We analyze the spatial discretization errors associated with solutions of onedimensional hyperbolic conservation laws by discontinuous Galerkin methods in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p+1 on each element. We also prove that the local and global discret...

We analyze the convergence behavior of collocation schemes applied to approximate solutions of BVPs in nonlinear index 1 DAEs, which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity in the inherent nonlinear ODE system. In particular, we focus on the case when the inherent ODE system is singular with a singularity of the first kind and apply p...

The accuracy of information transmission while solving domain decomposed problems is crucial to the smooth transition a solution around interface/overlapping region. This paper describes systematical study on an accuracy-enhancing interface treatment algorithm based back and forth error compensation correction method (BFECC). By repetitively employing low order interpolation technique (usually ...

In this paper, we propose a novel gradient recovery method for elliptic interface problem using body-fitted mesh in two dimension. Due to the lack of regularity of solution at interface, standard gradient recovery methods fail to give superconvergent results, and thus will lead to overrefinement when served as a posteriori error estimator. This drawback is overcome by designing an immersed grad...

In this paper, we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in oneand two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementa...

In this paper, we investigate the discretization of general convex optimal control problem using the mixed finite element method. The state and co-state are discretized by the lowest order Raviart-Thomas element and the control is approximated by piecewise constant functions. We derive error estimates for both the control and the state approximation. Moreover, we present the superconvergence an...

We establish interior velocity superconvergence estimates for mixed finite element approximations of second order elliptic problems on non-matching rectangular and quadrilateral grids. Both mortar and non-mortar methods for imposing the interface conditions are considered. In both cases it is shown that a discrete L2-error in the velocity in a compactly contained subdomain away from the interfa...

The supercloseness and superconvergence property of stabilized finite element methods applied to the Stokes problem are studied. We consider consistent residual based stabilization methods as well as nonconsistent local projection type stabilizations. Moreover, we are able to show the supercloseness of the linear part of the MINI-element solution which has been previously observed in practical ...