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作者： Kasey Bray University of Kentucky

学位级别：博士

Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

关键词： Accuracy biharmonic operator eigenvalue problem preconditioning multigrid algorithms rigorous Fourier analysis roundoff errors

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multigrid algorithms FOR hp-DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ELLIPTIC PROBLEMS

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2015年第1期53卷 598-618页

作者： Antonietti, Paola F. Sarti, Marco Verani, Marco Politecn Milan Dipartimento Matemat MOX Lab Modeling & Sci Comp I-20133 Milan Italy

We present W-cycle h-, p-, and hp-multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order p(2+mu), where p is the polynomial approximation degree and mu = 0, 1. A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.

关键词： hp-version discontinuous Galerkin multigrid algorithms elliptic problems geometric multigrid

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A refined convergence analysis of multigrid algorithms for elliptic equations

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ANALYSIS AND APPLICATIONS 2015年第3期13卷 255-290页

作者： Chang, Qianshun Jia, Rong-Qing Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada

multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss-Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

关键词： multigrid algorithms elliptic equations multilevel approximation approximation conditions smoothing conditions finite element approximation

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APPLICATIONS OF multigrid algorithms TO FINITE DIFFERENCE SCHEMES FOR ELLIPTIC EQUATIONS WITH VARIABLE COEFFICIENTS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第3期36卷 A1140-A1162页

作者： Jia, Rong-Qing Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada

This paper is devoted to a study of multigrid algorithms applied to finite difference schemes. If the elliptic equation has variable coefficients, the analysis of multigrid algorithms in the existent literature only gave a convergence rate depending on the number of levels. In this paper, for multigrid algorithms applied to finite difference schemes for elliptic equations with variable coefficients, we establish a convergence rate independent of the number of levels. Our convergence analysis does not require any additional regularity of the solution and is valid for commonly used smoothing operators including the standard Gauss-Seidel method. Under guidance of the general theory, we give details of implementation of the inherited multigrid V(1,1) algorithm. Furthermore, we will provide numerical examples to illustrate the general theory and demonstrate that the inherited multigrid algorithm is efficient for numerical solutions of elliptic equations with variable coefficients. In particular, we will consider elliptic equations on an L-shaped domain whose solutions do not have full regularity and show that the multigrid V(1,1) algorithm performs well in such situations.

关键词： multigrid algorithms finite difference schemes multilevel approximation elliptic equations

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Optimized parametric inference for the inner loop of the multigrid Ensemble Kalman Filter

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JOURNAL OF COMPUTATIONAL PHYSICS 2022年 471卷

作者： Moldovan, G. Lehnasch, G. Cordier, L. Meldi, M. Univ Poitiers Inst Pprime CNRS ISAEENSMA 11 Bd Marie & Pierre CurieSite FuturoscopeTSA 41 F-86073 Poitiers 9 France

Essential features of the multigrid Ensemble Kalman Filter (Moldovan et al. (2021) [24]) recently proposed for Data Assimilation of fluid flows are investigated and assessed in this article. The analysis is focused on the improvement in performance due to the inner loop. In this step, data from solutions calculated on the higher resolution levels of the multigrid approach are used as surrogate observations to improve the model prediction on the coarsest levels of the grid. The latter represents the level of resolution used to run the ensemble members for global Data Assimilation. The method is tested over two classical one-dimensional problems, namely the linear advection problem and the Burgers' equation. The analyses encompass a number of different aspects such as different grid resolutions. The results indicate that the contribution of the inner loop is essential in obtaining accurate flow reconstruction and global parametric optimization. These findings open exciting perspectives of application to grid-dependent reduced-order models extensively used in fluid mechanics applications for complex flows, such as Large Eddy Simulation (LES).(c) 2022 Elsevier Inc. All rights reserved.

关键词： Kalman Filter Data Assimilation multigrid algorithms

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Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell's Equations

Application of Helmholtz/Hodge Decomposition to Finite Eleme...

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作者： Nan, Zhe Louisiana State University

学位级别：博士

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell's equations. We begin with the introduction of Maxwell's equations and a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell's equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments.

关键词： multigrid algorithms Helmholtz/Hodge decomposition Maxwell's equations interface problems extraction formulas finite element methods

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Automatic coarsening of three dimensional anisotropic unstructured meshes for multigrid applications

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APPLIED MATHEMATICS AND COMPUTATION 2012年第21期218卷 10500-10519页

作者： Mesri, Youssef Guillard, Herve Coupez, Thierry IFP Energies Nouvelles F-92852 Rueil Malmaison France Inria F-06902 Sophia Antipolis France Univ Nice Sophia Antipolis Lab Jean Akexandre Dieudonne F-06108 Nice France CNRS UMR 7635 Ecole Mines Paris CEMEF F-06904 Sophia Antipolis France

This paper describes an algorithm designed for the automatic coarsening of three-dimensional unstructured simplicial meshes. This algorithm can handle very anisotropic meshes like the ones typically used to capture the boundary layers in CFD with Low Reynolds turbulence modeling that can have aspect ratio as high as 10(4). It is based on the concept of mesh generation governed by metrics and on the use of a natural metric mapping the initial (fine) mesh into an equilateral one. The paper discusses and compares several ways to define node based metric from element based metric. Then the semi-coarsening algorithm is described. Several application examples are presented, including a full three-dimensional complex model of an aircraft with extremely high anisotropy. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Mesh generation Tetrahedral meshes Coarsening Anisotropy multigrid algorithms CFD Boundary layers

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HP-multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge-Kutta smoother

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JOURNAL OF COMPUTATIONAL PHYSICS 2012年第22期231卷 7564-7583页

作者： van der Vegt, J. J. W. Rhebergen, S. Univ Twente Dept Appl Math NL-7500 AE Enschede Netherlands Univ Minnesota Sch Math Minneapolis MN 55455 USA

Using a detailed multilevel analysis of the complete hp-multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multi-grid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained. (C) 2012 Elsevier Inc. All rights reserved.

关键词： multigrid algorithms Discontinuous Galerkin methods Higher order accurate discretizations Space-time methods Runge-Kutta methods Fourier analysis Multilevel analysis

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Reliable performance prediction for multigrid software on distributed memory systems

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ADVANCES IN ENGINEERING SOFTWARE 2011年第5期42卷 247-258页

作者： Romanazzi, Giuseppe Jimack, Peter K. Goodyer, Christopher E. Univ Coimbra Dept Matemat CMUC P-3001454 Coimbra Portugal Univ Leeds Sch Comp Leeds LS2 9JT W Yorkshire England

We propose a model for describing and predicting the parallel performance of a broad class of parallel numerical software on distributed memory architectures. The purpose of this model is to allow reliable predictions to be made for the performance of the software on large numbers of processors of a given parallel system, by only benchmarking the code on small numbers of processors. Having described the methods used, and emphasized the simplicity of their implementation, the approach is tested on a range of engineering software applications that are built upon the use of multigrid algorithms. Despite their simplicity, the models are demonstrated to provide both accurate and robust predictions across a range of different parallel architectures, partitioning strategies and multigrid codes. In particular, the effectiveness of the predictive methodology is shown for a practical engineering software implementation of an elastohydrodynamic lubrication solver. (C) 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

关键词： Performance prediction Parallel engineering software multigrid algorithms Partial differential equations Distributed memory architectures Parallel distributed algorithms

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multigrid methods for Maxwell's equations

Multigrid methods for Maxwell's equations

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作者： Cui, Jintao Louisiana State University

学位级别：博士

In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell's equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation ar

关键词： Maxwell's equations nonconforming finite element methods discontinuous Galerkin methods graded meshes multigrid algorithms

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