Convergence analysis for Backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system

In this paper we investigate the basic ingredients for global superconvergence strategy of streamline diffusion (SD) and discontinuous Galerkin (DG) finite element approximations in $H^{1}$ and $W^{1,\infty}$-norms (see \cite{Adams:75}) for the solution of the Vlasov--Poisson--Fokker--Planck system. This study is an extension of the results in \cite{Asadzadeh:90}-\cite{Asadzadeh.Sopasakis:2007}, to finite element schemes including discretizations of the Poisson term, where we also introduce results of an extension of the $h$-versions of SD and DG to the corresponding $hp$-versions. Optimal convergence results presented in the paper relay on the estimates for the regularized Green's functions with memory terms where some interpolation postprocessing techniques play important roles, see \cite{Baouendi.Grisvard:86}.

Error analysis of a novel discontinuous Galerkin method for the two-dimensional Poisson's equation

A BDDC algorithm for a class of staggered discontinuous Galerkin methods

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

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 error bounds are derived based on employing the generalization of the standard dual-weighted-residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

Chapter 11 - Finite Element Approximation with Splines

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on ...

For elliptic interface problems in two and three dimensions, this paper studies a priori and residual-based a posteriori error estimations for the Crouzeix--Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with no assumption on the distribution of the diffusion coefficient.

This is a further development of [9] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either continuous or piecewise constant functions. The spaces are nested, but due to the penalty term in the DG method the corresponding forms are not inherited. For the continuous finite element spaces we show that the variable V-cycle provides an optimal preconditioner for the DG system. The piece-wise constant functions do not have approximation property so in order to control the energy growth of the inter-level transfer operator we apply W–cycle MG. Finally, we present a number of numerical experiments that support the theoretical findings.

High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximations using optimal control

This paper discusses implementation and adaptivity of the Discontinuous Galerkin (DG) finite element method as applied to linear and non-linear structural dynamic problems. By the DG method, both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels. Both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived. They are third-order accurate and, while the implicit procedure is unconditionally stable, the explicit one is conditionally stable. An h-adaptive procedure based on the Zienkiewicz–Zhu error estimate using the SPR technique is applied. Numerical examples are presented to show the suitability of the DG method for both linear and non-linear structural dynamic analysis. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

It is shown that the interelement discontinuities in a discontinuous Galerkin finite element approximation are subordinate to the error measured in the broken $H^1$-seminorm. One consequence is that the DG-norm of the error is equivalent to the broken energy seminorm. Computable a posteriori error bounds are obtained for the error measured in both the DG-norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms. The estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error.