Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneralized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements
79
0
0.0
(
0
)
Added by
Cuiyu He
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The well-known Prager-Synge identity is valid in $H^1(Omega)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise $H^1(Omega)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; Omega)$ through a local element-wise scheme and that recovers a gradient in $H(curl;Omega)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.
rate research
Read More
Grid adaptation for implicit Large Eddy Simulation (LES) is a non-trivial challenge due to the inherent coupling of the modeling and numerical errors. An attempt to address the challenge first requires a comprehensive assessment and then the development of error estimators to highlight regions that require refinement. Following the work of Schranner et al., a novel approach to estimate the numerical dissipation of the turbulent kinetic energy (TKE) equations is proposed. The presented approach allows the computation of the local numerical dissipation for arbitrary curvilinear grids through a post-processing procedure. This method, as well as empirical and kinetic-energy-based approaches, are employed to estimate the inherent numerical TKE. We incorporate the numerical TKE to evaluate an effective eddy viscosity, an effective Kolmogorov length scale, and an effective TKE to build a family of Index Quality (IQ) based error estimators. The proposed IQ based estimators are then assessed and utilized to show their effectiveness through an application of grid adaptation for the periodic hill test case and transitional flow over the SD 7003 airfoil. Numerical results are validated through a comparison against reference LES and experimental data. Flow over the adapted grids appear better abled to capture pertinent flow features and integrated functions, such as the lift and drag coefficients.
We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;{Omega}) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.
We establish a new H2 Korns inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle [41] and the NZT tetrahedron [45] are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct new regularized interpolation estimate and the enriching operator for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results are consistent with the theoretical prediction.
Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing high-contrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile in 2018 (SIAM J. Numer. Anal., 56, 3121--3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available online.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
