A posteriori error estimation for conforming, non-conforming and discontinuous finite element schemes are discussed within a single framework. By dealing with three ostensibly different schemes under the same umbrella, the same common underlying principles at work in each case are highlighted leading to a clearer understanding of the issues involved. The ideas are presented in the context of pi...

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori e...

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the ...

Article history: Received 9 August 2012 Received in revised form 12 December 2012 Accepted 24 January 2013 Available online 14 February 2013

Abstract. An adaptive algorithm, based on residual type a posteriori indicators of errors measured in L∞(L2) and L2(L2) norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local s...

Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when...

We investigate an a posteriori error analysis of adaptive finite element approximations of linear-quadratic boundary optimal control problems under bilateral bound constraints, which act on a Neumann boundary condition. We use a symmetric interior penalty Galerkin (SIPG) method as discretization method. An efficient and reliable residual-type error estimator is introduced by invoking data oscil...

A convergence analysis to the weak solution is derived for interior penalty discontinuous Galerkin methods applied to the heat equation in two and three dimensions under general mixed boundary conditions. Strong convergence is established in the DG norm, as well as in the L norm, in space and in the L norm in time.