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A robust prolongation operator for non-nested finite element methods

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2015年第3期69卷 235-246页

作者： Hu, Jun Peking Univ LMAM Beijing 100871 Peoples R China Peking Univ Sch Math Sci Beijing 100871 Peoples R China

The paper is to present a general framework for designing a prolongation operator for general nonnested finite element methods. The main idea is to take the inclusion interpolation in the unrefined region that consists of all unrefined elements, and the canonical interpolation of the quasi-interpolation in the quasi-strict refined region that consists of children elements which are not neighbors of the unrefined region, and define a careful mixture in the small layer between these two regions. This leads to a robust prolongation operator that has both local projection and uniform boundedness properties for most of nonnested finite element methods in literature. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： prolongation operator Nonnested finite element

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Learning-based local weighted least squares for algebraic multigrid method

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年第1期493卷

作者： Wang, Fan Gu, Xiang Sun, Jian Xu, Zongben Xi An Jiao Tong Univ Sch Math & Stat 28 Xianning West Rd Xian 710049 Shaanxi Peoples R China

Algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problemdependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system.

关键词： Algebraic multigrid method prolongation operator Local weighted least squares model Deep learning

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An efficient algebraic multigrid method for second-order elliptic equation on polygonal domains

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2021年第0期

作者： Li, Ming Honghe Univ Dept Math & Stat Mengzi 661100 Yunnan Peoples R China

Based on a coarsening strategy of adjacency matrix, a new algebraic prolongation operator is developed for standard V-cycle multigrid method to accelerate the whole process. An efficient algebraic multigrid (EAMG) method is proposed for solving large-scale linear system, which arise from finite element (FE) discretization of second-order elliptic boundary value problem. Numerical experiments on polygonal domains are conducted to demonstrate that the EAMG computation is more efficient than standard method.

关键词： algebraic multigrid method finite element method polygonal domain prolongation operator

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A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme

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MATHEMATICS AND COMPUTERS IN SIMULATION 2015年 117卷 54-67页

作者： Cheng, Dongsheng Liu, Zhiyong Wu, Tingting Shenzhen Inst Informat Technol Sch Software Engn Shenzhen 518172 Peoples R China Shenzhen Polytech Ind Ctr Shenzhen 518055 Peoples R China Shandong Normal Univ Dept Math Jinan Peoples R China

In this paper, a preconditioned iterative method is developed to solve the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The complex shifted-Laplacian is generalized to precondition the Helmholtz-PML equation, which is discretized by an optimal 25-point finite difference scheme that we presented in Chen et al. (2011). A spectral analysis is given for the discrete preconditioned system from the perspective of linear fractal mapping, and Bi-CGSTAB is used to solve it. The multigrid method is employed to invert the preconditioner approximately, and a new matrix-based prolongation operator is constructed in the multigrid cycle. Numerical experiments are presented to illustrate the efficiency of the multigrid-based preconditioned Bi-CGSTAB method with the new prolongation operator. Numerical results are also given to compare the performance of the new prolongation operator with that of the prolongation operator based on the algebraic multigrid (AMG) principle. (C) 2015 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Helmholtz equation Preconditioner Multigrid prolongation operator

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A multiscale two-point flux-approximation method

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JOURNAL OF COMPUTATIONAL PHYSICS 2014年 275卷 273-293页

作者： Moyner, Olav Lie, Knut-Andreas SINTEF ICT Dept Appl Math N-0314 Oslo Norway

A large number of multiscale finite-volume methods have been developed over the past decade to compute conservative approximations to multiphase flow problems in heterogeneous porous media. In particular, several iterative and algebraic multiscale frameworks that seek to reduce the fine-scale residual towards machine precision have been presented. Common for all such methods is that they rely on a compatible primal-dual coarse partition, which makes it challenging to extend them to stratigraphic and unstructured grids. Herein, we propose a general idea for how one can formulate multiscale finite-volume methods using only a primal coarse partition. To this end, we use two key ingredients that are computed numerically: (i) elementary functions that correspond to flow solutions used in transmissibility upscaling, and (ii) partition-of-unity functions used to combine elementary functions into basis functions. We exemplify the idea by deriving a multiscale two-point flux-approximation (MsTPFA) method, which is robust with regards to strong heterogeneities in the permeability field and can easily handle general grids with unstructured fine- and coarse-scale connections. The method can easily be adapted to arbitrary levels of coarsening, and can be used both as a standalone solver and as a preconditioner. Several numerical experiments are presented to demonstrate that the MsTPFA method can be used to solve elliptic pressure problems on a wide variety of geological models in a robust and efficient manner. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Multiscale methods Iterative multiscale methods Unstructured grids prolongation operator Reservoir simulation

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Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models

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SPE JOURNAL 2012年第2期17卷 523-539页

作者： Zhou, H. Tchelepi, H. A. Stanford Univ Dept Energy Resources Engn Stanford CA 94305 USA

An efficient Two-Stage Algebraic Multiscale Solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU), or the Additive Schwarz (AS) method is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigen-values of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

关键词： scaling method BILU reservoir simulation boundary condition Upstream Oil & Gas prolongation operator coarse grid iteration matrix matrix preconditioner Reservoir Simulation Scaling methods

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A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2012年第24期231卷 8152-8175页

作者： Chen, Zhongying Cheng, Dongsheng Wu, Tingting Sun Yat Sen Univ Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Guangdong Peoples R China Shandong Normal Univ Sch Math Sci Jinan 250014 Peoples R China

In this paper, a new 27-point finite difference method is presented for solving the 3D Helmholtz equation with perfectly matched layer (PML), which is a second order scheme and pointwise consistent with the equation. An error analysis is made between the numerical wavenumber and the exact wavenumber, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. A full-coarsening multigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear system stemming from the Helmholtz equation with PML by the finite difference scheme. The shifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectral analysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method, with a multigrid method used to invert the preconditioner approximately. Full-coarsening multigrid is employed, and a new matrix-based prolongation operator is constructed accordingly. Numerical results are presented to demonstrate the efficiency of both the new 27-point finite difference scheme with refined parameters, and the preconditioned Bi-CGSTAB method with the 3D full-coarsening multigrid. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Helmholtz equation Perfectly matched layer Finite difference method Preconditioner Bi-CGSTAB Shifted-Laplacian Multigrid prolongation operator

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operator-based multiscale method for compressible flow

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SPE JOURNAL 2008年第2期13卷 267-273页

作者： Zhou, H. Tchelepi, H. A. Stanford Univ Dept Energy Resources Engn Stanford CA 94305 USA

Multiscale methods have been developed for accurate and efficient numerical solution of flow problems in large-scale heterogeneous reservoirs. A scalable and extendible operator-Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework. It is natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two operators are constructed: prolongation and restriction. The prolongation operator is constructed by assembling the multiscale basis functions. The specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finitevolume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators to the fine-scale flow equations. Solving the coarse-scale equation results in a high-quality coarse-scale pressure. The finescale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport (saturation) equation. We describe the OBMM approach for Multiscale modeling of compressible multiphase flow. We show that extension from incompressible to compressible flows is straightforward. No special treatment for compressibility is required. The efficiency of multiscale formulations over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrated using several numerical examples including a challenging depletion problem in a strongly heterogeneous permeability field (SPE 10).

关键词： Artificial Intelligence operator-based multiscale method prolongation operator scaling method compressibility OBMM saturation Upstream Oil & Gas boundary condition Multiscale Method

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An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier-Stokes equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2008年第8期227卷 4018-4037页

作者： Chiu, P. H. Sheu, Tony W. H. Lin, R. K. Natl Taiwan Univ Dept Engn Sci & Ocean Engn Taipei 106 Taiwan

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier-Stokes equations. This proposed method solves a smaller system of nonlinear Navier-Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier-Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection-diffusion-reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier-Stokes solutions. (C) 2007 Elsevier Inc. All rights reserved.

关键词： two-level method Navier-Stokes equations prolongation operator convection-diffusion-reaction explicit pressure gradient discretization dispersion-relation-preserving

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