Evaluation of numerical methods for elliptic partial differential equations

Houstis, Elias N.; Lynch, Robert E.; Rice, John R.; Papatheodorou, Theodore S.

doi:10.1016/0021-9991(78)90014-1

Cited by 56 publications

(42 citation statements)

References 9 publications

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“…There exist well-established performance evaluation methodologies for scientific software [Houstis et al 1978;1983;Boisvert et al 1979;Rice 1983;1990;Dyksen et al 1984;Moore et al 1990]. While there are many important factors that contribute to the quality of numerical software, we illustrate our ideas using speed and accuracy.…”

Section: Performance Evaluationmentioning

confidence: 99%

Pythia-Ii

Houstis

Catlin

Rice

et al. 2000

ACM Trans. Math. Softw.

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Often scientists need to locate appropriate software for their problems and then select from among many alternatives. We have previously proposed an approach for dealing with this task by processing performance data of the targeted software. This approach has been tested using a customized implementation referred to as PYTHIA. This experience made us realize the complexity of the algorithmic discovery of knowledge from performance data and of the management of these data together with the discovered knowledge. To address this issue, we created PYTHIA-II-a modular framework and system which combines a general knowledge discovery in databases (KDD) methodology and recommender system technologies to provide advice about scientific software/hardware artifacts. The functionality and effectiveness of the system is demonstrated for two existing performance studies using sets of software for solving partial differential equations. From the end-user perspective, PYTHIA-II allows users to specify the problem to be solved and their computational objectives. In turn, PYTHIA-II (i) selects the software available for the user's problem, (ii) suggests parameter values, and (iii) assesses the recommendation provided. PYTHIA-II provides all the necessary facilities to set up database schemas for testing suites and associated performance data in order to test sets of software. Moreover, it allows easy interfacing of alternative data mining and recommendation facilities. PYTHIA-II is an open-ended system implemented on public domain software and

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Section: Performance Evaluationmentioning

confidence: 99%

Pythia-Ii

Houstis

Catlin

Rice

et al. 2000

ACM Trans. Math. Softw.

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“…We therefore refer to (13) as the weighted residual formulation. We note that the variational statement of (13) can also be obtained directly from the method of weighted residuals (see for example [7,8,64,95]).…”

Section: Integration By Parts and The Weighted Residual Formmentioning

confidence: 99%

“…The first step in constructing the numerical hp-collocation scheme is the special choice of quadrature points to evaluate the integrals of (13). In the framework of hp-collocation we use the Gauss-Lobatto quadrature rule, which reads for a function f (ξ) over the one-dimensional parametric domain ξ ∈ [−1, 1]…”

Section: Reduced Quadrature At the Gauss-lobatto Pointsmentioning

confidence: 99%

“…The variational formulation (13) requires that on each element Ω e the residual of the original differential equation stated in (1), the forces over the element boundaries (flux terms) and possible external traction boundary conditions need to be in equilibrium in a weighted average sense. We therefore refer to (13) as the weighted residual formulation.…”

Section: Integration By Parts and The Weighted Residual Formmentioning

confidence: 99%

“…The main advantage of collocation as compared to Galerkin methods is a significant reduction of the computational cost for the formation and assembly 1 of stiffness matrices and residual vectors [13][14][15][16]. Collocation methods attracted wide attention during the 1970's and 80's (see for example [9,10,[13][14][15][17][18][19]), and have been widely applied in spectral element methods [1,2,11,12,20], in meshfree methods [21][22][23][24][25][26], and most recently in isogeometric analysis [16,[27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Schillinger

Evans

Frischmann

et al. 2014

Numerical Meth Engineering

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We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. AbstractWe demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, i.e. weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocation and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.

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Practical Finite Element Modeling in Earth Science Using Matlab

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Evaluation of numerical methods for elliptic partial differential equations

Cited by 56 publications

References 9 publications

Pythia-Ii

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

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Evaluation of numerical methods for elliptic partial differential equations

Cited by 56 publications

References 9 publications

Pythia-Ii

Pythia-Ii

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

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A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics