A-Posteriori Error Estimation for the Legendre Spectral Galerkin Method in One-Dimension

In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.

A-posteriori error estimation for second order mechanical systems

One important issue for the simulation of flexible multibody systems is the reduction of the flexible bodies de- grees of freedom. As far as safety questions are concerned knowledge about the error introduced by the reduction of the flexible degrees of freedom is helpful and very important. In this work, an a-posteriori error estimator for linear first order systems is extended for error estimation of me- chanical second order systems. Due to the special second order structure of mechanical systems, an improvement of the a-posteriori error estimator is achieved. A major advan- tage of the a-posteriori error estimator is that the estimator is independent of the used reduction technique. Therefore, it can be used for moment-matching based, Gramian matrices based or modal based model reduction techniques. The capability of the proposed technique is demon- strated by the a-posteriori error estimation of a mechanical system, and a sensitivity analysis of the parameters involved in the error estimation process is conducted.

EXTRAPOLATION AND A-POSTERIORI ERROR ESTIMATORS OF PETROV-GALERKIN METHODS FOR NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initial value problem for a nonlinear Volterra integro-differential equation. As by-products, we will also show that these enhanced approximations can be used to form a class of posteriori estimators for this Petrov-Galerkin finite element method. Numerical examples are supplied to illustrate the theoretical results.

3-D direct current resistivity forward modeling by adaptive multigrid finite element method

Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.

1 Tb/s Nyquist-WDM PM-RZ-QPSK Superchannel Transmission over 1000 km SMF-28 with MAP Equalization

In this paper, we evaluate transmission in a 1 Tb/s （10 × 112 Gb/s） Nyquist WDM PMRZQPSK superchannel over a widelydeployed SMF-28 fiber with and without maximum aposteriori （MAP） equalization. Over 1000 km can be reached with BER below the HD FEC limit and with a spectral efficiency of 4 b/s/Hz.

A Diagnostic Approach towards the Causes of Energy Balance Closure Problem

Hydrometeorological models are often evaluated and optimized on the basis of micrometeorological measurements. However, it has been known for more than three decades that surface measurements of sensible and latent heat energy (LE) are systematically underestimated. We studied this problem using six years of eddy-correlation measurements for four fields (corn, soybean, and prairie) in central Iowa, USA. We recorded major components of the energy equation (i.e. net radiation, sensible heat flux, LE, and soil heat flux, photosynthesis), and indirectly estimated most of the minor components of energy balance (namely storage in the soil, canopy and air). Storage in the canopy was related to leaf area index (LAI) acquired from Moderate Resolution Imaging Spectrometer (MODIS). In this paper, a diagnostic approach is investigated where systematic error is identified first. Three dimensional (3D) plots of the residual of energy equation vs. potential variables indicated the imbalance was largest mainly during the cold non-growing season when the soil was dry. Correlations between energy balance residual (EBR) and energy components showed that soil storage was not precisely estimated. Finally, an a-posteriori analysis (constrained linear multiple regression (CMLR)) was conducted to quantify the contribution of major/minor components of the energy equation towards EBR. The result highlights that the contribution of pertinent components of energy to EBR is mainly controlled by prevailing monthly hydrometeorological conditions;however, precise quantification of causes of imbalance is site-specific. A comparison between the a-posteriori analysis technique and the Bowen-ratio method demonstrates that the Bowen-ratio basically presumes a higher level of underestimation in LE. The results obtained in this study suggest that a-posteriori analysis may offer a superior methodology to correct measured eddy-correlation measurements. Furthermore, the overall trends in the correction of LE measurements suggest that there is a potential for rough monthly corrections of LE, irrespective of the type of crop.

Error Estimator Using Higher Order FEM for an Interface Problem

A higher order finite element method is considered to treat an interface problem. The polynomial degree is allowed to be arbitrary but it is fixed for the FEM-variational formulation. We propose an error estimator which turns out to be efficient and reliable. We demonstrate upper and lower bounds of the error estimator with respect to the exact accuracy. For the transmission problem, the coefficients for the internal and external domains can be highly dissimilar. One major difficulty is the characteristic of the estimator at the interface. The a-posteriori error estimates can be computed very efficiently element by element. To corroborate the theoretical analysis, we report on a few numerical results.