31 research outputs found
\u3ci\u3eRecent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations\u3c/i\u3e
No full textEditors: Xiaobing Feng, Ohannes Karakashian, and Yulong Xing
Chapter, Adaptivity and Error Estimation for Discontinuous Galerkin Methods, co-authored by Mahboub Baccouch, UNO faculty member.
The field of discontinuous Galerkin finite element methods has attracted considerable recent attention from scholars in the applied sciences and engineering. This volume brings together scholars working in this area, each representing a particular theme or direction of current research. Derived from the 2012 Barrett Lectures at the University of Tennessee, the papers reflect the state of the field today and point toward possibilities for future inquiry. The longer survey lectures, delivered by Franco Brezzi and Chi-Wang Shu, respectively, focus on theoretical aspects of discontinuous Galerkin methods for elliptic and evolution problems. Other papers apply DG methods to cases involving radiative transport equations, error estimates, and time-discrete higher order ALE functions, among other areas. Combining focused case studies with longer sections of expository discussion, this book will be an indispensable reference for researchers and students working with discontinuous Galerkin finite element methods and its applications.https://digitalcommons.unomaha.edu/facultybooks/1308/thumbnail.jp
Computational Fluid Dynamics
No full textThis book comprehensively explores numerical methods and their applications across diverse fields, strongly focusing on computational fluid dynamics (CFD) and advanced modeling techniques. Starting with numerical approaches for solving the viscid and inviscid Burgers equations establishes a foundation for understanding complex fluid dynamics. Subsequent chapters delve into cutting-edge topics, including Large Eddy Simulations (LES) for turbulence modeling, heat transfer analysis, and the influence of working fluids on vortex dynamics in industrial pipelines. The book also explores emerging areas such as nanoscale simulations, plasmonic excitations, and biomedical applications like hemodynamics in atrial fibrillation. Real-world case studies and practical examples demonstrate the versatility of CFD in addressing challenges in engineering, biology, and energy systems. This book combines theoretical rigour with practical insights and is designed for advanced undergraduate and graduate students, researchers, and professionals. It bridges the gap between numerical theory and real-world applications, providing readers with the tools to solve complex problems across various scientific and engineering domains. Whether you’re looking to deepen your understanding of numerical methods, enhance your CFD expertise, or explore innovative applications, this book is a valuable resource for gaining actionable insights and fostering innovation in computational modeling
Influence of Working Fluid on the Mean Flow Vortex Structure in Industrial Flow Pipelines of Arbitrary Bend Angle via RANS Simulation
Get PDFNumerical simulations of flow through industrial pipelines is important for design purposes in commercial and domestic transport of oil and gas. In this paper we use steady Computational Fluid Dynamics (CFD) to investigate the effect of different working fluids and streamwise bend angle on the spatial distribution, wall shear stress and turbulence kinetic energy of the fully-developed Dean vortex. Using Ansys Fluent, steady Reynolds-Averaged Navier Stokes (RANS) simulations are performed in a parametric study for various gas flows (air, methane and carbon dioxide) and pipe elbow bend angles (90°,60°,45°,30°,0°). Our results show the flow properties (velocity streamlines, turbulence kinetic energy and induced wall shear stress) of the Dean vortex for inflow velocities ranging from 5 to 15 m/s at Dean numbers of O(104) corresponding to working fluids having the same inflow mass flux. Reducing the pipe bend angle was found to minimize the wall shear stress and subsequent vortex magnitude for all gas flows types. We discuss these results quantitatively in terms of turbulent kinetic energy/the degree of secondary flow comparing to experimental literature. Reference to pipeline longevity is made
The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations
Get PDFWe present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is th order convergent in the -norm, when the space of piecewise polynomials of degree is used. A th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of the DG error is proportional to the -degree right Radau polynomial. These results allow us to develop a residual-based a posteriori error estimator which is computationally simple, efficient and asymptotically exact. The proposed a posteriori error estimator is proved to converge to the actual error in the -norm with order . Computational results indicate that the theoretical orders of convergence are optimal. Finally, a local adaptive mesh refinement procedure that makes use of our local a posteriori error estimate is also presented. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement
Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations
No full textAbstractWe develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(hp+1) convergence rate in the L2-norm when p-degree piecewise polynomials with p≥1 are used. We further prove that the DG solution is O(h2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(hp+2) super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the (p+1)-degree right Radau polynomial and the less significant part converges at O(hp+2) rate in the L2-norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori error estimates and prove that they converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we prove that the global effectivity index in the L2-norm converges to unity at O(h) rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented
Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation in one space dimension
No full textSuperconvergence of the local discontinuous galerkin method applied to the one‐dimensional second‐order wave equation
No full textA stochastic local discontinuous Galerkin method for stochastic two-point boundary-value problems driven by additive noises
No full text
Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems
No full text
