The finite element methods are important and powerful tools for solving partial differential equations on complex geometries, but their high-order generalizations pose significant challenges in terms of robustness and mesh generation. In this paper, we introduce a high-order finite element method, which is insensitive to element quality and hence is more robust, and which requires only linear elements even for curved geometries and hence significantly simplifies meshing. Based on our recent work on Adaptive Extended Stencil FEM, or AES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002 nme.5246), the proposed method replaces the traditional interpolatory FEM basis functions with generalized Lagrange polynomial basis functions, constructed via local weighted least-squares approximations. In this work, we show that AES-FEM can achieve high-order accuracy using linear elements to discretize the geometry and using the standard FEM hat functions as test functions even for curved geometries. This is in contrast to most high-order generalizations of FEM, which require high-order curved elements for curved geometries. We describe the selection of stencils and the treatment of boundary conditions for high-order AES-FEM, and show that even-degree polynomial basis functions are preferred for even-order PDEs. We present numerical results in 2D and 3D for the Poisson equation and the convection-diffusion equation and demonstrate up to sixth order accuracy with AES-FEM. Overall, AES-FEM is much easier to use than FEM for curved boundaries, while being more accurate, more stable, and more efficient in terms of runtime versus error over relatively fine meshes.
Another class of high-order FEM is the discontinuous Galerkin (DG) methods. Unlike in isoparametric or spectral element methods, the basis functions in DG are not continuous between elements. Instead, a numerical flux is used at the element boundaries. DG is most conveniently applied to hyperbolic PDEs, such as the Navier-Stokes and compressible Euler equations @cite_6 @cite_31 , while extensions to parabolic and elliptic problems have also been developed @cite_29 . DG can achieve high-order accuracy and can be used on complex geometries @cite_28 , but high-quality curved elements are required for curved boundaries. Other methods related to DG include the interior penalty methods for elliptical problems, which can be unified with DG under one framework @cite_25 , the compact DG @cite_35 , which has a more compact stencil than local DG methods, and hybridized DG @cite_9 , which has a lower number of degrees of freedom.
@cite_35 @cite_28 @cite_29 @cite_9 @cite_6 @cite_31 @cite_25
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Discontinuous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. This book covers both theory and computation as it focuses on three primal DG methods--the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation is divided into three parts: Part I focuses on the application of DG methods to second order elliptic problems in one dimension and in higher dimensions. Part II presents the time-dependent parabolic problems without and with convection. Part III contains applications of DG methods to solid mechanics (linear elasticity), fluid dynamics (Stokes and Navier Stokes), and porous media flow (two-phase and miscible displacement). Appendices contain proofs and MATLAB code for one-dimensional problems for elliptic equations and routines written in C that correspond to algorithms for the implementation of DG methods in two or three dimensions. Audience: This book is intended for numerical analysts, computational and applied mathematicians interested in numerical methods for partial differential equations or who study the applications discussed in the book, and engineers who work in fluid dynamics and solid mechanics and want to use DG methods for their numerical results. The book is appropriate for graduate courses in finite element methods, numerical methods for partial differential equations, numerical analysis, and scientific computing. Chapter 1 is suitable for a senior undergraduate class in scientific computing. Contents: List of Figures; List of Tables; List of Algorithms; Preface; Part I: Elliptic Problems; Chapter 1: One-dimensional problem; Chapter 2: Higher dimensional problem; Part II: Parabolic Problems; Chaper 3: Purely parabolic problems; Chapter 4: Parabolic problems with convection; Part III: Applications; Chapter 5: Linear elasticity; Chapter 6: Stokes flow; Chapter 7: Navier-Stokes flow; Chapter 8: Flow in porous media; Appendix A: Quadrature rules; Appendix B: DG codes; Appendix C: An approximation result; Bibliography; Index. We present a high-order discontinuous Galerkin (DG) method for solving the time-dependent Maxwell equations on non-conforming hybrid meshes. The hybrid mesh combines unstructured tetrahedra for the discretization of irregularly shaped objects with a hexahedral mesh for the rest of the computational domain. The transition between tetrahedra and hexahedra is completely non-conform, that is, no pyramidal or prismatic elements are introduced to link these elements. Within each mesh element, the electromagnetic field components are approximated by a arbitrary order nodal polynomial and a centered approximation is used for the evaluation of numerical fluxes at inter-element boundaries. The time integration of the associated semi-discrete equations is achieved by a fourth-order leap-frog scheme. The method is described and discussed, including algorithm formulation, stability, and practical implementation issues such as the hybrid mesh generation and the computation of flux matrices with cubature rules. We illustrate the performance of the proposed method on several two- and three-dimensional examples involving comparisons with DG methods on single element-type meshes. The results show that the use of non-conforming hybrid meshes in DG methods allows for a notable reduction in computing time without sacrificing accuracy. We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units. We present an energy-stable low-storage high order discontinuous Galerkin method for the wave equation.We describe algorithms for efficient implementation on Graphics Processing Units (GPUs).We derive order-dependent global and local timestep restrictions for DG on hybrid meshes.We verify high order convergence and computational efficiency with numerical experiments. This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or ‘coarse’ space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C0-conforming functions, piecewise linear functions that are continuous at the centroids of the edges faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3-D model problem showing uniform convergence of the constructed methods. Copyright © 2006 John Wiley & Sons, Ltd. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.-W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440-2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility. Compared to the BR2 [F. Bassi and S. Rebay, J. Comput. Phys., 131 (1997), pp. 267-279] and IP [J. Douglas, Jr., and T. Dupont, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Phys. 58, Springer, Berlin, 1976, pp. 207-216] methods, which are known to be compact, the present method produces fewer nonzero elements in the matrix and is computationally more efficient. A graph G is δ-hyperbolic if for any four vertices u,v,x,y of G the two larger of the three distance sums dG(u,v) + dG(x,y), dG(u,x) + dG(v,y), dG(u,y) + dG(v,x) differ by at most δ, and the smallest δ ≥ 0 for which G is δ-hyperbolic is called the hyperbolicity of G. In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log2n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to ne. We also show a lower bound of Ω(log log n) on the approximation factor, namely every s-multiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n). We study a high order Discontinuous Galerkin Time Domain (DGTD) method for solving the system of Maxwell equations. This method is formulated on a non-conforming multi-element mesh mixing an unstructured triangulation for the discretization of irregularly shaped objects with a structured (Cartesian) quadrangulation for the rest of the computational domain. Within each element, the electromagnetic field components are approximated by a high order nodal polynomial. The DG method proposed here makes use of a centered scheme for the definition of the numerical traces of the electric and magnetic fields at element interfaces, associated to a second order or fourth order leap-frog scheme for the time integration of the associated semi-discrete equations. We formulate this DGTD method in 3D and study its theoretical properties. In particular, an a priori convergence estimation is elaborated. Finally, we present numerical results for the application of the method to 2D test problems. The discrete Gaussian @math is the distribution that assigns to each vector @math in a shifted lattice @math probability proportional to @math . It has long been an important tool in the study of lattices. More recently, algorithms for discrete Gaussian sampling (DGS) have found many applications in computer science. In particular, polynomial-time algorithms for DGS with very high parameters @math have found many uses in cryptography and in reductions between lattice problems. And, in the past year, Aggarwal, Dadush, Regev, and Stephens-Davidowitz showed @math -time algorithms for DGS with a much wider range of parameters and used them to obtain the current fastest known algorithms for the two most important lattice problems, the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). Motivated by its increasing importance, we investigate the complexity of DGS itself and its relationship to CVP and SVP. Our first result is a polynomial-time dimension-preserving reduction from DGS to CVP. There is a simple reduction from CVP to DGS, so this shows that DGS is equivalent to CVP. Our second result, which we find to be more surprising, is a polynomial-time dimension-preserving reduction from centered DGS (the important special case when @math ) to SVP. In the other direction, there is a simple reduction from @math -approximate SVP for any @math , and we present some (relatively weak) evidence to suggest that this might be the best achievable approximation factor. We also show that our CVP result extends to a much wider class of distributions and even to other norms. This paper deals with the problem of linear programming with inexact data represented by real closed intervals. Optimization problems with interval data arise in practical computations and they are of theoretical interest for more than forty years. We extend the concept of duality gap (DG), the difference between the primal and its dual optimal value, into interval linear programming. We consider two situations: First, DG is zero for every realization of interval parameters (the so called strongly zero DG) and, second, DG is zero for at least one realization of interval parameters (the so called weakly zero DG). We characterize strongly and weakly zero DG and its special case where the matrix of coefficients is real. We discuss computational complexity of testing weakly and strongly zero DG for commonly used types of interval linear programs and their variants with the real matrix of coefficients. We distinguish the NP-hard cases and the cases that are efficiently decidable. Based on DG conditions, we extend previous results about the bounds of the optimal value set given by Rohn. We provide equivalent statements for the bounds The problem of domain generalization is to learn from multiple training domains, and extract a domain-agnostic model that can then be applied to an unseen domain. Domain generalization (DG) has a clear motivation in contexts where there are target domains with distinct characteristics, yet sparse data for training. For example recognition in sketch images, which are distinctly more abstract and rarer than photos. Nevertheless, DG methods have primarily been evaluated on photo-only benchmarks focusing on alleviating the dataset bias where both problems of domain distinctiveness and data sparsity can be minimal. We argue that these benchmarks are overly straightforward, and show that simple deep learning baselines perform surprisingly well on them. In this paper, we make two main contributions: Firstly, we build upon the favorable domain shift-robust properties of deep learning methods, and develop a low-rank parameterized CNN model for end-to-end DG learning. Secondly, we develop a DG benchmark dataset covering photo, sketch, cartoon and painting domains. This is both more practically relevant, and harder (bigger domain shift) than existing benchmarks. The results show that our method outperforms existing DG alternatives, and our dataset provides a more significant DG challenge to drive future research. Recently, there has been an increasing interest in using distributed generators (DGs) not only to inject power into the grid but also to enhance the power quality. In this paper, a stationary-frame control method for voltage unbalance compensation in an islanded microgrid is proposed. This method is based on the proper control of DGs interface converters. The DGs are properly controlled to autonomously compensate for voltage unbalance while sharing the compensation effort and also active and reactive powers. The control system of the DGs mainly consists of active and reactive power droop controllers, a virtual impedance loop, voltage and current controllers, and an unbalance compensator. The design approach of the control system is discussed in detail, and simulation and experimental results are presented. The results demonstrate the effectiveness of the proposed method in the compensation of voltage unbalance. Smart grid, as the next generation of power grid characterized by “two-way” communications, has been paid great attention to realizing green, reliable, and efficient electricity delivery for our future lives. In order to support the two-way communications in smart grid, a large number of smart meters (SMs) should be deployed to customers to report their near real-time data to control center for monitoring purpose. However, this kind of real-time report could disclose users’ privacy, bringing down the users’ willingness to participate in smart grid. In order to address the challenge, in this paper, by considering the lifetime of SMs as exponential distribution, we propose a diverse grouping-based aggregation protocol with error detection (DG-APED), which employs differential privacy technique into grouping-based private stream aggregation for secure smart grid communications. DG-APED can not only achieve privacy-preserving aggregation, but also perform error detection efficiently when some SMs are malfunctioning. Detailed security analysis shows that DG-APED can guarantee the security and privacy requirements of smart grid communications. In addition, extensive performance evaluation also verifies the effectiveness and efficiency of the proposed DG-APED. This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier?Stokes equations. We extend a discontinuous finite element discretization originally considered for hyperbolic systems such as the Euler equations to the case of the Navier?Stokes equations by treating the viscous terms with a mixed formulation. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. As a consequence the method is ideally suited to compute high-order accurate solution of the Navier?Stokes equations on unstructured grids. The performance of the proposed method is illustrated by computing the compressible viscous flow on a flat plate and around a NACA0012 airfoil for several flow regimes using constant, linear, quadratic, and cubic elements. We give a randomized @math -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic @math -time and @math -space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample @math vectors from the discrete Gaussian distribution at any parameter in @math time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a @math -time algorithm that approximates the Closest Vector Problem to within a factor of @math . In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate @math discrete Gaussian samples in just @math time and space. Among other things, this implies a @math -time and space algorithm for @math -approximate decision SVP. Among the ancillary services, the operating reserves are important researching aspects, which deal with supplying way and selling-buying prices. Currently, distributed generation (DG) rises as a new participant in providing ancillary services, therefore, it is of main importance in operating reserves subject. This work shows some advantages and technical drawbacks by using DG when operating reserves are supplied. Different sceneries were computed where the electric generation was performed in a centralized and non-centralized way respectively. A methodology for identifying the maximum DG, which can be allocated, is proposed; this maximum level can be assessed without carrying out a negative impact in the network. The operative reserves price provide by generation distribution is analyzed. Diverse DG technologies, which present better performance in supplying such reserves, are also presented. Microgrids are Low Voltage distribution networks comprising various distributed generators (DG), storage devices and controllable loads that can operate either interconnected or isolated from the main distribution grid as a controlled entity. This paper describes the operation of a Central Controller for Microgrids. The controller aims to optimize the operation of the Microgrid during interconnected operation, i.e. maximize its value by optimizing production of the local DGs and power exchanges with the main distribution grid. Two market policies are assumed including Demand Side Bidding options for controllable loads. The developed optimization algorithms are applied on a typical LV study case network operating under various market policies and assuming realistic spot market prices and DG bids reflecting realistic operational costs. The effects on the Microgrid and the Distribution network operation are presented and discussed. Due to its reduced communication overhead and robustness to failures, distributed energy management is of paramount importance in smart grids, especially in microgrids, which feature distributed generation (DG) and distributed storage (DS). Distributed economic dispatch for a microgrid with high renewable energy penetration and demand-side management operating in grid-connected mode is considered in this paper. To address the intrinsically stochastic availability of renewable energy sources (RES), a novel power scheduling approach is introduced. The approach involves the actual renewable energy as well as the energy traded with the main grid, so that the supply-demand balance is maintained. The optimal scheduling strategy minimizes the microgrid net cost, which includes DG and DS costs, utility of dispatchable loads, and worst-case transaction cost stemming from the uncertainty in RES. Leveraging the dual decomposition, the optimization problem formulated is solved in a distributed fashion by the local controllers of DG, DS, and dispatchable loads. Numerical results are reported to corroborate the effectiveness of the novel approach. We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, nonconforming, and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric, and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain, which are then automatically coupled. Finally, the framework brings about a new point of view, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom. Domain shift refers to the well known problem that a model trained in one source domain performs poorly when applied to a target domain with different statistics. Domain Generalization (DG) techniques attempt to alleviate this issue by producing models which by design generalize well to novel testing domains. We propose a novel meta-learning method for domain generalization. Rather than designing a specific model that is robust to domain shift as in most previous DG work, we propose a model agnostic training procedure for DG. Our algorithm simulates train test domain shift during training by synthesizing virtual testing domains within each mini-batch. The meta-optimization objective requires that steps to improve training domain performance should also improve testing domain performance. This meta-learning procedure trains models with good generalization ability to novel domains. We evaluate our method and achieve state of the art results on a recent cross-domain image classification benchmark, as well demonstrating its potential on two classic reinforcement learning tasks. The increasing presence of single-phase distributed generators and unbalanced loads in the electric power system may lead to unbalance of the three phase voltages, resulting in increased losses and heating. The distribution network operators (DNOs) are increasingly being challenged to maintain the required power quality. To reduce voltage unbalance DNOs are seeking to connect larger DG units to the three phases instead of a single-phase connection. The three-phase connection can be realised by three single-phase inverters or by a three-phase inverter. Each inverter topology can be implemented with different control strategies. The control can be equiped with active power filtering functions which can improve the power quality. In this paper, the effect of connecting DG units by means of a three-phase connection instead of a single-phase connection on voltage unbalance is studied. Besides two commonly used control strategies, two other control strategies that combine DG and active power filtering functions are implemented and their effect on voltage unbalance is studied. The last two control strategies lead to the reduction of voltage unbalance such that the voltage requirements are maintained.