A unifying theory of a posteriori error control for discontinuous Galerkin FEM

Abstract

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617-637, 2005) and Carstensen and Hu (J Numer Math 107(3):473-502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

“…Theorem 4.1 Let u ∈ H 2 0 (Ω) be the solution to (1), (2), u h ∈ S r h be the approximation obtained by the DG method and σ and τ as in (9). Then there exists a positive constant C, independent of h, u and u h , so that…”

“…Using this recovery operator, in conjunction with the inconsistent formulation for the IPDG presented in [17] (which ensures that the weak formulation of the problem is defined under minimal regularity assumptions on the analytical solution), we derive efficient and reliable a posteriori estimates of residual type for the IPDG method in the corresponding energy norm. Some ideas from a posteriori analyses for the Poisson problem presented in [4,21,20,1,9] are also implicitly utilized here in the context of fourth order problems.…”

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

“…Recently, Discontinuous Galerkin (DG) methods [14,24,34] have been increasingly applied to wave propagation problems in general [13] and the Helmholtz equation in particular [2,3,18,19,20,21] including hybridized DG approximations [23]. An a posteriori error analysis of DG methods for standard second order elliptic boundary value problems has been performed in [1,8,10,26,31,35], and a convergence analysis has been 1 provided in [9,25,32]. However, to the best of our knowledge a convergence analysis for adaptive DG discretizations of the Helmholtz equation is not yet available in the literature.…”

“…Using, e.g., the unified approach to the a posteriori error control of IPDG methods [10], the reliability of the estimator η h can be easily established.…”

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

“…Numerical examples were investigated by many authors in the one dimensional case [21,21,22,23,24,34]. But in three dimensional case, many numerical methods are unstable [7,8,9,10]. Therefore, we need to give a stable method for solving this class of equation in three dimensions.…”