113:, FEM can be used for detecting large flaws in a structure, but as the size of the flaw is reduced there is a need to use a high-frequency wave. In order to simulate the propagation of a high-frequency wave, the FEM mesh required is very fine resulting in increased computational time. On the other hand, SEM provides good accuracy with fewer degrees of freedom. Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also useful for adopting a central difference method (CDM). The disadvantages of SEM include difficulty in modeling complex geometry, compared to the flexibility of FEM.
933:. Since not all interior basis functions need to be present, the p-version finite element method can create a space that contains all polynomials up to a given degree with fewer degrees of freedom. However, some speedup techniques possible in spectral methods due to their tensor-product character are no longer available. The name
820:
Development of the most popular LGL form of the method is normally attributed to Maday and Patera. However, it was developed more than a decade earlier. First, there is the Hybrid-Collocation-Galerkin method (HCGM), which applies collocation at the interior
Lobatto points and uses a Galerkin-like
116:
Although the method can be applied with a modal piecewise orthogonal polynomial basis, it is most often implemented with a nodal tensor product
Lagrange basis. The method gains its efficiency by placing the nodal points at the Legendre-Gauss-Lobatto (LGL) points and performing the Galerkin method
108:
over non-uniformly spaced nodes. In SEM computational error decreases exponentially as the order of approximating polynomial increases, therefore a fast convergence of solution to the exact solution is realized with fewer degrees of freedom of the structure in comparison with FEM. In
626:
39:
as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see
Development History)
362:
914:, differential quadrature method, and G-NI are different names for the same method. These methods employ global rather than piecewise polynomial basis functions. The extension to a piecewise FEM or SEM basis is almost trivial.
821:
integral procedure at element interfaces. The
Lobatto-Galerkin method described by Young is identical to SEM, while the HCGM is equivalent to these methods. This earlier work is ignored in the spectral literature.
764:
1079:
Komatitsch, D. and
Villote, J.-P.: “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geologic Structures,” Bull. Seismological Soc. America, 88, 2, 368-392 (1998)
1117:
Maday, Y. and Patera, A. T., “Spectral
Element Methods for the Incompressible Navier-Stokes Equations” In State-of-the-Art Surveys on Computational Mechanics, A.K. Noor, editor, ASME, New York (1989).
100:. The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. Such polynomials are usually orthogonal
1126:
Diaz, J., “A Collocation-Galerkin Method for the Two-point
Boundary Value Problem Using Continuous Piecewise Polynomial Spaces,” SIAM J. Num. Anal., 14 (5) 844-858 (1977) ISSN 0036-1429
233:
1210:
870:
830:
G-NI or SEM-NI are the most used spectral methods. The
Galerkin formulation of spectral methods or spectral element methods, for G-NI or SEM-NI respectively, is modified and
98:
1091:
Wheeler, M.F.: “A C0-Collocation-Finite
Element Method for Two-Point Boundary Value and One Space Dimension Parabolic Problems,” SIAM J. Numer. Anal., 14, 1, 71-90 (1977)
484:
121:
using the same nodes. With this combination, simplifications result such that mass lumping occurs at all nodes and a collocation procedure results at interior points.
787:
812:
The Hybrid-Collocation-Galerkin possesses some superconvergence properties. The LGL form of SEM is equivalent, so it achieves the same superconvergence properties.
412:
185:
890:
807:
668:
648:
452:
432:
385:
158:
490:
1016:
Muradova, Aliki D. (2008). "The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions".
241:
1675:
1301:
1135:
Young, L.C., “A Finite-Element Method for
Reservoir Simulation,” Soc. Petr. Engrs. J. 21(1) 115-128, (Feb. 1981), paper SPE 7413 presented Oct. 1978,
1251:
1680:
1227:
1201:
454:
is no larger than the degree of the piecewise polynomial basis. Similar results can be obtained to bound the error in stronger topologies. If
1685:
1525:
1295:
1595:
1452:
1307:
680:
1172:
1156:
1068:
1520:
1503:
1654:
1440:
899:
SEM is a Galerkin based FEM (finite element method) with Lagrange basis (shape) functions and reduced numerical integration by
1690:
1421:
1410:
1387:
831:
17:
1063:
Karniadakis, G. and Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford Univ. Press, (2013),
1393:
56:
series, a chief advantage being that the resulting method is of a very high order. This approach relies on the fact that
1510:
1475:
110:
1515:
1105:
Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019),
124:
The most popular applications of the method are in computational fluid dynamics and modeling seismic wave propagation.
1194:
1632:
1617:
1493:
1259:
1241:
1279:
1602:
1488:
1218:
57:
53:
1264:
1644:
1622:
1607:
1590:
1498:
1483:
981:
Patera, A. T. (1984). "A spectral element method for fluid dynamics - Laminar flow in a channel expansion".
190:
1564:
1335:
1187:
1167:
P. Ĺ olĂn, K. Segeth, I. DoleĹľel: Higher-order finite element methods, Chapman & Hall/CRC Press, 2003.
911:
907:
929:
spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for
1612:
1458:
1374:
893:
29:
1179:
840:
1649:
1322:
990:
101:
67:
1416:
1330:
930:
922:
900:
118:
105:
1639:
1580:
1043:
937:
means that accuracy is increased by increasing the order of the approximating polynomials (thus,
457:
1269:
1168:
1152:
1064:
671:
137:
61:
772:
1585:
1575:
1464:
1432:
1033:
1025:
998:
390:
163:
1627:
1570:
1559:
133:
49:
621:{\displaystyle \|u-u_{N}\|_{H^{k}(\Omega )}\leq C_{s,k}N^{k-1-s}\|u\|_{H^{s+1}(\Omega )}}
994:
1405:
1352:
1148:
918:
875:
792:
653:
633:
437:
417:
370:
143:
1669:
1274:
1002:
835:
357:{\displaystyle \|u-u_{N}\|_{H^{1}(\Omega )}\leqq C_{s}N^{-s}\|u\|_{H^{s+1}(\Omega )}}
1047:
1446:
1363:
1340:
1357:
1235:
21:
1029:
36:
1106:
33:
650:, we can also increase the degree of the basis functions. In this case, if
1151:, Finite element analysis, John Wiley & Sons, Inc., New York, 1991.
1543:
1038:
1382:
953:
926:
387:
is related to the discretization of the domain (ie. element length),
1136:
759:{\displaystyle \|u-u_{N}\|_{H^{1}(\Omega )}\leqq C\exp(-\gamma N)}
1537:
1531:
1346:
1183:
921:
space spanned by nodal basis functions associated with
834:
is used instead of integrals in the definition of the
878:
843:
795:
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683:
656:
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460:
440:
420:
393:
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244:
193:
166:
146:
70:
1211:
Numerical methods for partial differential equations
964:
refinements to obtain exponential convergence rates.
1552:
1474:
1431:
1373:
1321:
1288:
1250:
1226:
1217:
884:
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801:
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92:
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8:
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323:
316:
265:
245:
1223:
1202:
1188:
1180:
1037:
892:. Their convergence is a consequence of
877:
842:
794:
774:
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682:
655:
635:
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562:
546:
522:
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273:
268:
258:
243:
204:
192:
171:
165:
145:
75:
69:
941:) rather than decreasing the mesh size,
140:holds here and it can be shown that, if
973:
160:is the solution of the weak equation,
228:{\displaystyle u\in H^{s+1}(\Omega )}
7:
1453:Moving particle semi-implicit method
1364:Weighted essentially non-oscillatory
1101:
1099:
1097:
1087:
1085:
1059:
1057:
917:The spectral element method uses a
1302:Finite-difference frequency-domain
721:
610:
531:
346:
282:
219:
84:
14:
1107:doi.org/10.1016/j.cma.2018.10.019
956:) combines the advantages of the
1676:Numerical differential equations
983:Journal of Computational Physics
865:{\displaystyle a(\cdot ,\cdot )}
187:is the approximate solution and
1655:Method of fundamental solutions
1441:Smoothed-particle hydrodynamics
927:p-version finite element method
1681:Partial differential equations
1296:Alternating direction-implicit
859:
847:
753:
741:
724:
718:
613:
607:
534:
528:
349:
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285:
279:
222:
216:
93:{\displaystyle L^{2}(\Omega )}
87:
81:
28:(SEM) is a formulation of the
18:partial differential equations
1:
1308:Finite-difference time-domain
16:In the numerical solution of
1686:Computational fluid dynamics
1347:Advection upstream-splitting
1003:10.1016/0021-9991(84)90128-1
117:integrations with a reduced
111:structural health monitoring
32:(FEM) that uses high-degree
1358:Essentially non-oscillatory
1341:Monotonic upstream-centered
1707:
1618:Infinite difference method
1236:Forward-time central-space
923:Gauss–Lobatto points
1521:Poincaré–Steklov operator
1280:Method of characteristics
1030:10.1007/s10444-007-9050-7
832:Gauss-Lobatto integration
479:{\displaystyle k\leq s+1}
58:trigonometric polynomials
1538:Tearing and interconnect
1532:Balancing by constraints
132:The classic analysis of
119:Gauss-Lobatto quadrature
52:expands the solution in
1645:Computer-assisted proof
1623:Infinite element method
1411:Gradient discretisation
1137:doi.org/10.2118/7413-PA
952:finite element method (
782:{\displaystyle \gamma }
128:A-priori error estimate
26:spectral element method
1633:Petrov–Galerkin method
1394:Discontinuous Galerkin
912:orthogonal collocation
886:
872:and in the functional
866:
803:
783:
760:
664:
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622:
480:
448:
428:
408:
381:
358:
229:
181:
154:
94:
1691:Finite element method
1613:Isogeometric analysis
1459:Material point method
908:pseudospectral method
903:using the same nodes.
887:
867:
804:
784:
761:
665:
645:
623:
481:
449:
429:
409:
407:{\displaystyle C_{s}}
382:
359:
230:
182:
180:{\displaystyle u_{N}}
155:
102:Chebyshev polynomials
95:
30:finite element method
1650:Integrable algorithm
1476:Domain decomposition
1024:(2): 179–206, 2008.
876:
841:
793:
773:
681:
654:
634:
491:
458:
438:
418:
414:is independent from
391:
371:
242:
191:
164:
144:
106:Lagrange polynomials
68:
1494:Schwarz alternating
1417:Loubignac iteration
995:1984JCoPh..54..468P
931:numerical stability
925:. In contrast, the
816:Development History
104:or very high order
1640:Validated numerics
901:Lobatto quadrature
882:
862:
799:
779:
756:
660:
640:
618:
476:
444:
424:
404:
377:
354:
225:
177:
150:
90:
1663:
1662:
1603:Immersed boundary
1596:Method of moments
1511:Neumann–Dirichlet
1504:abstract additive
1489:Fictitious domain
1433:Meshless/Meshfree
1317:
1316:
1219:Finite difference
885:{\displaystyle F}
802:{\displaystyle u}
672:analytic function
663:{\displaystyle u}
643:{\displaystyle N}
447:{\displaystyle s}
427:{\displaystyle N}
380:{\displaystyle N}
153:{\displaystyle u}
62:orthonormal basis
1698:
1608:Analytic element
1591:Boundary element
1484:Schur complement
1465:Particle-in-cell
1400:Spectral element
1224:
1204:
1197:
1190:
1181:
1175:
1165:
1159:
1147:Barna SzabĂł and
1145:
1139:
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789:depends only on
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134:Galerkin methods
99:
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1705:
1701:
1700:
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1696:
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1666:
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1664:
1659:
1628:Galerkin method
1571:Method of lines
1548:
1516:Neumann–Neumann
1470:
1427:
1369:
1336:High-resolution
1313:
1284:
1246:
1213:
1208:
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1125:
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1074:
1062:
1055:
1018:Adv Comput Math
1015:
1014:
1010:
980:
979:
975:
971:
874:
873:
839:
838:
827:
825:Related methods
818:
791:
790:
771:
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703:
693:
679:
678:
652:
651:
632:
631:
630:As we increase
591:
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50:spectral method
46:
12:
11:
5:
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1565:Pseudospectral
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1375:Finite element
1371:
1370:
1368:
1367:
1361:
1355:
1353:Riemann solver
1350:
1344:
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1260:Lax–Friedrichs
1256:
1254:
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1247:
1245:
1244:
1242:Crank–Nicolson
1239:
1232:
1230:
1221:
1215:
1214:
1209:
1207:
1206:
1199:
1192:
1184:
1177:
1176:
1160:
1140:
1128:
1119:
1110:
1093:
1081:
1072:
1053:
1008:
989:(3): 468–488.
972:
970:
967:
966:
965:
946:
919:tensor product
915:
904:
897:
894:Strang's lemma
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1323:Finite volume
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1173:1-58488-438-X
1170:
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1157:0-471-50273-1
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1069:9780199671366
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947:
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928:
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916:
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850:
844:
837:
836:bilinear form
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829:
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54:trigonometric
51:
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20:, a topic in
19:
1447:Peridynamics
1399:
1265:Lax–Wendroff
1163:
1143:
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1122:
1113:
1075:
1021:
1017:
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986:
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934:
819:
811:
768:
629:
487:
366:
131:
123:
115:
47:
25:
15:
1581:Collocation
1149:Ivo Babuška
138:CĂ©a's lemma
37:polynomials
22:mathematics
1670:Categories
1270:MacCormack
1252:Hyperbolic
1039:1885/56758
44:Discussion
1586:Level-set
1576:Multigrid
1526:Balancing
1228:Parabolic
935:p-version
857:⋅
851:⋅
777:γ
748:γ
745:−
739:
730:≦
722:Ω
705:‖
691:−
685:‖
611:Ω
588:‖
581:‖
573:−
567:−
540:≤
532:Ω
515:‖
501:−
495:‖
465:≤
347:Ω
324:‖
317:‖
309:−
291:≦
283:Ω
266:‖
252:−
246:‖
220:Ω
198:∈
85:Ω
34:piecewise
1560:Spectral
1499:additive
1422:Smoothed
1388:Extended
1048:46564029
1544:FETI-DP
1424:(S-FEM)
1343:(MUSCL)
1331:Godunov
991:Bibcode
60:are an
1553:Others
1540:(FETI)
1534:(BDDC)
1406:Mortar
1390:(XFEM)
1383:hp-FEM
1366:(WENO)
1349:(AUSM)
1310:(FDTD)
1304:(FDFD)
1289:Others
1275:Upwind
1238:(FTCS)
1171:
1155:
1067:
1046:
954:hp-FEM
769:where
670:is an
434:, and
367:where
24:, the
1567:(DVR)
1528:(BDD)
1467:(PIC)
1461:(MPM)
1455:(MPS)
1443:(SPH)
1413:(GDM)
1402:(SEM)
1360:(ENO)
1298:(ADI)
1044:S2CID
969:Notes
1449:(PD)
1396:(DG)
1169:ISBN
1153:ISBN
1065:ISBN
960:and
948:The
906:The
136:and
64:for
48:The
1034:hdl
1026:doi
999:doi
736:exp
1672::
1096:^
1084:^
1056:^
1042:.
1032:.
1022:29
1020:.
997:.
987:54
985:.
950:hp
910:,
809:.
674::
235::
1203:e
1196:t
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962:p
958:h
945:.
943:h
939:p
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860:)
854:,
848:(
845:a
797:u
754:)
751:N
742:(
733:C
725:)
719:(
714:1
710:H
699:N
695:u
688:u
658:u
638:N
614:)
608:(
603:1
600:+
597:s
593:H
584:u
576:s
570:1
564:k
560:N
554:k
551:,
548:s
544:C
535:)
529:(
524:k
520:H
509:N
505:u
498:u
474:1
471:+
468:s
462:k
442:s
422:N
400:s
396:C
375:N
350:)
344:(
339:1
336:+
333:s
329:H
320:u
312:s
305:N
299:s
295:C
286:)
280:(
275:1
271:H
260:N
256:u
249:u
223:)
217:(
212:1
209:+
206:s
202:H
195:u
173:N
169:u
148:u
88:)
82:(
77:2
73:L
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