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Uniform Subspace Correction preconditioners for Discontinuous Galerkin Methods with hp‑Refnement

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Communications on Applied Mathematics and Computation 2022年第2期4卷 697-727页

作者： Will Pazner Tzanio Kolev Lawrence Livermore National Laboratory Center for Applied Scientifc ComputingLivermoreCAUSA

In this paper,we develop subspace correction preconditioners for discontinuous Galerkin(DG)discretizations of elliptic problems with *** preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace,and a set of small nonconforming edge *** conforming subspace is preconditioned using a matrix-free low-order refned technique,which in this work,we extend to the hprefnement context using a variational restriction *** condition number of the resulting linear system is independent of the granularity of the mesh h,and the degree of the polynomial approximation *** method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial *** examples are shown on several test cases involving adaptively and randomly refned meshes,using both the symmetric interior penalty method and the second method of Bassi and Rebay(BR2).

关键词： Discontinuous Galerkin preconditioners Domain decomposition hprefnement

Explicit Iterative Methods of Second Order and Approximate Inverse preconditioners for Solving Complex Computational Problems

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Applied Mathematics 2020年第4期11卷 307-327页

作者： Anastasia-Dimitra Lipitakis Department of Informatics and Telematics Harokopio University Athens Greece

Explicit Exact and Approximate Inverse preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.

关键词： Approximate Inverse preconditioners Iterative Methods Second Order Iterative Schemes Exact Inverse Methods Approximate Inverse Explicit Preconditioning Conjugate Gradients Convergence Analysis

Efficient Solution of a Generalized Eigenvalue Problem Arising in a Thermoconvective Instability

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Communications in Computational Physics 2008年第2期3卷 308-329页

作者： M.C.Navarro H.Herrero A.M.Mancho A.Wathen Departamento de Matematicas Facultad de Ciencias QuimicasUniversidad de Castilla-La Mancha13071 Ciudad RealSpain Departamento deMatematicas IMAFFConsejo Superior de Investigaciones Cientificas28006MadridSpain Numerical Analysis Group Computing LaboratoryOxford UniversityOxford OX13QDUnited Kingdom.

The aim of this paper is to develop an efficient numerical method to compute the eigenvalues of the stability analysis of a problem describing the motion of a fluid within a cylindrical container heated non-homogeneously from *** axisymmetric stationary motion settles in,at certain values of the external parameters appearing in the set of partial differential equations modeling the *** basic solution is computed by discretizing the equations with a Chebyshev collocation *** linear stability is formulated with a generalized eigenvalue *** numerical approach(generalized Arnoldi method)uses the idea of preconditioning the eigenvalue problem with a modified Cayley transformation before applying the Arnoldi *** works have dealt with transformations requiring regularity to one of the *** this article we extend those results to the case in which that submatrix is *** method allows a fast computation of the critical eigenvalues which determine whether the steady flow is stable or *** algorithm based on this method is compared to the QZ method and is found to be computationally more *** reliability of the computed eigenvalues in terms of stability is confirmed via pseudospectra calculations.

关键词： Rayleigh-Benard convection Chebyshev collocation preconditioners Arnoldi method Hopf bifurcation.

An Improved Wavelet Based Preconditioner for Sparse Linear Problems

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Applied Mathematics 2010年第5期1卷 370-376页

作者： Arikera Padmanabha Reddy Nagendrapp M. Bujurke 不详

In this paper, we present the construction of purely algebraic Daubechies wavelet based preconditioners for Krylov subspace iterative methods to solve linear sparse system of equations. Effective preconditioners are designed with DWTPerMod algorithm by knowing size of the matrix and the order of Daubechies wavelet. A notable feature of this algorithm is that it enables wavelet level to be chosen automatically making it more robust than other wavelet based preconditioners and avoids user choosing a level of transform. We demonstrate the efficiency of these preconditioners by applying them to several matrices from Tim Davis collection of sparse matrices for restarted GMRES.

关键词： Discrete Wavelet Transform preconditioners Sparse Matrices Krylov Subspace Iterative Methods