Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.

ON MAXIMUM NORM ESTIMATES FOR RITZ-VOLTERRAPROJECTION WITH APPLICATIONS TO SOME TIME DEPENDENT PROBLEMS

The stability in L∞-norm is considered for the Ritz-Volterra projection and some applications are presented in this paper. As a result, point-wise error estimates are established for the finite element approximation for the parabolic integrodifferential equation, Sobolev equations, and a diffusion equation with non-local boundary value problem.

Notes on the Norm Estimates for the Sum of Two Matrices

This is a lecture note of my joint work with Chi-Kwong Li concerning various results on the norm structure of n 2 n matrices (as Hilbert-space operators). The main result says that the triangle inequality serves as the ultimate norm estimate for the upper bounds of summation of two matrices. In the case of summation of two normal matrices, the result turns out to be a norm estimate in terms of the spectral variation for normal matrices.

THE BEST L2 NORM ERROR ESTIMATE OF LOWER ORDER FINITE ELEMENT METHODS FOR THE FOURTH ORDER PROBLEM

In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？

Parameter Method Data Processing for CPⅢ Precise Trigonometric Leveling Network

In view of the limitation of the difference method,the adjustment model of CPⅢprecise trigonometric leveling control network based on the parameter method was proposed in the present paper.The experiment results show that this model has a simple algorithm and high data utilization,avoids the negative influences caused by the correlation among the data acquired from the difference method and its accuracy is improved compared with the difference method.In addition,the strict weight of CPⅢprecise trigonometric leveling control network was also discussed in this paper.The results demonstrate that the ranging error of trigonometric leveling can be neglected when the vertical angle is less than 3 degrees.The accuracy of CPⅢprecise trigonometric leveling control network has not changed significantly before and after strict weight.

L_(p) Quasi Norm State Estimator for Power Systems

This paper proposes an L_(p)(0<p<1)quasi norm state estimator for power system static state estimation.Compared with the existing L1 and L2 norm estimators,the proposed estimator can suppress the bad data more effectively.The robustness of the proposed estimator is discussed,and an analysis shows that its ability to suppress bad data increases as p decreases.Moreover,an algorithm is suggested to solve the nonconvex state estimation problem.By introducing a relaxation factor in the mathematical model of the proposed estimator,the algorithm can prevent the solution from converging to a local optimum as much as possible.Finally,simulations on a 3-bus DC system,the IEEE 14-bus and IEEE 300-bus systems as well as a 1204-bus provincial system verify the high computation efficiency and robustness of the proposed estimator.

Optimal L_∞ Estimates for Galerkin Methods for Nonlinear Singular Two-point Boundary Value Problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems

We study the enhancement of accuracy,by means of the convolution postprocessing technique,for discontinuous Galerkin(DG)approximations to hyperbolic problems.Previous investigations have focused on the superconvergence obtained by this technique for elliptic,time-dependent hyperbolic and convection-diffusion problems.In this paper,we demonstrate that it is possible to extend this postprocessing technique to the hyperbolic problems written as the Friedrichs’systems by using an upwind-like DG method.We prove that the L2-error of the DG solution is of order k+1/2,and further the post-processed DG solution is of order 2k+1 if Qkpolynomials are used.The key element of our analysis is to derive the(2k+1)-order negative norm error estimate.Numerical experiments are provided to illustrate the theoretical analysis.

Nonlinear Reduced DNN Models for State Estimation

We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providingdata-to-state maps, represented in terms of Deep Neural Networks. A major constituentis a sensor-induced decomposition of a model-compliant Hilbert space warranting approximation in problem relevant metrics. It plays a similar role as in a ParametricBackground Data Weak framework for state estimators based on Reduced Basis concepts. Extensive numerical tests shed light on several optimization strategies that areto improve robustness and performance of such estimators.

A Mixed-finite Volume Element Coupled with the Method of Characteristic Fractional Step Difference for Simulating Transient Behavior of Semiconductor Device of Heat Conductor And Its Numerical Analysis

The mathematical system is formulated by four partial differential equations combined with initial- boundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids （coarse partition and refined partition）, piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.