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 ...
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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.
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 pro...
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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.
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 arti...
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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.
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 simil...
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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.
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 ...
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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.
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 diffi...
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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 int...
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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.
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 t...
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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 relaxat...
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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...
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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.
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