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Error Analysis of Nodal Meshless Methods 1

引用

8th International Workshop on Meshfree Methods for Partial Differential Equations

作者： Schaback, Robert Univ Gottingen Inst Numer & Angew Math Lotzestr 16-18 D-37083 Gottingen Germany

ISBN: (数字)9783319519548

ISBN: (纸本)9783319519548;9783319519531

There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes x(1), . . . , x(M) is an element of Omega subset of R-d. If u* is the true solution in some Sobolev space S allowing enough smoothness for the problem in question, and if the calculated approximate values at the nodes are denoted by (u) over tilde (1), . . . ,(u) over tilde (M), the canonical form of error bounds is max (1 <= j <= M) vertical bar u* (x(j)) - (u) over tilde (j)vertical bar <= is an element of parallel to u*parallel to(S) where is an element of depends crucially on the problem and the discretization, but not on the solution. This contribution shows how to calculate such is an element of numerically and explicitly, for any sort of discretization of strong problems via nodal values, may the discretization useMoving Least Squares, unsymmetric or symmetric RBF collocation, or localized RBF or polynomial stencils. This allows users to compare different discretizations with respect to error bounds of the above form, without knowing exact solutions, and admitting all possible ways to set up generalized stiffness matrices. The error analysis is proven to be sharp under mild additional assumptions. As a byproduct, it allows to construct worst cases that push discretizations to their limits. All of this is illustrated by numerical examples.

关键词： Stiffness matrix

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Memory-Saving Technique for the Sakurai-Sugiura Eigenvalue Solver Using the Shifted Block Conjugate Gradient Method 1

引用

2nd International Workshop on Eigenvalue Problems - Algorithms, Software and Applications in Petascale Computing (EPASA)

作者： Futamura, Yasunori Sakurai, Tetsuya Univ Tsukuba 1-1-1 Tennohdai Tsukuba Ibaraki Japan

ISBN: (数字)9783319624266

ISBN: (纸本)9783319624266;9783319624242

In recent years, a numerical quadrature-based sparse eigensolver-the so-called Sakurai-Sugiura method-and its variants have attracted attention because of their highly coarse-grained parallelism. In this paper, we propose a memory-saving technique for a variant of the Sakurai-Sugiura method. The proposed technique can be utilized when inner linear systems are solved with the shifted block conjugate gradient method. Using our technique, eigenvalues and residual norms can be obtained without the explicit need to compute the eigenvector. This technique saves a considerable amount of memory space when eigenvectors are unnecessary. Our technique is also beneficial in cases where eigenvectors are necessary, because the residual norms of the target eigenpairs can be cheaply computed and monitored during each iteration step of the inner linear solver.

关键词： Conjugate gradient method

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Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation 1

引用

2nd International Workshop on Eigenvalue Problems - Algorithms, Software and Applications in Petascale Computing (EPASA)

作者： Yokota, Rio Ibeid, Huda Keyes, David Tokyo Inst Technol Meguro Ku 2-12-1 O Okayama Tokyo Japan King Abdullah Univ Sci & Technol 4700 KAUST Thuwal Saudi Arabia

ISBN: (数字)9783319624266

ISBN: (纸本)9783319624266;9783319624242

There has been a large increase in the amount of work on hierarchical low-rank approximation methods, where the interest is shared by multiple communities that previously did not intersect. This objective of this article is two-fold;to provide a thorough review of the recent advancements in this field from both analytical and algebraic perspectives, and to present a comparative benchmark of two highly optimized implementations of contrasting methods for some simple yet representative test cases. The first half of this paper has the form of a survey paper, to achieve the former objective. We categorize the recent advances in this field from the perspective of compute-memory tradeoff, which has not been considered in much detail in this area. Benchmark tests reveal that there is a large difference in the memory consumption and performance between the different methods.

关键词： Benchmarking

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Laminar Boundary Layer Flow with DBD Plasma Actuation: A Similarity Equation 1

引用

Conference on Boundary and Interior Layers - computational and Asymptotic Methods (BAIL)

作者： de Oliveira, Gael Kotsonis, Marios van Oudheusden, Bas Delft Univ Technol Fac Aerosp Engn Dept Aerodynam Wind Energy & Prop AWEP Kluyverweg 1 NL-2629 HS Delft Netherlands

ISBN: (数字)9783319672021

ISBN: (纸本)9783319672021;9783319672014

The framework of self-similar laminar boundary layer flow solutions is extended to include the effect of actuation with body force fields resembling those generated by DBD plasma actuators. The deduction line is similar to previous work investigating the effect of porous wall suction on laminar boundary layers. The starting point of the analysis is a generalised form of the Boundary Layer Partial Differential Equations (BL-PDEs) that includes volume force terms. Actuation force distributions are defined such that the volume force term of the BL-PDE equations conforms to the requirements of similarity. New similarity parameters for the plasma strength and thickness are identified. The procedure yields a general similarity equation which includes the effect of pressure gradients, wall transpiration and DBD plasma actuation. Select numerical solutions of the new similarity equation are presented to develop instinctive understanding and prompt a discussion on the construction of new closure relations for integral boundary layer models.

关键词： Laminar boundary layer

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Numerical Studies of Higher Order Variational Time Stepping Schemes for Evolutionary Navier-Stokes Equations 1

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Conference on Boundary and Interior Layers - computational and Asymptotic Methods (BAIL)

作者： Ahmed, Naveed Matthies, Gunar Weierstrass Inst Appl Anal & Stochast WIAS Mohrenstr 39 D-10117 Berlin Germany Tech Univ Dresden Inst Numer Math D-01062 Dresden Germany

ISBN: (数字)9783319672021

ISBN: (纸本)9783319672021;9783319672014

We present in this paper numerical studies of higher order variational time stepping schemes combined with finite element methods for simulations of the evolutionary Navier-Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin-Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.

关键词： Navier Stokes equations

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SHEM: An Optimal Coarse Space for RAS and Its Multiscale Approximation 23rd

SHEM: An Optimal Coarse Space for RAS and Its Multiscale App...

引用

23rd International Conference on Domain Decomposition Methods

作者： Gander, Martin J. Loneland, Atle Univ Geneva Sect Math CH-1211 Geneva 4 Switzerland Univ Bergen Dept Informat N-5020 Bergen Norway

ISBN: (纸本)9783319523897;9783319523880

In domain decomposition methods, coarse spaces are traditionally added to make the method scalable. Coarse spaces can however do much more: they can act on other error components that the subdomain iteration has difficulties with, and thus accelerate the overall solution process. We identify here the optimal coarse space for RAS, where optimal does not refer to scalable, but to best possible. This coarse space leads to convergence of the subdomain iterative method in two steps. Since this coarse space is very rich, we propose an approximation which turns out to be also very effective for multiscale problems. © Springer International Publishing AG 2017.

关键词： Domain decomposition methods

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Building a Numerical Framework to Model Gas-Liquid-Solid Interactions Using Meshfree Interpolation Methods 1

引用

8th International Workshop on Meshfree Methods for Partial Differential Equations

作者： Wang, Chu Zhang, Lucy T. Convergent Sci Madison WI USA Rensselaer Polytech Inst Troy NY 12180 USA

ISBN: (数字)9783319519548

ISBN: (纸本)9783319519548;9783319519531

In this work, we present a numerical framework that can model and simulate gas-liquid-solid three-phase interactions. A non-boundary-fitted approach is developed to simultaneously accommodate the moving gas-liquid interfaces and deforming solid. The connectivity-free front tracking method (CFFT) is adopted to track the gas-liquid interface, where an approximation-correction step is used to construct an indicator field without requiring the connectivity of the interfacial points. Therefore, topological change such as free surfaces with bubble breaking up and coalescing can be handled more easily and robustly. The fluid-solid interactions are modeled using the modified immersed finite element method (mIFEM). A more realistic and accurate solid movement and deformation are achieved by solving the solid dynamics, rather than been imposed as in the original IFEM. The coupling of the two algorithms is achieved using a meshfree interpolation function, the reproducing kernel particle method. The concept of constructing the indicator function to distinguish gas from liquid and fluid from solid naturally combines the CFFT and mIFEM algorithms together, and simulate the complex 3-phase physical system in a cohesive manner.

关键词： Gases

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Closed Form Inverse of Local Multi-Trace Operators 23rd

Closed Form Inverse of Local Multi-Trace Operators

引用

23rd International Conference on Domain Decomposition Methods

作者： Ayala, Alan Claeys, Xavier Dolean, Victorita Gander, Martin J. Univ Paris 06 Paris France INRIA Paris Paris France Univ Strathclyde Glasgow Lanark Scotland Univ Geneva Geneva Switzerland

ISBN: (纸本)9783319523897;9783319523880

Local multi-trace formulations are a way to express transmission problems. They are based on integral formulations of the solution on each subdomain, and between the subdomains both the known jumps in the traces and the fluxes of the transmission problem are imposed, and thus the method contains multiple traces in its formulation. We show in this paper that it is possible to derive a closed form inverse for local multi-trace operators for an elliptic model problem. © Springer International Publishing AG 2017.

关键词： Inverse problems

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On Nilpotent Subdomain Iterations 23rd

On Nilpotent Subdomain Iterations

引用

23rd International Conference on Domain Decomposition Methods

作者： Chaouqui, Faycal Gander, Martin J. Santugini-Repiquet, Kevin Univ Geneva Sect Math Geneva Switzerland Univ Bordeaux INRIA Bordeaux Sud Ouest IMBCNRS UMR5251MC2 Bordeaux France

ISBN: (纸本)9783319523897;9783319523880

Subdomain iterations which lead to a nilpotent iteration operator converge in a finite number of steps, and thus are equivalent to direct solvers. We determine in this paper for which type of decomposition and relaxation parameters Dirichlet-Neumann, Neumann-Neumann, and Optimal Schwarz methods can be nilpotent. We start using a simple one dimensional model problem, and show that Neumann-Neumann and Dirichlet-Neumann cannot lead to a nilpotent iteration matrix in general. Optimal Schwarz methods can be nilpotent for an arbitrary number of subdomains. © Springer International Publishing AG 2017.

关键词： Iterative methods

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Comparison of Tridiagonalization Methods Using High-Precision Arithmetic with MuPAT 1

引用

2nd International Workshop on Eigenvalue Problems - Algorithms, Software and Applications in Petascale Computing (EPASA)

作者： Ino, Ryoya Asami, Kohei Ishiwata, Emiko Hasegawa, Hidehiko Tokyo Univ Sci Dept Math Informat Sci Shinjuku Ku 1-3 Kagurazaka Tokyo Japan Univ Tsukuba Fac Lib Informat & Media Sci Kasuga 1-2 Tsukuba Ibaraki Japan

ISBN: (数字)9783319624266

ISBN: (纸本)9783319624266;9783319624242

In general, when computing the eigenvalues of symmetric matrices, a matrix is tridiagonalized using some orthogonal transformation. The Householder transformation, which is a tridiagonalization method, is accurate and stable for dense matrices, but is not applicable to sparse matrices because of the required memory space. The Lanczos and Arnoldi methods are also used for tridiagonalization and are applicable to sparse matrices, but these methods are sensitive to computational errors. In order to obtain a stable algorithm, it is necessary to apply numerous techniques to the original algorithm, or to simply use accurate arithmetic in the original algorithm. In floating-point arithmetic, computation errors are unavoidable, but can be reduced by using high-precision arithmetic, such as double-double (DD) arithmetic or quad-double (QD) arithmetic. In the present study, we compare double, double-double, and quad-double arithmetic for three tridiagonalization methods;the Householder method, the Lanczos method, and the Arnoldi method. To evaluate the robustness of these methods, we applied them to dense matrices that are appropriate for the Householder method. It was found that using high-precision arithmetic, the Arnoldi method can produce good tridiagonal matrices for some problems whereas the Lanczos method cannot.

关键词： Digital arithmetic

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