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A new mixed finite-element method for H² elliptic problems

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年 128卷 300-319页

作者： Farrell, Patrick E. Hamdan, Abdalaziz MacLachlan, Scott P. Univ Oxford Math Inst Oxford England Imperial Coll London Dept Math London England Mem Univ Newfoundland Dept Math & Stat St John NL Canada

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite -element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

关键词： Mixed finite-element methods Biharmonic equation multigrid methods Saddle-point problems

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Local multigrid solvers for adaptive isogeometric analysis in hierarchical spline spaces

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IMA JOURNAL OF NUMERICAL ANALYSIS 2022年第3期42卷 2429-2458页

作者： Hofreither, Clemens Mitter, Ludwig Speleers, Hendrik Austrian Acad Sci Johann Radon Inst Computat & Appl Math Altenberger Str 69 A-4040 Linz Austria Johannes Kepler Univ Linz Inst Computat Math Altenberger Str 69 A-4040 Linz Austria Univ Roma Tor Vergata Dept Math Via Ric Sci 1 I-00133 Rome Italy

We propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical discretization space under the assumption that the hierarchical mesh satisfies an admissibility condition, i.e., the number of interacting mesh levels is uniformly bounded. We also provide several numerical experiments. The main analytical tools are quasi-interpolators for THB-splines and the abstract convergence theory of subspace correction methods.

关键词： isogeometric analysis multigrid methods adaptivity local multigrid solvers hierarchical spline spaces THB-splines HB-splines

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Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2022年第5期29卷 e2436-e2436页

作者： Donatelli, Marco Krause, Rolf Mazza, Mariarosa Semplice, Matteo Trotti, Ken Insubria Univ Dept Sci & High Technol Como Italy Univ Italian Switzerland Fac Informat Lugano Switzerland

In this article, we focus on a two-dimensional conservative steady-state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)-based discretization approach. Precisely, we use both classical FVs and the so-called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first-order discretizations. By exploiting the Toeplitz-like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second-order FV discretization. This same information is leveraged to discuss parameter-free symbol-based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well.

关键词： banded preconditioning finite volume methods fractional diffusion equations multigrid methods spectral distribution Toeplitz matrices

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Fast Geometric multigrid Preconditioned Finite-Difference Frequency-Domain Method for Borehole Electromagnetic Sensing

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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS 2022年 19卷 1页

作者： Sun, Qingtao Hu, Yunyun Duke Univ Dept Elect & Comp Engn Durham NC 27708 USA

To model the responses of borehole electromagnetic sensing in complicated geological environments, the geometric multigrid preconditioned finite-difference frequency-domain (FDFD) method is examined on its numerical performance as a fast forward modeling solver in this work. The geometric multigrid method is employed as a preconditioner for the biconjugate gradient stabilized method to efficiently solve the linear system resulting from the finite-difference method based on the staggered Yee's cell. The Gauss-Seidel method is utilized as both presmoother and postsmoother for the multigrid method, and the bilinear approach is employed to construct the restriction and prolongation operators between coarse and fine grids. The F-cycle coarsening scheme is used to balance the convergence and computational efficiency of the multigrid method. Numerical examples of borehole electromagnetic sensing are included to examine the performance of the method, which not only demonstrates its accuracy but also shows its capability as a fast solver for practical application. The geometric multigrid preconditioned FDFD method can work as a good candidate for 3-D modeling-based interpretation and evaluation for borehole electromagnetic sensing.

关键词： Electromagnetics Computational modeling Numerical models Conductivity multigrid methods Sensors Solid modeling Biconjugate gradient stabilized (BiCGStab) method borehole electromagnetic sensing finite-difference frequency-domain (FDFD) method geometric multigrid preconditioner

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Numerical Partial Differential Equations

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丛书名： Computational Science & Engineering

2025年

作者： James H. Adler Hans De Sterck Scott MacLachlan Luke Olson

Get 10% off by pre-ordering this book.This item has not yet published. Pre-order now and we will ship and process payment when the book becomes available.To pre-order contact Customer Service:800-447-SIAM (US &amp... 详细信息

ISBN: (数字)9781611978285

ISBN: (纸本)9781611978278

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This comprehensive textbook focuses on numerical methods for approximating solutions to partial differential equations (PDEs). The authors present a broad survey of these methods, introducing readers to the central concepts of various families of discretizations and solution algorithms and laying the foundation needed to understand more advanced material. The authors include over 100 well-established definitions, theorems, corollaries, and lemmas and summaries of and references to in-depth treatments of more advanced mathematics when needed.

Numerical Partial Differential Equations is divided into four parts:

Part I covers basic background on PDEs and numerical methods.
Part II introduces the three main classes of numerical methods for PDEs that are the book's focus (finite-difference, finite-element, and finite-volume methods).
Part III discusses linear solvers and finite-element and finite-volume methods at a more advanced level.
Part IV presents further high-level topics on discretizations and solvers.

关键词： Finite-element methods finite-difference methods finite-volume methods iterative methods preconditioning hyperbolic conservation laws elliptic equations diffusion equations multigrid methods consistency stability accuracy of discretizations partial differential equations

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Substructured two-grid and multi-grid domain decomposition methods

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NUMERICAL ALGORITHMS 2022年第1期91卷 413-448页

作者： Ciaramella, G. Vanzan, T. Politecn Milan MOX Lab Milan Italy Ecole Polytecn Fed Lausanne CSQI Chair Lausanne Switzerland

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

关键词： Domain decomposition methods Schwarz methods Substructured methods Two-level methods Coarse correction multigrid methods Elliptic equations

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On the numerical modeling of convection-diffusion problems by finite element multigrid preconditioning methods

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ADVANCES IN ENGINEERING SOFTWARE 2014年 68卷 56-69页

作者： Filelis-Papadopoulos, Christos K. Gravvanis, George A. Lipitakis, Elias A. Democritus Univ Thrace Sch Engn Dept Elect & Comp Engn GR-67100 Xanthi Greece Athens Univ Econ & Business Dept Informat GR-10434 Athens Greece

During the last decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. The effectiveness of the multigrid method can be also exploited by using the finite element method. Finite Element Approximate Inverses in conjunction with Richardon's iterative method could be used as smoothers in the multigrid method. Thus, a new class of smoothers based on approximate inverses can be derived. Effectiveness of explicit approximate inverses relies in the fact that they are close approximants to the inverse of the coefficient matrix and are fast to compute in parallel. Furthermore, the proposed class of finite element approximate inverses in conjunction with the explicit preconditioned Richardson method yield improved results against the classic smoothers such as Jacobi method. Moreover, a dynamic relaxation scheme is proposed based on the Dynamic Over/Under Relaxation (DOUR) algorithm. Furthermore, results for multigrid preconditioned Krylov subspace methods, such as GMRES(res), IDR(s) and BiCGSTAB based on approximate inverse smoothing and a dynamic relaxation technique are presented for the steady-state convection-diffusion equation. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Sparse linear systems Finite element method multigrid methods Approximate inverse smoothing DOUR algorithm PR2 refinement multigrid preconditioning

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multigrid methods FOR SECOND ORDER HAMILTON-JACOBI-BELLMAN AND HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第5期35卷 S323-S344页

作者： Han, Dong Wan, Justin W. L. Univ Waterloo Cheriton Sch Comp Sci Waterloo ON N2L 3G1 Canada

We propose multigrid methods for solving the discrete algebraic equations arising from the discretization of the second order Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. We propose a damped-relaxation method as a smoother for multigrid. In contrast with the standard policy iteration, the proposed damped-relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems with large jumps in control, we develop restriction and interpolation methods to capture the jumps on the coarse grids as well as during the coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.

关键词： multigrid methods full approximation scheme relaxation scheme policy iteration Hamilton-Jacobi-Bellman equations Hamilton-Jacobi-Bellman-Isaacs equations jump in control

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Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II: Numerical applications

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 437卷

作者： Adriani, Andrea Bianchi, Davide Ferrari, Paola Serra-Capizzano, Stefano Univ Insubria Dept Theoret & Appl Sci Via Dunant 3 I-21100 Varese Italy Shenzhen Univ Town Harbin Inst Technol Sch Sci Shenzhen 518055 Peoples R China Univ Wuppertal Sch Math & Nat Sci Gaussstr 20 D-42119 Wuppertal Germany Univ Insubria Dept Sci & High Technol Via Valleggio 11 I-22100 Como Italy Uppsala Univ Dept Informat Technol Uppsala Sweden

In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Omega subset of R-d, d >= 1. When Omega = [0, 1], such graphs include the standard Toeplitz graphs and, for Omega = [0, 1](d), the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure. (c) 2023 Elsevier B.V. All rights reserved.

关键词： Graphs Graph Laplacian Asymptotic spectra PDE discretizations Preconditioning multigrid methods

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Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

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COMPUTERS & STRUCTURES 2024年 291卷

作者： Herrero-Perez, D. Pico-Vicente, S. G. Tech Univ Cartagena Murcia 30202 Spain

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

关键词： Topology optimization Structural redundancy Fail-safe design Adaptivity multigrid methods High performance computing

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