158:) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger
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are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains. In other words, spectral methods take on a
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that encompasses a few standard or recent methods. It is based on the separate approximation of a function and of its gradient. Core properties allow the convergence of the method for a series of linear and nonlinear problems, and therefore all the methods that enter the GDM framework (conforming and
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are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by
Lagrange multipliers, judiciously chosen to preserve the accuracy
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The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which
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Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the
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219:, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a
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may be recast as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and
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do not require a mesh connecting the data points of the simulation domain. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort.
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with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for
231:. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are
405:, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on all neighboring subdomains by the same unknown. In dual methods, such as
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E. N. Sarmin, L. A. Chudov (1963), On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method,
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296:. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is
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112:(DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as
559:, and are well suited to problems in complicated geometries. Spectral methods are generally the most accurate, provided that the solutions are sufficiently smooth.
17:
104:(PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of
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or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like the
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The finite difference method is often regarded as the simplest method to learn and use. The finite element and finite volume methods are widely used in
625:. Series in computational and physical processes in mechanics and thermal sciences (3rd. ed.). Boca Raton: CRC Press, Taylor & Francis Group.
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In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
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of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by
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In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the
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a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s.
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exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a
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235:. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many
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by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A
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results. In the finite element community, a method where the degree of the elements is very high or increases as the grid parameter
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1001:: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.
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Numerical methods for solving partial differential equations : a comprehensive introduction for scientists and engineers
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272:. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a
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nonconforming finite element, mixed finite element, mimetic finite difference...) inherit these convergence properties.
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Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the
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In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as
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Ordinary and
Partial Differential Equation Routines in C, C++, Fortran, Java, Maple and Matlab
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The main idea of multigrid is to accelerate the convergence of a basic iterative method by
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A Compendium of
Partial Differential Equation Models: Method of Lines Analysis with Matlab
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List of numerical analysis topics#Numerical methods for partial differential equations
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finite element methods, and serve a basis for distributed, parallel computations.
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The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
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Anderson, Dale A.; Pletcher, Richard H.; Tannehill, John C. (2013).
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approach to multigrid. MG methods can be used as solvers as well as
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Multigrid finite element methods for electromagnetic field modeling
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European
Community on Computational Methods in Applied Sciences
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Non-overlapping domain decomposition methods are also called
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International
Council for Industrial and Applied Mathematics
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Chen, Shang-Ying; Wei, Jian-Yu; Hsu, Kuo-Chin (2023-10-01).
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710:, Cambridge: Cambridge University Press, pp. 309–336,
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correction from time to time, accomplished by solving a
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USSR Computational
Mathematics and Mathematical Physics
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Hamdi, S., W. E. Schiesser and G. W. Griffiths (2007),
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476:. They are an example of a class of techniques called
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in the form of algebraic equations . Similar to the
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Numerical methods for ordinary differential equations
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Numerical PDE Techniques for
Scientists and Engineers
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U. Trottenberg; C. W. Oosterlee; A. SchĂĽller (2001).
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Numerical treatment of partial differential equations
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Numerical methods for ordinary differential equations
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method is hybrid between a dual and a primal method.
360:. Domain decomposition methods are typically used as
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Société de Mathématiques
Appliquées et Industrielles
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Japan
Society for Industrial and Applied Mathematics
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Numerical methods for partial differential equations
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Numerical methods for partial differential equations
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Numerical Methods for Partial Differential Equations
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Numerical methods for partial differential equations
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Numerical methods for partial differential equations
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Numerical Methods for Partial Differential Equations
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1175:, open access Lectures and Codes for Numerical PDEs
702:Rubinstein, Jacob; Pinchover, Yehuda, eds. (2005),
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822:. J. M. Blackledge, P. Yardley. London: Springer.
708:An Introduction to Partial Differential Equations
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1049:Practical Fourier analysis for multigrid methods
857:. Hans-Görg Roos, M. Stynes. Berlin: Springer.
623:Computational fluid mechanics and heat transfer
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100:(MOL, NMOL, NUMOL) is a technique for solving
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687:: CS1 maint: location missing publisher (
1046:Roman Wienands; Wolfgang Joppich (2005).
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308:decreases to zero is sometimes called a
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18:Numerical partial differential equations
1100:Yu Zhu; Andreas C. Cangellaris (2006).
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44:that studies the numerical solution of
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51:In principle, specialized methods for
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957:Lee, H. J.; Schiesser, W. E. (2004).
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142:for finding approximate solutions to
1452:Moving particle semi-implicit method
1363:Weighted essentially non-oscillatory
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178:gradient discretization method (GDM)
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292:while finite element methods use a
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1301:Finite-difference frequency-domain
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1926:Stochastic differential equations
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268:, often involving the use of the
2500:Numerical differential equations
2242:Supersymmetric quantum mechanics
1133:Analysis of the multigrid method
422:iterative substructuring methods
392:abstract additive Schwarz method
207:for representing and evaluating
110:differential algebraic equations
2124:Stochastic variational calculus
1916:Ordinary differential equations
1654:Method of fundamental solutions
1440:Smoothed-particle hydrodynamics
1131:Shah, Tasneem Mohammad (1989).
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507:. This principle is similar to
106:ordinary differential equations
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1921:Partial differential equations
1794:Arbitrary-precision arithmetic
1295:Alternating direction-implicit
936:. Cambridge University Press.
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399:Balancing domain decomposition
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172:Gradient discretization method
166:Gradient discretization method
102:partial differential equations
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30:For the academic journal, see
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1809:Interactive geometry software
1307:Finite-difference time-domain
1135:(Thesis). Oxford University.
890:The Numerical Method of Lines
851:Grossmann, Christian (2007).
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529:separability of the equations
264:to numerically solve certain
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716:10.1017/cbo9780511801228.012
586:LeVeque, Randall J. (1992).
557:computational fluid dynamics
527:. They do not depend on the
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334:Domain decomposition methods
237:computational fluid dynamics
27:Branch of numerical analysis
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1861:Computational number theory
1824:Numerical-analysis software
1357:Essentially non-oscillatory
1340:Monotonic upstream-centered
592:. Basel: Birkhäuser Basel.
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515:in two or more dimensions.
340:Domain decomposition method
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1531:Balancing by constraints
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1644:Computer-assisted proof
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310:spectral element method
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2028:Multivariable calculus
1911:Differential equations
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1744:Computational geometry
1632:Petrov–Galerkin method
1393:Discontinuous Galerkin
816:Evans, Gwynne (2000).
797:encyclopediaofmath.org
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266:differential equations
223:term are converted to
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2292:Representation theory
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1933:Differential geometry
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1106:. Wiley. p. 132
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124:Finite element method
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2259:Algebraic structures
2208:in quantum mechanics
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1938:Differential forms
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1079:. Academic Press.
894:. Academic Press.
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2134:Mathematical
2112: /
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1960:Chaos theory
1943:Gauge theory
1871:Graph theory
1766:Cryptography
1446:Peridynamics
1264:Lax–Wendroff
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464:for solving
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366:Krylov space
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227:, using the
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2479:topics list
2413:Mathematics
2329:Game theory
2230:Topological
2196:Topological
2191:Statistical
2154:Hamiltonian
1580:Collocation
553:engineering
533:Lamé system
116:resources.
114:open source
108:(ODEs) and
2494:Categories
2385:Psychology
2349:Statistics
2149:Lagrangian
1776:Statistics
1712:Algorithms
1269:MacCormack
1251:Hyperbolic
1159:course at
802:2021-11-15
778:2021-11-15
754:2021-11-15
731:2021-11-15
675:1015215158
641:References
613:2021-11-15
547:Comparison
537:elasticity
462:algorithms
239:packages.
221:divergence
150:. It uses
53:hyperbolic
2390:Sociology
2380:Chemistry
2176:Effective
2171:Conformal
2166:Classical
2038:Geometric
2011:Geometric
1585:Level-set
1575:Multigrid
1525:Balancing
1227:Parabolic
1076:Multigrid
1033:1435-5663
873:191468303
683:cite book
470:hierarchy
278:sinusoids
57:parabolic
2465:Category
2114:analysis
2033:Exterior
2006:Exterior
1886:Analysis
1848:Discrete
1722:analysis
1559:Spectral
1498:additive
1421:Smoothed
1387:Extended
838:41572731
563:See also
468:using a
386:and the
347:solve a
48:(PDEs).
2475:outline
2406:Related
2375:Biology
2225:Bosonic
2186:Quantum
2136:physics
2102: (
1834:Solvers
1543:FETI-DP
1423:(S-FEM)
1342:(MUSCL)
1330:Godunov
1137:Bibcode
555:and in
539:or the
415:FETI-DP
63:exist.
2048:Vector
2043:Tensor
2021:Vector
2016:Tensor
1717:design
1552:Others
1539:(FETI)
1533:(BDDC)
1405:Mortar
1389:(XFEM)
1382:hp-FEM
1365:(WENO)
1348:(AUSM)
1309:(FDTD)
1303:(FDFD)
1288:Others
1274:Upwind
1237:(FTCS)
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501:global
413:. The
298:smooth
160:domain
2181:Gauge
1566:(DVR)
1527:(BDD)
1466:(PIC)
1460:(MPM)
1454:(MPS)
1442:(SPH)
1412:(GDM)
1401:(SEM)
1359:(ENO)
1297:(ADI)
377:GMRES
203:is a
180:is a
154:(the
138:is a
1448:(PD)
1395:(DG)
1112:ISBN
1081:ISBN
1054:ISBN
1029:ISSN
963:ISBN
938:ISBN
896:ISBN
869:OCLC
859:ISBN
834:OCLC
824:ISBN
720:ISBN
689:link
671:OCLC
661:ISBN
627:ISBN
602:ISBN
407:FETI
403:BDDC
401:and
364:for
260:and
199:The
176:The
146:for
134:The
96:The
1167:IMS
1021:doi
712:doi
594:doi
535:of
472:of
456:in
375:or
215:or
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