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Journal articles on the topic 'Partial differential equations, finite element method, Oseen equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Ying, Lung-an. "Book Review: Partial differential equations and the finite element method." Mathematics of Computation 76, no. 259 (September 1, 2007): 1693–94. http://dx.doi.org/10.1090/s0025-5718-07-02023-6.

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2

Ellerby, F. B., and C. Johnson. "Numerical Solutions of Partial Differential Equations by the Finite Element Method." Mathematical Gazette 73, no. 463 (March 1989): 59. http://dx.doi.org/10.2307/3618226.

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3

W., L. B., and Claes Johnson. "Numerical Solution of Partial Differential Equations by the Finite Element Method." Mathematics of Computation 52, no. 185 (January 1989): 247. http://dx.doi.org/10.2307/2008668.

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4

Balasundaram, S., and P. K. Bhattacharyya. "A Mixed Finite Element Method for Fourth Order Partial Differential Equations." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 66, no. 10 (1986): 489–99. http://dx.doi.org/10.1002/zamm.19860661019.

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5

Pani, Amiya K. "AnH1-Galerkin Mixed Finite Element Method for Parabolic Partial Differential Equations." SIAM Journal on Numerical Analysis 35, no. 2 (April 1998): 712–27. http://dx.doi.org/10.1137/s0036142995280808.

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6

Wrobel, L. C. "Numerical solution of partial differential equations by the finite element method." Engineering Analysis with Boundary Elements 9, no. 1 (January 1992): 106. http://dx.doi.org/10.1016/0955-7997(92)90133-r.

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7

Mredula, K. P., and D. C. Vakaskar. "Haar Wavelet Implementation to Various Partial Differential Equations." European Journal of Engineering Research and Science 2, no. 3 (March 30, 2017): 44. http://dx.doi.org/10.24018/ejers.2017.2.3.307.

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The article brings together a series of algorithms with the modification in formulation of solution to various partial differential equations. The algorithms are modified with implementation of Haar Wavelet. Test examples are considered for validation with few cases. Salient features of multi resolution is closely compared with different resolutions. The approach combines well known finite difference and finite element method with wavelets. A detailed description of algorithm is attempted for simplification of the approach.

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8

Hou, Lei, Jun Jie Zhao, and Han Ling Li. "Finite Element Convergence Analysis of Two-Scale Non-Newtonian Flow Problems." Advanced Materials Research 718-720 (July 2013): 1723–28. http://dx.doi.org/10.4028/www.scientific.net/amr.718-720.1723.

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The convergence of the first-order hyperbolic partial differential equations in non-Newton fluid is analyzed. This paper uses coupled partial differential equations (Cauchy fluid equations, P-T/T stress equation) on a macroscopic scale to simulate the free surface elements. It generates watershed by excessive tensile elements. The semi-discrete finite element method is used to solve these equations. These coupled nonlinear equations are approximated by linear equations. Its super convergence is proposed.

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9

Grande, Jörg, and Arnold Reusken. "A Higher Order Finite Element Method for Partial Differential Equations on Surfaces." SIAM Journal on Numerical Analysis 54, no. 1 (January 2016): 388–414. http://dx.doi.org/10.1137/14097820x.

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10

Sereno, C., A. Rodrigues, and J. Villadsen. "Solution of partial differential equations systems by the moving finite element method." Computers & Chemical Engineering 16, no. 6 (June 1992): 583–92. http://dx.doi.org/10.1016/0098-1354(92)80069-l.

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11

Yang, Qing, and Yirang Yuan. "An Approximation of Three-Dimensional Semiconductor Devices by Mixed Finite Element Method and Characteristics-Mixed Finite Element Method." Numerical Mathematics: Theory, Methods and Applications 8, no. 3 (August 2015): 356–82. http://dx.doi.org/10.4208/nmtma.2015.my12031.

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AbstractThe mathematical model for semiconductor devices in three space dimensions are numerically discretized. The system consists of three quasi-linear partial differential equations about three physical variables: the electrostatic potential, the electron concentration and the hole concentration. We use standard mixed finite element method to approximate the elliptic electrostatic potential equation. For the two convection-dominated concentration equations, a characteristics-mixed finite element method is presented. The scheme is locally conservative. The optimalL2-norm error estimates are derived by the aid of a post-processing step. Finally, numerical experiments are presented to validate the theoretical analysis.

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12

Baines, M. J., M. E. Hubbard, and P. K. Jimack. "Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations." Communications in Computational Physics 10, no. 3 (September 2011): 509–76. http://dx.doi.org/10.4208/cicp.201010.040511a.

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AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

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13

KC, Gokul, and Ram Prasad Dulal. "Adaptive Finite Element Method for Solving Poisson Partial Differential Equation." Journal of Nepal Mathematical Society 4, no. 1 (May 14, 2021): 1–18. http://dx.doi.org/10.3126/jnms.v4i1.37107.

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Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

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14

Gokhman, Alexander. "NEW METHOD FOR SOLVING PARTIAL AND ORDINARY DIFFERENTIAL EQUATIONS USING FINITE-ELEMENT TECHNIQUE." Numerical Heat Transfer, Part B: Fundamentals 18, no. 1 (July 1990): 1–22. http://dx.doi.org/10.1080/10407799008944939.

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15

Lehrenfeld, Christoph, Maxim A. Olshanskii, and Xianmin Xu. "A Stabilized Trace Finite Element Method for Partial Differential Equations on Evolving Surfaces." SIAM Journal on Numerical Analysis 56, no. 3 (January 2018): 1643–72. http://dx.doi.org/10.1137/17m1148633.

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16

Cheng, Y. M., D. Z. Li, N. Li, Y. Y. Lee, and S. K. Au. "Solution of some Engineering Partial Differential Equations Governed by the Minimal of a Functional by Global Optimization Method." Journal of Mechanics 29, no. 3 (May 1, 2013): 507–16. http://dx.doi.org/10.1017/jmech.2013.26.

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AbstractMany engineering problems are governed by partial differential equations which can be solved by analytical as well as numerical methods, and examples include the plasticity problem of a geotechnical system, seepage problem and elasticity problem. Although the governing differential equations can be solved by either iterative finite difference method or finite element, there are however limitations to these methods in some special cases which will be discussed in the present paper. The solutions of these governing differential equations can all be viewed as the stationary value of a functional. Using an approximate solution as the initial solution, the stationary value of the functional can be obtained easily by modern global optimization method. Through the comparisons between analytical solutions and fine mesh finite element analysis, the use of global optimization method will be demonstrated to be equivalent to the solutions of the governing partial differential equations. The use of global optimization method can be an alternative to the finite difference/ finite element method in solving an engineering problem, and it is particularly attractive when an approximate solution is available or can be estimated easily.

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17

Gunzburger, Max D., Clayton G. Webster, and Guannan Zhang. "Stochastic finite element methods for partial differential equations with random input data." Acta Numerica 23 (May 2014): 521–650. http://dx.doi.org/10.1017/s0962492914000075.

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The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

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18

Asle-Zaeem, M., and S. D. Mesarovic. "Investigation of Phase Transformation in Thin Film Using Finite Element Method." Solid State Phenomena 150 (January 2009): 29–41. http://dx.doi.org/10.4028/www.scientific.net/ssp.150.29.

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Cahn-Hilliard type of phase field model coupled with elasticity is used to derive governing equations for the stress-mediated diffusion and phase transformation in thin films. To solve the resulting equations, a finite element (FE) model is presented. The partial differential equations governing diffusion and mechanical equilibrium are of different orders; Mixed-order finite elements, with C0 interpolation functions for displacement, and C1 interpolation functions for concentration are implemented. To validate this new numerical solver for such coupled problems, we test our implementation on thin film diffusion couples.

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19

Papanikos, George, and Maria Ch Gousidou-Koutita. "A Computational Study with Finite Element Method and Finite Difference Method for 2D Elliptic Partial Differential Equations." Applied Mathematics 06, no. 12 (2015): 2104–24. http://dx.doi.org/10.4236/am.2015.612185.

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20

Sulaeman, Erwin, S. M. Afzal Hoq, Abdurahim Okhunov, and Marwan Badran. "Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations." International Journal of Engineering & Technology 7, no. 4.36 (December 9, 2018): 379. http://dx.doi.org/10.14419/ijet.v7i4.36.28146.

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Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid. The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix. To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements. For this purpose, analytical solution is derived in detailed for a particular heat conduction problem. The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and L¥.

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21

Abhyankar, N. S., E. K. Hall, and S. V. Hanagud. "Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations." Journal of Applied Mechanics 60, no. 1 (March 1, 1993): 167–74. http://dx.doi.org/10.1115/1.2900741.

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The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

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22

Shakeri, Fatemeh, and Mehdi Dehghan. "A high order finite volume element method for solving elliptic partial integro-differential equations." Applied Numerical Mathematics 65 (March 2013): 105–18. http://dx.doi.org/10.1016/j.apnum.2012.10.002.

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23

Adjerid, Slimane, and Joseph E. Flaherty. "A moving-mesh finite element method with local refinement for parabolic partial differential equations." Computer Methods in Applied Mechanics and Engineering 55, no. 1-2 (April 1986): 3–26. http://dx.doi.org/10.1016/0045-7825(86)90083-6.

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24

Oomens, C. W. J., M. Maenhout, C. H. van Oijen, M. R. Drost, and F. P. Baaijens. "Finite element modelling of contracting skeletal muscle." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 358, no. 1437 (August 22, 2003): 1453–60. http://dx.doi.org/10.1098/rstb.2003.1345.

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To describe the mechanical behaviour of biological tissues and transport processes in biological tissues, conservation laws such as conservation of mass, momentum and energy play a central role. Mathematically these are cast into the form of partial differential equations. Because of nonlinear material behaviour, inhomogeneous properties and usually a complex geometry, it is impossible to find closed-form analytical solutions for these sets of equations. The objective of the finite element method is to find approximate solutions for these problems. The concepts of the finite element method are explained on a finite element continuum model of skeletal muscle. In this case, the momentum equations have to be solved with an extra constraint, because the material behaves as nearly incompressible. The material behaviour consists of a highly nonlinear passive part and an active part. The latter is described with a two-state Huxley model. This means that an extra nonlinear partial differential equation has to be solved. The problems and solutions involved with this procedure are explained. The model is used to describe the mechanical behaviour of a tibialis anterior of a rat. The results have been compared with experimentally determined strains at the surface of the muscle. Qualitatively there is good agreement between measured and calculated strains, but the measured strains were higher.

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25

Cao, Zhoujian, Pei Fu, Li-Wei Ji, and Yinhua Xia. "Application of local discontinuous Galerkin method to Einstein equations." International Journal of Modern Physics D 28, no. 01 (January 2019): 1950014. http://dx.doi.org/10.1142/s0218271819500147.

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Finite difference and pseudo-spectral methods have been widely used in the numerical relativity to solve the Einstein equations. As the third major category method to solve partial differential equations, finite element method is less frequently used in numerical relativity. In this paper, we design a finite element algorithm to solve the evolution part of the Einstein equations. This paper is the second one of a systematic investigation of applying adaptive finite element method to the Einstein equations, especially aiming for binary compact objects simulations. The first paper of this series has been contributed to the constrained part of the Einstein equations for initial data. Since applying finite element method to the Einstein equations is a big project, we mainly propose the theoretical framework of a finite element algorithm together with local discontinuous Galerkin method for the Einstein equations in the current work. In addition, we have tested our algorithm based on the spherical symmetric spacetime evolution. In order to simplify our numerical tests, we have reduced the problem to a one-dimensional space problem by taking the advantage of the spherical symmetry. Our reduced equation system is a new formalism for spherical symmetric spacetime simulation. Based on our test results, we find that our finite element method can capture the shock formation which is introduced by numerical error. In contrast, such shock is smoothed out by numerical dissipation within the finite difference method. We suspect this is partly the reason that the accuracy of finite element method is higher than the finite difference method. At the same time, different kinds of formulation parameters setting are also discussed.

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26

Lai, Junjiang, and Hongyu Liu. "On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations." Mathematics 9, no. 16 (August 23, 2021): 2014. http://dx.doi.org/10.3390/math9162014.

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In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

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27

Hong, Chengyu, Xuben Wang, Gaishan Zhao, Zhao Xue, Fei Deng, Qinping Gu, Zhixiang Song, et al. "Discontinuous finite element method for efficient three-dimensional elastic wave simulation." Journal of Geophysics and Engineering 18, no. 1 (February 2021): 98–112. http://dx.doi.org/10.1093/jge/gxaa070.

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Abstract The existing discontinuous Galerkin (DG) finite element method (FEM) for the numerical simulation of elastic wave propagation is primarily implemented in two dimensions. Here, a discontinuous FEM (DFEM) for efficient three-dimensional (3D) elastic wave simulation is presented. First, the velocity–stress equations of 3D elastic waves in isotropic media are transformed into first-order coefficient-changed partial differential equations. A DG discretisation method for wave field values on a unit boundary is then defined using the local Lax–Friedrichs flux format. The equations are first transformed into equivalent integral equations, and subsequently into a spatial semi-discrete ordinary differential equation system using a hierarchical orthogonal basis function. The DFEM is extended to an arbitrary high-order accuracy in the time domain using the exponential integrator technique and the explicit optimal strong-stability-preserving Runge–Kutta method. Finally, an efficient method for selecting the calculation area of the geometry of the current shot record is realised. For the computation, a multi-node parallelism with improved resource utilisation and parallelisation efficiency is implemented. The numerical results show that the proposed method can improve both the accuracy of the simulation and the efficiency of the calculation compared with existing methods.

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28

Ko, Seungchan, Petra Pustějovská, and Endre Süli. "Finite element approximation of an incompressible chemically reacting non-Newtonian fluid." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (March 2018): 509–41. http://dx.doi.org/10.1051/m2an/2017043.

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We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiĭ operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

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29

Daneshmehr, A. R., S. Akbari, and A. Nateghi. "Dynamic Thermal Analysis of Laminated Cylinder with a Piezoelectric Layer Based on Three-Dimensional Elasticity Solution." Applied Mechanics and Materials 186 (June 2012): 16–25. http://dx.doi.org/10.4028/www.scientific.net/amm.186.16.

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Three-dimensional elasticity solution is presented for finite length, simply supported, laminated cylinder with a piezoelectric layer under dynamic thermal load and pressure. The piezoelectric layer can be used as an actuator or a sensor. The ordinary differential equations are obtained from partial differential equations of motion by means of trigonometric function expansion in longitudinal direction. Galerkin finite element method is used to solve the resulting ordinary differential equations. Finally numerical results are discussed for different situations.

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30

Gusev, A. A. "High-Accuracy Finite Element Method for Solving Boundary-Value Problems for Elliptic Partial Differential Equations." RUDN JOURNAL OF MATHEMATICS, INFORMATION SCIENCES AND PHYSICS 25, no. 3 (2017): 217–33. http://dx.doi.org/10.22363/2312-9735-2017-25-3-217-233.

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31

Dubos, T. "A conservative Fourier-finite-element method for solving partial differential equations on the whole sphere." Quarterly Journal of the Royal Meteorological Society 135, no. 644 (October 2009): 1877–89. http://dx.doi.org/10.1002/qj.487.

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32

Feng, Xiaobing, Junshan Lin, and Cody Lorton. "A MULTIMODES MONTE CARLO FINITE ELEMENT METHOD FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS." International Journal for Uncertainty Quantification 6, no. 5 (2016): 429–43. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.2016016805.

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33

Madureira, Alexandre L. "A multiscale finite element method for partial differential equations posed in domains with rough boundaries." Mathematics of Computation 78, no. 265 (January 1, 2009): 25. http://dx.doi.org/10.1090/s0025-5718-08-02159-5.

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34

Franklin, Scott R., Padmanabhan Seshaiyer, and Philip W. Smith. "A Three-Field Finite Element Method for Elliptic Partial Differential Equations Driven by Stochastic Loads." Stochastic Analysis and Applications 23, no. 4 (July 2005): 757–83. http://dx.doi.org/10.1081/sap-200064476.

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35

Zhao, Jingjun, Jingyu Xiao, and Yang Xu. "Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/857205.

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A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.

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36

Wang, Youming, Xuefeng Chen, and Zhengjia He. "A second-generation wavelet-based finite element method for the solution of partial differential equations." Applied Mathematics Letters 25, no. 11 (November 2012): 1608–13. http://dx.doi.org/10.1016/j.aml.2012.01.021.

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37

Nouy, A., A. Clément, F. Schoefs, and N. Moës. "An extended stochastic finite element method for solving stochastic partial differential equations on random domains." Computer Methods in Applied Mechanics and Engineering 197, no. 51-52 (October 2008): 4663–82. http://dx.doi.org/10.1016/j.cma.2008.06.010.

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38

Yas, M. H., Morteza Shakeri, and M. R. Saviz. "Elasticity Solution of Laminated Cylindrical Shell with Piezoelectric Layer under Local Ring Load." Key Engineering Materials 334-335 (March 2007): 917–20. http://dx.doi.org/10.4028/www.scientific.net/kem.334-335.917.

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Elasticity solution is presented for simply-supported, orthotropic, piezoelectric cylindrical shell with finite length under local ring load in the middle of shell and electrostatic excitation. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations(o.d.e.) with variable coefficients by means of trigonometric function expansion in longitudinal direction for displacement and external forces. The resulting ordinary differential equations are solved by Galerkin finite element method. Numerical examples are presented for [0/90/P] lamination with sensor and actuator for different thicknesses.

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39

Lugo Jiménez, Abdul Abner, Guelvis Enrique Mata Díaz, and Bladismir Ruiz. "A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems." Selecciones Matemáticas 8, no. 1 (June 30, 2021): 1–11. http://dx.doi.org/10.17268/sel.mat.2021.01.01.

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Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

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40

Boulbrachene, Messaoud. "Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs." Sultan Qaboos University Journal for Science [SQUJS] 24, no. 2 (January 19, 2020): 109. http://dx.doi.org/10.24200/squjs.vol24iss2pp109-121.

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In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L-error estimate for linear problems.

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41

Brandner, Philip, and Arnold Reusken. "Finite element error analysis of surface Stokes equations in stream function formulation." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (October 12, 2020): 2069–97. http://dx.doi.org/10.1051/m2an/2020044.

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We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ3 without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk–Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

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42

Song, Mi Tao, and Deng Qing Cao. "A Hybrid Finite Element-Analytical Approach for Dynamic Analysis of a Micro-Resonator Driven by Electrostatic Combs." Applied Mechanics and Materials 226-228 (November 2012): 708–13. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.708.

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Combining the finite element method and the analytical method, a hybrid finite element-analytical approach is established to calculate the nonlinear dynamic responses of a micro-resonator driven by electrostatic combs accurately for the purpose of programmed dynamical simulations and great shortening of workloads. The spatially discretized equations obtained by using the analytical undamped global mode functions to the nonlinear integro-partial differential equations and the ordinary differential equations of the micro-resonator, in which the coefficients are estimated by the discrete global mode shapes from the finite element method, are used to calculate the nonlinear dynamic responses of the micro-resonator. The results are compared with those merely based on the analytical mode functions of the micro-resonator, which shows that they can reach high accuracy when the elements in the micro-resonator are sufficiently small.

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Kalis, H., and I. Kangro. "SIMPLE METHODS OF ENGINEERING CALCULATION FOR SOLVING STATIONARY 2 –D HEAT TRANSFER PROBLEMS IN MULTILAYER MEDIA." Environment. Technology. Resources. Proceedings of the International Scientific and Practical Conference 1 (June 26, 2006): 359. http://dx.doi.org/10.17770/etr2003vol1.1991.

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There are well-known different numerical methods for solving the boundary value problems for partial differential equations. Some of them are: finite difference method (FDM), finite element method (FEM), boundary element methods (BEM), and others. In the given work two methods FDM and BEM for the mathematical model of stationary distribution of heat in the multilayer media are considered. These methods were used for the reduction of the two-dimensional heat transfer problem described by a partial differential equation to a boundary – value problem for a system of ordinary differential equations. (ODEs). Such a procedure allows obtaining simple engineering algorithms for solving heat transfer equation in mulyilayer domain. In the case of three layers the system of ODEs is possible for solving analytically.

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Boateng, Francis Ohene, Joseph Ackora-Prah, Benedict Barnes, and John Amoah-Mensah. "A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem." European Journal of Pure and Applied Mathematics 14, no. 3 (August 5, 2021): 706–22. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3893.

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In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear ellipticÂ partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

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Li, Yuguo, and Kerry Key. "2D marine controlled-source electromagnetic modeling: Part 1 — An adaptive finite-element algorithm." GEOPHYSICS 72, no. 2 (March 2007): WA51—WA62. http://dx.doi.org/10.1190/1.2432262.

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In Part 1 of this work, we develop an adaptive finite-element algorithm for forward modeling of the frequency-domain, marine controlled-source electromagnetic (CSEM) response of a 2D conductivity structure that is excited by a horizontal electric dipole source. After transforming the governing equations for the secondary electromagnetic fields into the wavenumber domain, the coupled system of two partial differential equations for the strike-parallel electric and magnetic fields is approximated using the finite-element method. The model domain is discretized using an unstructured triangular element grid that readily accommodates arbitrarily complex structures. A numerical solution of the system of linear equations is obtained using the quasi-minimal residual (QMR) method, which requiresmuch less storagethan full matrix inversion methods. We implement an automatedadaptive grid refinement algorithm in which the finite-element solution is computed iteratively on successively refined grids. Grid refinement is guided by an a posteriori error estimator based on a recently developed gradient recovery operator. The error estimator uses the solution to a dual problem in order to bias refinement toward elements that affect the solution at the electromagnetic (EM) receiver locations and enables the computation of asymptotically exact solutions to the 2.5D partial differential equations. We validate the finite element formulation against the canonical 1D reservoir model and study the performance of the adaptive refinement algorithm. An example model study of a complex offshore structure of interest for petroleum exploration illustrates the utility of the adaptive finite-element method for CSEM modeling.

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Liu, Zhiyong, and Qiuyan Xu. "A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations." Mathematics 7, no. 10 (October 13, 2019): 964. http://dx.doi.org/10.3390/math7100964.

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In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

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Gao, X. W., Huayu Liu, Miao Cui, Kai Yang, and Haifeng Peng. "Free element method and its application in CFD." Engineering Computations 36, no. 8 (October 7, 2019): 2747–65. http://dx.doi.org/10.1108/ec-10-2018-0471.

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Purpose The purpose of this paper is to propose a new strong-form numerical method, called the free element method, for solving general boundary value problems governed by partial differential equations. The main idea of the method is to use a locally formed element for each point to set up the system of equations. The proposed method is used to solve the fluid mechanics problems. Design/methodology/approach The proposed free element method adopts the isoparametric elements as used in the finite element method (FEM) to represent the variation of coordinates and physical variables and collocates equations node-by-node based on the newly derived element differential formulations by the authors. The distinct feature of the method is that only one independently formed individual element is used at each point. The final system of equations is directly formed by collocating the governing equations at internal points and the boundary conditions at boundary points. The method can effectively capture phenomena of sharply jumped variables and discontinuities (e.g. the shock waves). Findings a) A new numerical method called the FEM is proposed; b) the proposed method is used to solve the compressible fluid mechanics problems for the first time, in which the shock wave can be naturally captured; and c) the method can directly set up the system of equations from the governing equations. Originality/value This paper presents a completely new numerical method for solving compressible fluid mechanics problems, which has not been submitted anywhere else for publication.

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Burman, Erik, Peter Hansbo, Mats G. Larson, and André Massing. "Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (November 2018): 2247–82. http://dx.doi.org/10.1051/m2an/2018038.

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We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in ℝd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in ℝ3.

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Rong, Yao, Yang Sun, LiQing Zhu, and Xiao Xiao. "Analysis of the Three-Dimensional Dynamic Problems by Using a New Numerical Method." Advances in Civil Engineering 2021 (May 4, 2021): 1–12. http://dx.doi.org/10.1155/2021/5555575.

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The problems of the consolidation of saturated soil under dynamic loading are very complex. At present, numerical methods are widely used in the research. However, some traditional methods, such as the finite element method, involve more degrees of freedom, resulting in low computational efficiency. In this paper, the scaled boundary element method (SBFEM) is used to analyze the displacement and pore pressure response of saturated soil due to consolidation under dynamic load. The partial differential equations of linear problems are transformed into ordinary differential equations and solved along the radial direction. The coefficients in the equations are determined by approximate finite elements on the circumference. As a semianalytical method, the application of scaled boundary element method in soil-structure interaction is extended. Dealing with complex structures and structural nonlinearity, it can simulate two-phase saturated soil-structure dynamic interaction in infinite and finite domain, which has an important engineering practical value. Through the research, some conclusions are obtained. The dimension of the analytical problem can be reduced by one dimension if only the boundary surface is discretized. The SBFEM can automatically satisfy the radiation conditions at infinite distances. The 3D scaled boundary finite element equation for dynamic consolidation of saturated soils is not only accurate in finite element sense but also convenient in mathematical processing.

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Cano, María José, Eliseo Chacon-Vera, and Francisco Esquembre. "Simulation of partial differential equations models in Java." Engineering Computations 34, no. 3 (May 2, 2017): 800–813. http://dx.doi.org/10.1108/ec-05-2015-0111.

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Purpose Computer simulations improve the knowledge of physical models and are widely used in teaching and research. Key aspects are to understand their solutions and to make interactive changes to the models, observing their effects in real-time. The drawback of creating interactive simulations of physical models is the high level of programming expertise required. The purpose of this study is to facilitate this task. Design/methodology/approach Java is the perfect language for this task; it yields high-quality graphics and is widely spread in the scientific community. Because many important physical models are described by means of partial differential equations (PDEs), the combination of Java with FreeFem++, a C++ PDE solver based on the finite element method, is considered. Findings In this study, a Java library is introduced to numerically solve PDE equations via a run-time connection with FreeFem++. The solution is encapsulated into Java objects that are ready to be used in different programming tasks. The library also includes new Java visualization elements for solutions and meshes in the context of the Open Source Physics project library. Together, the connection features and the visualization elements facilitate the creation of Java simulations by programming researchers. For those with less programming capabilities, this work has been included into Easy Java Simulations, a tool to further ease the creation of interactive simulations. Originality/value The present study approach allows simulating models given PDEs. The equations are solved either in local or in remote mode (e.g. by a network accessible to a high-performance computer) and visualized locally, providing a high degree of interactivity to the end user.

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