Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Brix, Kolja; Canuto, Claudio; Dahmen, Wolfgang

doi:10.1007/s00211-014-0691-4

Cited by 4 publications

(8 citation statements)

References 8 publications

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“…The main challenge associated with constructing effective preconditioners for A LOR is the high aspect ratio associated with the low-order refined mesh T LOR . Because the Gauss-Lobatto points are clustered near the endpoints of the interval, the resulting Cartesian product mesh consists of parallelepipeds with aspect ratios that scale like p [11]. As a result, the mesh T LOR is not shape-regular with respect to p, and standard multigrid-type methods will not result in uniform convergence under p-refinement.…”

Section: Consider a Scalar Poisson Or Helmholtz Problemmentioning

confidence: 99%

“…projection or fractional step) methods, or unsplit methods. For the unsplit methods, we use the method of lines to first discretize in space and then temporally discretize the resulting system of ordinary differential equations (11) and (5). Here, we use diagonally implicit Runge-Kutta (DIRK) schemes as our time-integration method [2].…”

Section: Temporal Discretizationmentioning

confidence: 99%

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High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

et al. 2020

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We present a matrix-free flow solver for high-order finite element discretizations of the incompressible Navier-Stokes and Stokes equations with GPU acceleration. For high polynomial degrees, assembling the matrix for the linear systems resulting from the finite element discretization can be prohibitively expensive, both in terms of computational complexity and memory. For this reason, it is necessary to develop matrix-free operators and preconditioners, which can be used to efficiently solve these linear systems without access to the matrix entries themselves. The matrix-free operator evaluations utilize GPU-accelerated sum-factorization techniques to minimize memory movement and maximize throughput. The preconditioners developed in this work are based on a low-order refined methodology with parallel subspace corrections, as described for diffusion problems in [39]. The saddle-point Stokes system is solved using block-preconditioning techniques, which are robust in mesh size, polynomial degree, time step, and viscosity. For the incompressible Navier-Stokes equations, we make use of projection (fractional step) methods, which require Helmholtz and Poisson solves at each time step. The performance of our flow solvers is assessed on several benchmark problems in two and three spatial dimensions.

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Section: Consider a Scalar Poisson Or Helmholtz Problemmentioning

confidence: 99%

Section: Temporal Discretizationmentioning

confidence: 99%

High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

et al. 2020

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“…The main challenge in designing a preconditioner for the low-order refined operator K h is that the resulting mesh is anisotropic. The spacing of Gauss-Lobatto points near the interval endpoints scales like 1/p 2 , and the spacing of Gauss-Lobatto points near the midpoint of the interval scales like 1/p [13]. Thus, the aspect ratio of the parallelepipeds making up the Cartesian grid generated from the one-dimensional Gauss-Lobatto points scales like p. As a consequence, the low-order refined meshes T h are not shape regular in p, and we can expect degraded convergence of multigrid-type algorithms [31].…”

Section: Proposition 1 ([17 Propositions 21 and 22]mentioning

confidence: 99%

“…We are now interested in studying the speed of convergence of the preconditioned system BA h , where the additive Schwarz preconditioner B is defined by (13). Note that the condition number κ(BA h ) is bounded by c 1 /c 0 , where c 1 and c 0 are constants satisfying ( 16)…”

Section: 2mentioning

confidence: 99%

Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods

Pazner

2019

Preprint

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In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in p, requiring specialized solvers to treat the anisotropy. We make use of a structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and BR2 discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

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“…Titik evaluasi yang digunakan pada Gauss-Legendre didapatkan dari akar penyelesaian polinom Legendre pada persamaan (3) tergantung berapa titik yang digunakan (Brix, et al, 2013). Sebagai contoh jika 2 buah titik yang digunakan pada metode ini maka nilai 0 dan 1 merupakan akar penyelesaian dari 2 = 1 2 3 2 − 1 yaitu 0 =-0,577350629 dan 1 =0,577350629.…”

Section: Polinomial Legendreunclassified

Perbandingan Metode Gauss- Legendre, Gauss-Lobatto, dan Gauss-Kronrod pada Integrasi Numerik Fungsi Eksponensial

Darmawan

2016

jmpm. jurnal. matematika. dan. pendidik. matematika.

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AbstrakIntegrasi numerik merupakan merupakan suatu metode untuk menetukan nilai integrasi dari suatu fungsi dimana jika suatu fungsi tersebut sulit diselesaikan secara analitik menggunakan metode baku yang ada pada ilmu kalkulus. Solusi yang didapatkan oleh integrasi numerik ini adalah nilai hampiran atau aproksimasi sehingga akan muncul error . Terdapat dua metode integrasi numerik yaitu metode Newon-Coates (equally space) dan metode Gauss Kuadratur (unequally space). Pada artikel ini akan dikaji integrasi numerik dengan metode Gauss Kuadratur yaitu metode Gauss-Legendre, Gauss-Lobatto, dan Gauss-Kronroad yang akan diterapakan untuk menentukan nilai hampiran integrasi dari fungsi eksponensial. Sehingga akan dilakukan analisis error untuk menetukan metode mana yang memiliki akurasi paling bagus yang mendekati nilai eksaknya. Newon-Coates (equally space) and Quadrature Gauss (unequally space). In this paper will be reviewed numerical integration of Quadrature Gauss method is the Gauss-Legendre, Gauss-Lobatto, and Gauss-Kronroad will be used to determine the approximation value of the integration on exponential function. Therefore the error analysis will be done to determine which method has the most excellent accuracy to approach exact value. Kata kunci: Integrasi Numerik, Gauss-Legendre, Gauss-Lobatto, Gauss-Kronrod, Fungsi Eksponensial Abstract Numerical integration is a method to determine the value of the integration of a function where if a function is difficult to be solved analytically using standard methods exist in calculus. The solution obtained by numerical integration is the approximation value so error will be appear. There are two methods of numerical integration methods that are

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Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Cited by 4 publications

References 8 publications

High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods

Perbandingan Metode Gauss- Legendre, Gauss-Lobatto, dan Gauss-Kronrod pada Integrasi Numerik Fungsi Eksponensial

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