Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDGk_{k} method

In our earlier work [8], we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method

Postprocessing finite-element methods for the Navier–Stokes Equations: the Fully discrete case

An accuracy-enhancing postprocessing technique for finite-element discretizations of the Navier–Stokes equations is analyzed. The technique had been previously analyzed only for semidiscretizations, and fully discrete methods are addressed in the present paper. We show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time-stepping procedure. Further refined bounds are obtained when the timestepping procedure is either the backward Euler method or the two-step backward differentiation formula

Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations

We consider two-grid mixed-finite element schemes for the spatial discretization of the incompressible Navier-Stokes equations. A standard mixed-finite element method is applied over the coarse grid to approximate the nonlinear Navier-Stokes equations while a linear evolutionary problem is solved over the fine grid. The previously computed Galerkin approximation to the velocity is used to linearize the convective term. For the analysis we take into account the lack of regularity of the solutions of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. Optimal error bounds are obtained

The Postprocessed Mixed Finite-Element Method for the Navier–Stokes Equations: Refined Error Bounds

A postprocessing technique for mixed finite-element methods for the incompressible Navier–Stokes equations is analyzed. The postprocess, which amounts to solving a (linear) Stokes problem, is shown to increase the order of convergence of the method to which it is applied by one unit (times the logarithm of the mesh diameter). In proving the error bounds, some superconvergence results are also obtained. Contrary to previous analysis of the postprocessing technique, in the present paper we take into account the loss of regularity suffered by the solutions of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data.Ministerio de Educación y Ciencia MTM2006- 0084