A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.

A posteriori error estimates of hp spectral element method for parabolic optimal control problems

<abstract><p>In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.</p></abstract>

A posteriori error analysis of an enriched Galerkin method of order one for the Stokes problem

We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.