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A Fast Direct Algorithm for Implementing a High-Order Finite Element Method on Rectangles as Applied to Boundary Value Problems for the Poisson Equation
Doklady Mathematics. 2017. Vol. 95. No. 2. P. 129-135.
Zlotnik A.A., Zlotnik I.A.
A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for an nth-order FEM based on the fast discrete Fourier transform. The amount of arithmetic operations is logarithmically optimal in the theory and is rather attractive in practice. The algorithm admits numerous further applications (including the multidimensional case).
Keywords: boundary value problemsFFTFast direct algorithmhigh order finite element methodPoisson equation
Publication based on the results of:
А.А. Злотник, И.А. Злотник, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 473 № 2 С. 131-137
Представлен новый быстрый прямой алгоритм реализации метода конечных элементов (МКЭ) порядка n>=2 на прямоугольниках для решения различных краевых задач для уравнений типа Пуассона. Он обобщает хорошо известный алгоритм для случая разностных схем или билинейных КЭ (n=1), использующий версии быстрого дискретного преобразования Фурье (БДПФ). Его ядром являются быстрые прямой и обратный алгоритмы разложения по собственным векторам ...
Added: October 25, 2016
Alexander Zlotnik, Ilya Zlotnik, / Cornell University. Series "Working papers by Cornell University". 2017. No. 1701.03967.
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson
kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion in eigenvectors of the 1D ...
Added: January 17, 2017
Alexander Zlotnik, Ilya Zlotnik, / Cornell University. Series "Working papers by Cornell University". 2016. No. 1609.07758.
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion in eigenvectors ...
Added: September 29, 2016
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We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor products finite element method (FEM) based on the tensor products of the 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds for solving the $N$-dimensional Poisson type equation $-\Delta u+\alpha u=f$ ($N\geq 2$) with the Dirichlet boundary conditions. They are based ...
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Злотник А.А., Злотник И.А., Журнал вычислительной математики и математической физики 2020 Т. 60 № 2 С. 234-252
Представлены прямые логарифмически оптимальные в теории и быстрые на практике алгоритмы реализации
метода конечных элементов (МКЭ) на основе тензорных произведений 1D пространств МКЭ высокого порядка
на многомерных прямоугольных параллелепипедах для решения уравнения типа Пуассона. Они основаны на хорошо известных Фурье-подходах. Ключевыми новыми элементами являются детальное описание собственных пар 1D задач на собственные значения для МКЭ высокого порядка и быстрые ...
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Zlotnik A., Zlotnik I. A., Journal of Computational and Applied Mathematics 2018
We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion ...
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Added: May 25, 2016
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Added: November 13, 2012
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Added: March 24, 2014
Barcelona : IEEE, 2017
International Conference on Control, Decision and Information Technologies. ...
Added: January 17, 2018
Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70
A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...
Added: July 19, 2014
Goldengorin B. I., European Journal of Operational Research 2009 Vol. 198 No. 1 P. 102-112
Added: July 31, 2012
М. : ИКИ РАН, 2011
Added: March 26, 2013
Sirotkin D., Malyshev D., Дискретная математика 2017 Т. 29 № 3 С. 114-125
Задача о независимом множестве для заданного обыкновенного графа состоит в вычислении размера наибольшего множества его попарно несмежных вершин. Предлагается новый способ редукции графов. С его помощью получено новое доказательство NP-полноты задачи о независимом множестве в классе планарных графов и доказана NP-полнота данной задачи в классе плоских графов, имеющих только треугольные внутренние грани, с максимальной степенью ...
Added: September 7, 2017
М. : Физматлит, 2013
Conference is devoted to application of the integrated models and soft computing in artificial intelligence. ...
Added: May 26, 2013
Shmid A., Novopashin M. A., Berezin A. A., IOSR Journal of Computer Engineering (IOSR-JCE) 2017 Vol. 19 No. 3 P. 113-121
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Springer, 2020
This volume offers a collection of carefully selected, peer-reviewed papers presented at the BIOMAT 2019 International Symposium, which was held at the University of Szeged, Bolyai Institute and the Hungarian Academy of Sciences, Hungary, October 21st-25th, 2019. The topics covered in this volume include tumor and infection modeling; dynamics of co-infections; epidemic models on networks; ...
Added: March 11, 2021
Malyshev D., Alekseev V., Дискретный анализ и исследование операций 2008 Т. 15 № 1 С. 3-10
Доказывается полиномиальная разрешимость задачи о независимом множестве для бесконечного семейства подмножеств класса планарных графов. ...
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Added: March 13, 2020
Malyshev D., Journal of Applied and Industrial Mathematics (перевод журналов "Сибирский журнал индустриальной математики" и "Дискретный анализ и исследование операций") 2020 Vol. 14 No. 4 P. 706-721
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Boissard E., Le Gouic T., Loubes J., Bernoulli: a journal of mathematical statistics and probability 2015 P. 740-759
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The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
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The history of logic and analytic philosophy in Central and Eastern Europe is still known to very few people. As an exception to the rule, only two scientific schools became internationally popular: the Vienna Circle and the Lvov-Warsaw School. Nevertheless, the countries included in this region have not only joint history, but also joint cultural ...
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