Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;\Omega)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.Comment: arXiv admin note: substantial text overlap with arXiv:1407.437

Recovery-Based Error Estimators for Diffusion Problems: Explicit Formulas

We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e., requiring solutions of global problems with mass matrices). In this paper, we develop fully explicit recovery-based error estimators for lower order conforming, mixed, and non- conforming finite element approximations to diffusion problems with full coefficient tensor. When the diffusion coefficient is piecewise constant scalar and its distribution is local quasi-monotone, it is shown theoretically that the estimators developed in this paper are robust with respect to the size of jumps. Numerical experiments are also performed to support the theoretical results

Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (ii): The Main Results

This paper is the second one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for Maxwell's equations with jump coefficients. In this paper, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients. We show that the HX preconditioner possesses fast convergence, which not only is nearly optimal with respect to the finite element mesh size but also is independent of the jumps in the coefficients across the interface between two neighboring subdomains.Comment: with 25 pages, 2 figure

Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verf¨urth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313–348; Petzoldt, Adv. Comput. Math. 16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps

Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients

Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media, the parameters of the equation are spatially discontinuous. Specifically, a scenario with coupled advection- and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. For the numerical approximation of the solution, an adaptive, pathwise discretization scheme based on a Finite Element approach is introduced. To stabilize the numerical approximation and accelerate convergence, the discrete space-time grid is chosen with respect to the varying discontinuities in each sample of the coefficients, leading to a stochastic formulation of the Galerkin projection and the Finite Element basis

A Posteriori Error Estimators for Elliptic Equations with Discontinuous Coefficients

We consider linear elliptic equations with discontinuous coefficients in two and three space dimensions with varying boundary conditions. The problem is discretized with linear finite elements. An adaptive procedure based on a posteriori error estimators for the treatment of singularities is proposed. Within the class of quasi-monotonically distributed coefficients we derive a posteriori error estimators with bounds that are independent of the variation of the coefficients. In numerical test cases we confirm the robustness of the error estimators and observe that on adaptively refined meshes the reduction of the error is optimal with respect to the number of unknowns