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Journal articles on the topic 'High-Order finite element methods'

Author: Grafiati

Published: 8 June 2024

Last updated: 31 July 2025

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1

Abreu, Eduardo, Ciro Díaz, Juan Galvis, and Marcus Sarkis. "On high-order conservative finite element methods." Computers & Mathematics with Applications 75, no. 6 (2018): 1852–67. http://dx.doi.org/10.1016/j.camwa.2017.10.020.

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2

Harari, Isaac, and Danny Avraham. "High-Order Finite Element Methods for Acoustic Problems." Journal of Computational Acoustics 05, no. 01 (1997): 33–51. http://dx.doi.org/10.1142/s0218396x97000046.

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The goal of this work is to design and analyze quadratic finite elements for problems of time-harmonic acoustics, and to compare the computational efficiency of quadratic elements to that of lower-order elements. Non-reflecting boundary conditions yield an equivalent problem in a bounded region which is suitable for domain-based computation of solutions to exterior problems. Galerkin/least-squares technology is utilized to develop robust methods in which stability properties are enhanced while maintaining higher-order accuracy. The design of Galerkin/least-squares methods depends on the order

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3

Bagheri, Babak, L. Ridgway Scott, and Shangyou Zhang. "Implementing and using high-order finite element methods." Finite Elements in Analysis and Design 16, no. 3-4 (1994): 175–89. http://dx.doi.org/10.1016/0168-874x(94)90063-9.

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4

Lin, Qun, and Junming Zhou. "Superconvergence in high-order Galerkin finite element methods." Computer Methods in Applied Mechanics and Engineering 196, no. 37-40 (2007): 3779–84. http://dx.doi.org/10.1016/j.cma.2006.10.027.

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5

Larson, Mats G., and Sara Zahedi. "Stabilization of high order cut finite element methods on surfaces." IMA Journal of Numerical Analysis 40, no. 3 (2019): 1702–45. http://dx.doi.org/10.1093/imanum/drz021.

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Abstract We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The

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6

Winther, Kaibo Hu &. Ragnar. "Well-Conditioned Frames for High Order Finite Element Methods." Journal of Computational Mathematics 39, no. 3 (2021): 333–57. http://dx.doi.org/10.4208/jcm.2001-m2018-0078.

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7

Dobrev, Veselin A., Tzanio V. Kolev, and Robert N. Rieben. "High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics." SIAM Journal on Scientific Computing 34, no. 5 (2012): B606—B641. http://dx.doi.org/10.1137/120864672.

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8

Yurun, Fan, and M. J. Crochet. "High-order finite element methods for steady viscoelastic flows." Journal of Non-Newtonian Fluid Mechanics 57, no. 2-3 (1995): 283–311. http://dx.doi.org/10.1016/0377-0257(94)01338-i.

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9

Opschoor, Joost A. A., Philipp C. Petersen, and Christoph Schwab. "Deep ReLU networks and high-order finite element methods." Analysis and Applications 18, no. 05 (2020): 715–70. http://dx.doi.org/10.1142/s0219530519410136.

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Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “[Formula: see text]-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN

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10

Jund, Sébastien, and Stéphanie Salmon. "Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping." International Journal of Applied Mathematics and Computer Science 17, no. 3 (2007): 375–93. http://dx.doi.org/10.2478/v10006-007-0031-2.

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Arbitrary High-Order Finite Element Schemes and High-Order Mass LumpingComputers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of orderkand show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also constr

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11

Akrivis, Georgios. "High-order finite element methods for the Kuramoto-Sivashinsky equation." ESAIM: Mathematical Modelling and Numerical Analysis 30, no. 2 (1996): 157–83. http://dx.doi.org/10.1051/m2an/1996300201571.

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12

Vidal-Ferràndiz, A., S. González-Pintor, D. Ginestar, G. Verdú, M. Asadzadeh, and C. Demazière. "Use of discontinuity factors in high-order finite element methods." Annals of Nuclear Energy 87 (January 2016): 728–38. http://dx.doi.org/10.1016/j.anucene.2015.06.021.

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13

Xiao, Yuanming, Jinchao Xu, and Fei Wang. "High-order extended finite element methods for solving interface problems." Computer Methods in Applied Mechanics and Engineering 364 (June 2020): 112964. http://dx.doi.org/10.1016/j.cma.2020.112964.

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14

Rank, Ernst, Zohar Yosibash, and Alexander Düster. "HOFEM07 – International workshop on high-order finite element methods, 2007." Computer Methods in Applied Mechanics and Engineering 198, no. 13-14 (2009): 1125. http://dx.doi.org/10.1016/j.cma.2009.01.002.

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15

MIURA, Shinichiro. "214 Turbulent channel flow Analysis of Finite Element Methods with High-Order Element." Proceedings of Conference of Kansai Branch 2006.81 (2006): _2–20_. http://dx.doi.org/10.1299/jsmekansai.2006.81._2-20_.

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16

Lu, Qiukai, Mark S. Shephard, Saurabh Tendulkar, and Mark W. Beall. "Parallel mesh adaptation for high-order finite element methods with curved element geometry." Engineering with Computers 30, no. 2 (2013): 271–86. http://dx.doi.org/10.1007/s00366-013-0329-7.

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17

Kohno, Haruhiko. "An efficient, high-order finite element method using the nodal averaging technique for incompressible fluid flows." Computer Physics Communications 195 (March 15, 2019): 68–76. https://doi.org/10.5281/zenodo.2594035.

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A new finite element method is presented for use of quadrilateral nine-node elements in the solution of the incompressible Navier–Stokes equations. In a conventional predictor–corrector scheme, the method applies the nodal averaging technique to discretize the Poisson equation used for the simultaneous relaxation of velocity and pressure. Additionally, efficient approximation procedures are devised to increase the speed of computation without deteriorating solution accuracy. The proposed numerical schemes are evaluated on two-dimensional test problems including a classical lid-driv

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18

Cui, Ming, Yanxin Su, and Dong Liang. "High-Order Finite Volume Methods for Aerosol Dynamic Equations." Advances in Applied Mathematics and Mechanics 8, no. 2 (2016): 213–35. http://dx.doi.org/10.4208/aamm.2013.m362.

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AbstractAerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.

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19

Świrydowicz, Kasia, Noel Chalmers, Ali Karakus, and Tim Warburton. "Acceleration of tensor-product operations for high-order finite element methods." International Journal of High Performance Computing Applications 33, no. 4 (2019): 735–57. http://dx.doi.org/10.1177/1094342018816368.

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20

Jiang, Yingjun, and Jingtang Ma. "High-order finite element methods for time-fractional partial differential equations." Journal of Computational and Applied Mathematics 235, no. 11 (2011): 3285–90. http://dx.doi.org/10.1016/j.cam.2011.01.011.

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21

Wildey, Tim, Sriramkrishnan Muralikrishnan, and Tan Bui-Thanh. "Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods." SIAM Journal on Scientific Computing 41, no. 5 (2019): S172—S195. http://dx.doi.org/10.1137/18m1193505.

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22

Li, Long-yuan, and Peter Bettess. "Adaptive Finite Element Methods: A Review." Applied Mechanics Reviews 50, no. 10 (1997): 581–91. http://dx.doi.org/10.1115/1.3101670.

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The adaptive finite element method (FEM) was developed in the early 1980s. The basic concept of adaptivity developed in the FEM is that, when a physical problem is analyzed using finite elements, there exist some discretization errors caused owing to the use of the finite element model. These errors are calculated in order to assess the accuracy of the solution obtained. If the errors are large, then the finite element model is refined through reducing the size of elements or increasing the order of interpolation functions. The new model is re-analyzed and the errors in the new model are recal

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23

Yi, Tae-Hyeong, and Francis X. Giraldo. "Vertical Discretization for a Nonhydrostatic Atmospheric Model Based on High-Order Spectral Elements." Monthly Weather Review 148, no. 1 (2019): 415–36. http://dx.doi.org/10.1175/mwr-d-18-0283.1.

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Abstract This study addresses the treatment of vertical discretization for a high-order, spectral element model of a nonhydrostatic atmosphere in which the governing equations of the model are separated into horizontal and vertical components by introducing a coordinate transformation, so that one can use different orders and types of approximations in both directions. The vertical terms of the decoupled governing equations are discretized using finite elements based on either Lagrange or basis-spline polynomial functions in the sigma coordinate, while maintaining the high-order spectral eleme

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24

Gao, Yichao, Feng Jin, Xiang Wang, and Jinting Wang. "Finite Element Analysis of Dam-Reservoir Interaction Using High-Order Doubly Asymptotic Open Boundary." Mathematical Problems in Engineering 2011 (2011): 1–23. http://dx.doi.org/10.1155/2011/210624.

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The dam-reservoir system is divided into the near field modeled by the finite element method, and the far field modeled by the excellent high-order doubly asymptotic open boundary (DAOB). Direct and partitioned coupled methods are developed for the analysis of dam-reservoir system. In the direct coupled method, a symmetric monolithic governing equation is formulated by incorporating the DAOB with the finite element equation and solved using the standard time-integration methods. In contrast, the near-field finite element equation and the far-field DAOB condition are separately solved in the pa

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25

Benítez, Marta, and de Castro Alfredo Bermúdez. "A second order characteristics finite element scheme for natural convection problems." Journal of computational and applied mathematics 235, no. 11 (2011): 3270–84. https://doi.org/10.5281/zenodo.10631650.

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© 2011. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/(opens innew tab/window) https://www.elsevier.com/about/policies-and-standards/sharing Link to publisher version: https://doi.org/10.1016/j.cam.2011.01.007

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26

Bradji, Abdallah, and Jürgen Fuhrmann. "Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method." Mathematica Bohemica 139, no. 2 (2014): 125–36. http://dx.doi.org/10.21136/mb.2014.143843.

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27

Keith, Brendan. "A priori error analysis of high-order LL* (FOSLL*) finite element methods." Computers & Mathematics with Applications 103 (December 2021): 12–18. http://dx.doi.org/10.1016/j.camwa.2021.10.015.

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28

Opschoor, Joost A. A., and Christoph Schwab. "Deep ReLU networks and high-order finite element methods II: Chebyšev emulation." Computers & Mathematics with Applications 169 (September 2024): 142–62. http://dx.doi.org/10.1016/j.camwa.2024.06.008.

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29

Diosady, Laslo T., and Scott M. Murman. "Scalable tensor-product preconditioners for high-order finite-element methods: Scalar equations." Journal of Computational Physics 394 (October 2019): 759–76. http://dx.doi.org/10.1016/j.jcp.2019.04.047.

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30

Ainsworth, Mark. "Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods." Journal of Computational Physics 198, no. 1 (2004): 106–30. http://dx.doi.org/10.1016/j.jcp.2004.01.004.

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31

Adjerid, Slimane, Mohammed Aiffa, and Joseph E. Flaherty. "High-Order Finite Element Methods for Singularly Perturbed Elliptic and Parabolic Problems." SIAM Journal on Applied Mathematics 55, no. 2 (1995): 520–43. http://dx.doi.org/10.1137/s0036139993269345.

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32

Giani, Stefano. "High-order/ $$hp$$ -adaptive discontinuous Galerkin finite element methods for acoustic problems." Computing 95, S1 (2012): 215–34. http://dx.doi.org/10.1007/s00607-012-0253-5.

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33

Bonito, Andrea, Claudio Canuto, Ricardo H. Nochetto, and Andreas Veeser. "Adaptive finite element methods." Acta Numerica 33 (July 2024): 163–485. http://dx.doi.org/10.1017/s0962492924000011.

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This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori erro

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34

Iskandarani, M., J. C. Levin, B. J. Choi, and D. B. Haidvogel. "Comparison of advection schemes for high-order h–p finite element and finite volume methods." Ocean Modelling 10, no. 1-2 (2005): 233–52. http://dx.doi.org/10.1016/j.ocemod.2004.09.005.

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35

Turusbekova, U. K., M. M. Muratbekov, and S. A. Altynbek. "RESEARCH OF ALGORITHMS FOR SEARCHING PRIMITIVE ELEMENTS OF A FINITE FIELD OF HIGH ORDER." Herald of the Kazakh-British technical university 21, no. 1 (2024): 85–93. http://dx.doi.org/10.55452/1998-6688-2024-21-1-85-93.

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One of the most important unsolved and notoriously difficult problems in computational finite field theory is the development of a fast algorithm for constructing primitive roots in a finite field. It is known that for many applications, instead of a primitive root, just an element of high multiplicative order is sufficient. Such applications include, but are not limited to, cryptography, coding theory, pseudorandom number generation, and combinatorial schemes. Explicit constructions of high-order elements usually rely on combinatory methods that can provide a provable lower bound on the order

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36

Lu, Hongqiang, Kai Cao, Lechao Bian, and Yizhao Wu. "High-Order Mesh Generation for Discontinuous Galerkin Methods Based on Elastic Deformation." Advances in Applied Mathematics and Mechanics 8, no. 4 (2016): 693–702. http://dx.doi.org/10.4208/aamm.2014.m618.

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AbstractIn this paper, a high-order curved mesh generation method for Discontinuous Galerkin methods is introduced. First, a regular mesh is generated. Second, the solid surface is re-constructed using cubic polynomial. Third, the elastic governing equations are solved using high-order finite element method to provide a fully or partly curved grid. Numerical tests indicate that the intersection between element boundaries can be avoided by carefully defining the elasticity modulus.

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37

Dziuk, Gerhard, and Charles M. Elliott. "Finite element methods for surface PDEs." Acta Numerica 22 (April 2, 2013): 289–396. http://dx.doi.org/10.1017/s0962492913000056.

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In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with il

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38

Jones, Derrick, and Xu Zhang. "A high order immersed finite element method for parabolic interface problems." ITM Web of Conferences 29 (2019): 01007. http://dx.doi.org/10.1051/itmconf/20192901007.

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We present a high order immersed finite element (IFE) method for solving 1D parabolic interface problems. These methods allow the solution mesh to be independent of the interface. Time marching schemes including Backward-Eulerand Crank-Nicolson methods are implemented to fully discretize the system. Numerical examples are provided to test the performance of our numerical schemes.

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39

Nshimiyimana, J. D., F. Plumier, C. Ndagije, J. Gyselinck, and C. Geuzain. "High Order Relaxation Methods for Co-simulation of Finite Element and Circuit Solvers." Advanced Electromagnetics 9, no. 1 (2020): 49–58. http://dx.doi.org/10.7716/aem.v9i1.1245.

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Coupled problems result in very stiff problems whose char- acteristic parameters differ with several orders in magni- tude. For such complex problems, solving them monolithi- cally becomes prohibitive. Since nowadays there are op- timized solvers for particular problems, solving uncoupled problems becomes easy since each can be solved indepen- dently with its dedicated optimized tools. Therefore the co-simulation of the sub-problems solvers is encouraged. The design of the transmission coupling conditions between solvers plays a fundamental role. The current paper ap- plies the waveform relaxa

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40

Zhang, Xiaodi. "High order interface-penalty finite element methods for elasticity interface problems in 3D." Computers & Mathematics with Applications 114 (May 2022): 161–70. http://dx.doi.org/10.1016/j.camwa.2022.03.044.

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41

Lehrenfeld, Christoph. "High order unfitted finite element methods on level set domains using isoparametric mappings." Computer Methods in Applied Mechanics and Engineering 300 (March 2016): 716–33. http://dx.doi.org/10.1016/j.cma.2015.12.005.

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42

Sehlhorst, H. G., R. Jänicke, A. Düster, E. Rank, H. Steeb, and S. Diebels. "Numerical investigations of foam-like materials by nested high-order finite element methods." Computational Mechanics 45, no. 1 (2009): 45–59. http://dx.doi.org/10.1007/s00466-009-0414-3.

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43

Li, Maojun, and Aimin Chen. "High order central discontinuous Galerkin-finite element methods for the Camassa–Holm equation." Applied Mathematics and Computation 227 (January 2014): 237–45. http://dx.doi.org/10.1016/j.amc.2013.11.016.

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44

Deng, Quanling, Victor Ginting, and Bradley McCaskill. "Construction of locally conservative fluxes for high order continuous Galerkin finite element methods." Journal of Computational and Applied Mathematics 359 (October 2019): 166–81. http://dx.doi.org/10.1016/j.cam.2019.03.049.

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45

Gawlik, Evan S., and Adrian J. Lew. "High-order finite element methods for moving boundary problems with prescribed boundary evolution." Computer Methods in Applied Mechanics and Engineering 278 (August 2014): 314–46. http://dx.doi.org/10.1016/j.cma.2014.05.008.

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46

Sherwin, Spencer J., and George Em Karniadakis. "A new triangular and tetrahedral basis for high-order (hp) finite element methods." International Journal for Numerical Methods in Engineering 38, no. 22 (1995): 3775–802. http://dx.doi.org/10.1002/nme.1620382204.

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47

Huang, Weijie, Weijun Ma, Liang Wei, and Zhiping Li. "High‐order dual‐parametric finite element methods for cavitation computation in nonlinear elasticity." Numerical Methods for Partial Differential Equations 36, no. 5 (2020): 1012–27. http://dx.doi.org/10.1002/num.22462.

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48

Attanayake, Champike, So-Hsiang Chou null, and Quanling Deng. "High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions." International Journal of Numerical Analysis and Modeling 20, no. 6 (2023): 870–95. http://dx.doi.org/10.4208/ijnam2023-1038.

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49

夏, 有伟. "High Order Central Discontinuous Galerkin-Finite Element Methods for the abcd Boussinesq System." Advances in Applied Mathematics 12, no. 10 (2023): 4288–99. http://dx.doi.org/10.12677/aam.2023.1210422.

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50

Guo, Yichen, Eric de Sturler, and Tim Warburton. "Stopping Criteria for the Conjugate Gradient Algorithm in High-Order Finite Element Methods." SIAM Journal on Scientific Computing 47, no. 1 (2025): A238—A267. https://doi.org/10.1137/23m157257x.

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