm are also derived. Finally, numerical examples are provided to validate the theoretical analysis and show the performance of the obtained a posteriori error estimators.
MSC:
| 65-XX | Numerical analysis |
| 35-XX | Partial differential equations |
| 65K10 | Numerical optimization and variational techniques |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
Keywords:
Stokes equation; Dirac measure; discontinuous Galerkin method; divergence-free; \(H(\mathrm{div})\)-conforming; a priori error estimate; a posteriori error estimateReferences:
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