1888:
106:. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors. It can also be solved by a combination of relaxation and appeal to still coarser grids. This recursive process is repeated until a grid is reached where the cost of direct solution there is negligible compared to the cost of one relaxation sweep on the fine grid. This multigrid cycle typically reduces all error components by a fixed amount bounded well below one, independent of the fine grid mesh size. The typical application for multigrid is in the numerical solution of
2405:
models in science and engineering described by partial differential equations. In view of the subspace correction framework, BPX preconditioner is a parallel subspace correction method where as the classic V-cycle is a successive subspace correction method. The BPX-preconditioner is known to be naturally more parallel and in some applications more robust than the classic V-cycle multigrid method. The method has been widely used by researchers and practitioners since 1990.
147:
2470:
as well as quasi-optimal spaces was derived. Also, they proved that, under appropriate assumptions, the abstract two-level AMG method converges uniformly with respect to the size of the linear system, the coefficient variation, and the anisotropy. Their abstract framework covers most existing AMG methods, such as classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.
1558:
2462:(AMG) construct their hierarchy of operators directly from the system matrix. In classical AMG, the levels of the hierarchy are simply subsets of unknowns without any geometric interpretation. (More generally, coarse grid unknowns can be particular linear combinations of fine grid unknowns.) Thus, AMG methods become black-box solvers for certain classes of
2616:
that appear in the nearly singular operator) independent convergence rate of the multigrid method applied to such nearly singular systems, i.e., in each grid, a space decomposition based on which the smoothing is applied, has to be constructed so that the null space of the singular part of the nearly
2466:. AMG is regarded as advantageous mainly where geometric multigrid is too difficult to apply, but is often used simply because it avoids the coding necessary for a true multigrid implementation. While classical AMG was developed first, a related algebraic method is known as smoothed aggregation (SA).
215:
There are many choices of multigrid methods with varying trade-offs between speed of solving a single iteration and the rate of convergence with said iteration. The 3 main types are V-Cycle, F-Cycle, and W-Cycle. These differ in which and how many coarse-grain cycles are performed per fine iteration.
2469:
In an overview paper by
Jinchao Xu and Ludmil Zikatanov, the "algebraic multigrid" methods are understood from an abstract point of view. They developed a unified framework and existing algebraic multigrid methods can be derived coherently. Abstract theory about how to construct optimal coarse space
2404:
Originally described in Xu's Ph.D. thesis and later published in
Bramble-Pasciak-Xu, the BPX-preconditioner is one of the two major multigrid approaches (the other is the classic multigrid algorithm such as V-cycle) for solving large-scale algebraic systems that arise from the discretization of
220:, F-Cycle takes 83% more time to compute than a V-Cycle iteration while a W-Cycle iteration takes 125% more. If the problem is set up in a 3D domain, then a F-Cycle iteration and a W-Cycle iteration take about 64% and 75% more time respectively than a V-Cycle iteration ignoring
243:
Any geometric multigrid cycle iteration is performed on a hierarchy of grids and hence it can be coded using recursion. Since the function calls itself with smaller sized (coarser) parameters, the coarsest grid is where the recursion stops. In cases where the system has a high
1565:
This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used. In other words, it can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns.
2611:
operator. There were many works to attempt to design a robust and fast multigrid method for such nearly singular problems. A general guide has been provided as a design principle to achieve parameters (e.g., mesh size and physical parameters such as
117:
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
2923:
2450:, that is, it adjusts the grid as the computation proceeds, in a manner dependent upon the computation itself. The idea is to increase resolution of the grid only in regions of the solution where it is needed.
3249:
Young-Ju Lee, Jinbiao Wu, Jinchao Xu and Ludmil
Zikatanov, Robust Subspace Correction Methods for Nearly Singular Systems, Mathematical Models and Methods in Applied Sciences, Vol. 17, No 11, pp. 1937-1963
216:
The V-Cycle algorithm executes one coarse-grain V-Cycle. F-Cycle does a coarse-grain V-Cycle followed by a coarse-grain F-Cycle, while each W-Cycle performs two coarse-grain W-Cycles per iteration. For a
2506:
Nearly singular problems arise in a number of important physical and engineering applications. Simple, but important example of nearly singular problems can be found at the displacement formulation of
2458:
Practically important extensions of multigrid methods include techniques where no partial differential equation nor geometrical problem background is used to construct the multilevel hierarchy. Such
2362:
time as well as in the case where the multigrid method is used as a solver. Multigrid preconditioning is used in practice even for linear systems, typically with one cycle per iteration, e.g., in
2120:
2211:
2373:
If the matrix of the original equation or an eigenvalue problem is symmetric positive definite (SPD), the preconditioner is commonly constructed to be SPD as well, so that the standard
1884:
1773:
1984:
2043:
3410:
2440:. These wavelet methods can be combined with multigrid methods. For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method.
1687:
2537:
2617:
singular operator has to be included in the sum of the local null spaces, the intersection of the null space and the local spaces resulting from the space decompositions.
2557:
2319:
2381:
can still be used. Such imposed SPD constraints may complicate the construction of the preconditioner, e.g., requiring coordinated pre- and post-smoothing. However,
1922:
1641:
1614:
2360:
2272:
2243:
98:
The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a
3305:
2510:
for nearly incompressible materials. Typically, the major problem to solve such nearly singular systems boils down to treat the nearly singular operator given by
1713:
2605:
2577:
2147:
1816:
1793:
1587:
232:, W-Cycle can show superiority in its rate of convergence per iteration over F-Cycle. The choice of smoothing operators are extremely diverse as they include
126:
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
3501:
3451:
2422:
87:
exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a
3880:
3427:
3401:
2414:
123:
2873:
Bramble, James H., Joseph E. Pasciak, and
Jinchao Xu. "Parallel multilevel preconditioners." Mathematics of Computation 55, no. 191 (1990): 1–22.
107:
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3343:
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3104:
3075:
3048:
3021:
2988:
2959:
2933:
2905:
2785:
2758:
2702:
2671:
2644:
3795:
3652:
3507:
3374:
2802:
Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method
248:, the correction procedure is modified such that only a fraction of the prolongated coarser grid solution is added onto the finer grid.
3300:
2952:
Hyperbolic problems: theory, numerics, applications: proceedings of the Ninth
International Conference on Hyperbolic Problems of 2002
225:
3313:
3200:
3720:
3703:
3854:
3640:
3621:
3610:
3587:
1561:
Assuming a 2-dimensional problem setup, the computation moves across grid hierarchy differently for various multigrid cycles.
2413:
Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of
3593:
2418:
2051:
190:
180:
3710:
3326:
R. P. Fedorenko (1964), The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, p. 227.
2155:
3675:
2580:
3715:
2882:
Xu, Jinchao. "Iterative methods by space decomposition and subspace correction." SIAM review 34, no. 4 (1992): 581-613.
2801:
3875:
3394:
3832:
1775:
is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and
3817:
3693:
3369:
113:
Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the
3459:
3441:
3360:
Repository for multigrid, multilevel, multiscale, aggregation, defect correction, and domain decomposition methods
3479:
2389:
217:
1569:
Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid
1070:% Recursive W-cycle multigrid for solving the Poisson equation (\nabla^2 phi = f) on a uniform grid of spacing h
575:% Recursive F-cycle multigrid for solving the Poisson equation (\nabla^2 phi = f) on a uniform grid of spacing h
281:% Recursive V-Cycle Multigrid for solving the Poisson equation (\nabla^2 phi = f) on a uniform grid of spacing h
3802:
3688:
3418:
2608:
2447:
1824:
76:
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2483:
1718:
170:
1930:
3844:
3822:
3807:
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3698:
3683:
3599:
2584:
2487:
1992:
135:
3764:
3535:
3387:
2366:. Its main advantage versus a purely multigrid solver is particularly clear for nonlinear problems, e.g.,
1887:
3812:
3658:
3574:
2479:
114:
64:
39:
3379:
1646:
3849:
3522:
3274:
3263:
2723:
2513:
221:
131:
1798:
The following recurrence relation is then obtained for the effort of obtaining the solution on grid
3616:
3530:
2430:
553:. This multigrid cycle is slower than V-Cycle per iteration but does result in faster convergence.
154:
There are many variations of multigrid algorithms, but the common features are that a hierarchy of
80:
2542:
1044:
Similarly the procedures can modified as shown in the MATLAB style pseudo code for 1 iteration of
3839:
3780:
2847:
2827:
2491:
2374:
119:
48:
3038:
3012:. Vol. 20 of Lecture Notes in Computational Science and Engineering. Springer. p. 140
2661:
2613:
2277:
102:
correction of the fine grid solution approximation from time to time, accomplished by solving a
2498:
parallel-in-time integration method can also be reformulated as a two-level multigrid in time.
2482:. Of particular interest here are parallel-in-time multigrid methods: in contrast to classical
2396:
for symmetric eigenvalue problems are all shown to be robust if the preconditioner is not SPD.
3885:
3469:
3339:
3309:
3196:
3156:
3150:
3129:
3100:
3092:
3071:
3044:
3017:
3005:
2984:
2955:
2943:
2929:
2901:
2893:
2864:
Xu, Jinchao. Theory of multilevel methods. Vol. 8924558. Ithaca, NY: Cornell
University, 1989.
2781:
2754:
2698:
2688:
2667:
2640:
2507:
2426:
2378:
127:
84:
3121:
3065:
2976:
2634:
3785:
3664:
3632:
3232:
3175:
Xu, J. and
Zikatanov, L., 2017. Algebraic multigrid methods. Acta Numerica, 26, pp.591-721.
2837:
2385:
2126:
1891:
Example of
Convergence Rates of Multigrid Cycles in comparison to other smoothing operators.
245:
229:
166:
88:
3275:
On the convergence of a relaxation method with natural constraints on the elliptic operator
1900:
1619:
1592:
3827:
3770:
3759:
2463:
2382:
2336:
2330:
2248:
2219:
233:
1692:
2727:
146:
3605:
3552:
2590:
2562:
2329:
A multigrid method with an intentionally reduced tolerance can be used as an efficient
2132:
1801:
1778:
1572:
237:
155:
103:
92:
72:
3869:
3474:
197:
3364:
3331:
2851:
2816:"Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods 1"
255:
These steps can be used as shown in the MATLAB style pseudo code for 1 iteration of
3646:
3563:
3540:
2947:
224:. Typically, W-Cycle produces similar convergence to F-Cycle. However, in cases of
2775:
2742:
1795:
is some constant modeling the effort of computing the result for one grid point.
17:
3557:
3435:
3280:
2842:
2815:
3188:
2367:
2333:
for an external iterative solver, e.g., The solution may still be obtained in
3270:
68:
60:
3330:
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007).
3236:
1557:
2898:
Numerical solution of partial differential equations on parallel computers
205:– interpolating a correction computed on a coarser grid into a finer grid.
2495:
2437:
150:
Visualization of iterative
Multigrid algorithm for fast O(n) convergence.
3743:
3320:
3284:
3176:
3582:
2814:
Bouwmeester, Henricus; Dougherty, Andrew; Knyazev, Andrew V. (2015).
2393:
3298:
William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000),
1385:% stop recursion at smallest grid size, otherwise continue recursion
1184:% stop recursion at smallest grid size, otherwise continue recursion
890:% stop recursion at smallest grid size, otherwise continue recursion
689:% stop recursion at smallest grid size, otherwise continue recursion
395:% stop recursion at smallest grid size, otherwise continue recursion
91:
approach to multigrid. MG methods can be used as solvers as well as
2690:
Multigrid finite element methods for electromagnetic field modeling
2832:
2363:
2804:. Electronic Transactions on Numerical Analysis, 15, 38–55, 2003.
3737:
3731:
3546:
2274:
i.e. W-cycle multigrid used on a 1D problem; it would result in
2245:
time. It should be mentioned that there is one exception to the
3383:
2981:
Multiscale and multiresolution methods: theory and applications
211:– Adding prolongated coarser grid solution onto the finer grid.
3332:"Section 20.6. Multigrid Methods for Boundary Value Problems"
3321:
A relaxation method for solving elliptic difference equations
3223:
Horton, Graham (1992). "The time-parallel multigrid method".
2478:
Multigrid methods have also been adopted for the solution of
2539:
robustly with respect to the positive, but small parameter
1048:
for an even superior rate of convergence in certain cases:
3006:"Wavelet-based numerical homogenization with applications"
165:– reducing high frequency errors, for example using a few
3285:
Multi-Level
Adaptive Solutions to Boundary-Value Problems
3193:
Computing Methods in Applied Sciences and Engineering, VI
2925:
Computational fluid dynamics: principles and applications
3359:
3008:. In Timothy J. Barth; Tony Chan; Robert Haimes (eds.).
2979:. In Timothy J. Barth; Tony Chan; Robert Haimes (eds.).
2944:"Multigrid for Atmospheric Data Assimilation: Analysis"
3338:(3rd ed.). New York: Cambridge University Press.
2892:
F. HĂĽlsemann; M. Kowarschik; M. Mohr; U. RĂĽde (2006).
2425:
is underway. Multigrid methods can also be applied to
75:. They are an example of a class of techniques called
3149:
U. Trottenberg; C. W. Oosterlee; A. SchĂĽller (2001).
3091:
U. Trottenberg; C. W. Oosterlee; A. SchĂĽller (2001).
3037:
U. Trottenberg; C. W. Oosterlee; A. SchĂĽller (2001).
2660:
U. Trottenberg; C. W. Oosterlee; A. SchĂĽller (2001).
2593:
2565:
2545:
2516:
2436:
Another set of multiresolution methods is based upon
2339:
2280:
2251:
2222:
2158:
2135:
2054:
1995:
1933:
1903:
1827:
1804:
1781:
1721:
1695:
1649:
1622:
1595:
1575:
3411:
Numerical methods for partial differential equations
3264:
An iterative method of solving elliptic net problems
2417:, or they can be applied directly to time-dependent
2115:{\displaystyle W_{1}=KN_{1}\sum _{p=0}^{n}\rho ^{p}}
3752:
3674:
3631:
3573:
3521:
3488:
3450:
3426:
3417:
2206:{\displaystyle W_{1}<KN_{1}{\frac {1}{1-\rho }}}
35:
3336:Numerical Recipes: The Art of Scientific Computing
2599:
2571:
2551:
2531:
2354:
2313:
2266:
2237:
2205:
2141:
2114:
2037:
1978:
1916:
1878:
1810:
1787:
1767:
1707:
1681:
1635:
1608:
1581:
3323:. USSR Comput. Math. Math. Phys. 1, p. 1092.
1616:. Assume furthermore that a solution on any grid
2636:Practical Fourier analysis for multigrid methods
158:(grids) is considered. The important steps are:
3126:Matrix-based multigrid: theory and applications
1897:And in particular, we find for the finest grid
3306:Society for Industrial and Applied Mathematics
2896:. In Are Magnus Bruaset; Aslak Tveito (eds.).
3395:
8:
2749:. McGraw-Hill Higher Education. p. 478
2747:Scientific Computing: An Introductory Survey
30:
3266:. USSR Comp. Math. Math. Phys. 11, 171–182.
3225:Communications in Applied Numerical Methods
2977:"Multiscale scientific computation: review"
1989:Combining these two expressions (and using
3423:
3402:
3388:
3380:
3277:. USSR Comp. Math. Math. Phys. 6, 101–13.
2841:
2831:
2633:Roman Wienands; Wolfgang Joppich (2005).
2592:
2564:
2544:
2515:
2423:hyperbolic partial differential equations
2421:. Research on multilevel techniques for
2338:
2279:
2250:
2221:
2185:
2179:
2163:
2157:
2134:
2106:
2096:
2085:
2075:
2059:
2053:
2029:
2013:
2000:
1994:
1970:
1951:
1938:
1932:
1908:
1902:
1879:{\displaystyle W_{k}=W_{k+1}+\rho KN_{k}}
1870:
1845:
1832:
1826:
1803:
1780:
1753:
1744:
1732:
1720:
1694:
1673:
1654:
1648:
1627:
1621:
1600:
1594:
1574:
2415:parabolic partial differential equations
1886:
1768:{\displaystyle \rho =N_{i+1}/N_{i}<1}
1556:
250:
145:
27:Method of solving differential equations
2687:Yu Zhu; Andreas C. Cangellaris (2006).
2625:
2216:that is, a solution may be obtained in
1979:{\displaystyle W_{1}=W_{2}+\rho KN_{1}}
108:elliptic partial differential equations
3010:Multiscale and Multiresolution Methods
2502:Multigrid for nearly singular problems
2494:in temporal direction. The well known
2038:{\displaystyle N_{k}=\rho ^{k-1}N_{1}}
29:
3067:Numerical Analysis of Wavelet Methods
3004:Björn Engquist; Olof Runborg (2002).
2942:Achi Brandt and Rima Gandlin (2003).
1643:may be obtained with a given effort
83:of behavior. For example, many basic
79:, very useful in problems exhibiting
7:
3653:Moving particle semi-implicit method
3564:Weighted essentially non-oscillatory
2777:An Introduction to Multigrid Methods
3502:Finite-difference frequency-domain
2743:"Section 11.5.7 Multigrid Methods"
1689:from a solution on a coarser grid
25:
2800:Andrew V Knyazev, Klaus Neymeyr.
2400:Bramble–Pasciak–Xu preconditioner
1682:{\displaystyle W_{i}=\rho KN_{i}}
1589:with a given grid point density
183:after the smoothing operation(s).
2720:Analysis of the multigrid method
3855:Method of fundamental solutions
3641:Smoothed-particle hydrodynamics
3155:. Academic Press. p. 417.
3099:. Academic Press. p. 356.
3093:"Chapter 9: Adaptive Multigrid"
2718:Shah, Tasneem Mohammad (1989).
2532:{\displaystyle A+\varepsilon M}
3881:Partial differential equations
3496:Alternating direction-implicit
3189:"Parabolic multi-grid methods"
2894:"Parallel geometric multigrid"
2419:partial differential equations
2349:
2343:
2308:
2305:
2299:
2284:
2261:
2255:
2232:
2226:
1391:smallest_grid_size_is_achieved
1190:smallest_grid_size_is_achieved
896:smallest_grid_size_is_achieved
695:smallest_grid_size_is_achieved
401:smallest_grid_size_is_achieved
1:
3508:Finite-difference time-domain
2722:(Thesis). Oxford University.
2409:Generalized multigrid methods
1472:% Prolongation and correction
1271:% Prolongation and correction
977:% Prolongation and Correction
776:% Prolongation and Correction
482:% Prolongation and Correction
124:separability of the equations
3547:Advection upstream-splitting
3370:Algebraic multigrid tutorial
3187:Hackbusch, Wolfgang (1985).
2552:{\displaystyle \varepsilon }
122:. They do not depend on the
3558:Essentially non-oscillatory
3541:Monotonic upstream-centered
3262:G. P. Astrachancev (1971),
2900:. Birkhäuser. p. 165.
2843:10.1016/j.procs.2015.05.241
2460:algebraic multigrid methods
2392:for SPD linear systems and
2129:, we then find (for finite
110:in two or more dimensions.
3902:
3818:Infinite difference method
3436:Forward-time central-space
3375:Links to AMG presentations
3289:Mathematics of Computation
2314:{\displaystyle O(Nlog(N))}
252:
3721:Poincaré–Steklov operator
3480:Method of characteristics
3304:(2nd ed.), Philadelphia:
2954:. Springer. p. 369.
2928:. Elsevier. p. 305.
2820:Procedia Computer Science
2639:. CRC Press. p. 17.
2474:Multigrid in time methods
2454:Algebraic multigrid (AMG)
2325:Multigrid preconditioning
1331:% Compute residual errors
1106:% Compute Residual Errors
836:% Compute residual errors
611:% Compute Residual Errors
549:The following represents
317:% Compute Residual Errors
3738:Tearing and interconnect
3732:Balancing by constraints
3319:R. P. Fedorenko (1961),
3128:. Springer. p. 66.
3070:. Elsevier. p. 44.
2983:. Springer. p. 53.
2490:methods, they can offer
2448:adaptive mesh refinement
1050:
555:
261:
193:error to a coarser grid.
3845:Computer-assisted proof
3823:Infinite element method
3611:Gradient discretisation
136:Navier-Stokes equations
77:multiresolution methods
3833:Petrov–Galerkin method
3594:Discontinuous Galerkin
3237:10.1002/cnm.1630080906
2601:
2573:
2553:
2533:
2480:initial value problems
2356:
2315:
2268:
2239:
2207:
2143:
2116:
2101:
2039:
1980:
1918:
1892:
1880:
1812:
1789:
1769:
1709:
1683:
1637:
1610:
1583:
1562:
151:
65:differential equations
3813:Isogeometric analysis
3659:Material point method
3122:"Algebraic multigrid"
3120:Yair Shapira (2003).
3064:Albert Cohen (2003).
2774:P. Wesseling (1992).
2693:. Wiley. p. 132
2602:
2574:
2554:
2534:
2429:, or for problems in
2357:
2316:
2269:
2240:
2208:
2144:
2117:
2081:
2040:
1981:
1919:
1917:{\displaystyle N_{1}}
1890:
1881:
1813:
1790:
1770:
1710:
1684:
1638:
1636:{\displaystyle N_{i}}
1611:
1609:{\displaystyle N_{i}}
1584:
1560:
149:
115:finite element method
40:Differential equation
3850:Integrable algorithm
3676:Domain decomposition
3301:A Multigrid Tutorial
2975:Achi Brandt (2002).
2946:. In Thomas Y. Hou;
2741:M. T. Heath (2002).
2591:
2583:operator with large
2563:
2543:
2514:
2355:{\displaystyle O(N)}
2337:
2278:
2267:{\displaystyle O(N)}
2249:
2238:{\displaystyle O(N)}
2220:
2156:
2133:
2052:
1993:
1931:
1901:
1825:
1802:
1779:
1719:
1693:
1647:
1620:
1593:
1573:
226:convection-diffusion
177:Residual Computation
3694:Schwarz alternating
3617:Loubignac iteration
2922:J. Blaz̆ek (2001).
2728:1989STIN...9123418S
2431:statistical physics
2390:flexible CG methods
1708:{\displaystyle i+1}
1542:Computational cost
236:methods and can be
228:problems with high
218:discrete 2D problem
189:– downsampling the
171:Gauss–Seidel method
120:boundary conditions
32:
3876:Numerical analysis
3840:Validated numerics
3365:Multigrid tutorial
3043:. Academic Press.
2666:. Academic Press.
2597:
2569:
2549:
2529:
2444:Adaptive multigrid
2427:integral equations
2375:conjugate gradient
2352:
2311:
2264:
2235:
2203:
2139:
2112:
2035:
1976:
1914:
1893:
1876:
1808:
1785:
1765:
1705:
1679:
1633:
1606:
1579:
1563:
1400:coarse_level_solve
1199:coarse_level_solve
905:coarse_level_solve
704:coarse_level_solve
410:coarse_level_solve
152:
85:relaxation methods
49:numerical analysis
3863:
3862:
3803:Immersed boundary
3796:Method of moments
3711:Neumann–Dirichlet
3704:abstract additive
3689:Fictitious domain
3633:Meshless/Meshfree
3517:
3516:
3419:Finite difference
3345:978-0-521-88068-8
3162:978-0-12-701070-0
3135:978-1-4020-7485-1
3106:978-0-12-701070-0
3077:978-0-444-51124-9
3050:978-0-12-701070-0
3023:978-3-540-42420-8
2990:978-3-540-42420-8
2961:978-3-540-44333-9
2935:978-0-08-043009-6
2907:978-3-540-29076-6
2787:978-0-471-93083-9
2760:978-0-07-112229-0
2704:978-0-471-74110-7
2673:978-0-12-701070-0
2646:978-1-58488-492-7
2609:positive definite
2600:{\displaystyle M}
2572:{\displaystyle A}
2508:linear elasticity
2379:iterative methods
2201:
2142:{\displaystyle n}
1811:{\displaystyle k}
1788:{\displaystyle K}
1582:{\displaystyle i}
1539:
1538:
1046:W-cycle multigrid
551:F-cycle multigrid
257:V-Cycle Multigrid
45:
44:
18:Multigrid methods
16:(Redirected from
3893:
3808:Analytic element
3791:Boundary element
3684:Schur complement
3665:Particle-in-cell
3600:Spectral element
3424:
3404:
3397:
3390:
3381:
3349:
3251:
3247:
3241:
3240:
3220:
3214:
3213:
3211:
3209:
3184:
3178:
3173:
3167:
3166:
3146:
3140:
3139:
3117:
3111:
3110:
3088:
3082:
3081:
3061:
3055:
3054:
3034:
3028:
3027:
3001:
2995:
2994:
2972:
2966:
2965:
2939:
2918:
2912:
2911:
2889:
2883:
2880:
2874:
2871:
2865:
2862:
2856:
2855:
2845:
2835:
2811:
2805:
2798:
2792:
2791:
2771:
2765:
2764:
2738:
2732:
2731:
2715:
2709:
2708:
2684:
2678:
2677:
2657:
2651:
2650:
2630:
2606:
2604:
2603:
2598:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2488:linear multistep
2386:steepest descent
2361:
2359:
2358:
2353:
2320:
2318:
2317:
2312:
2273:
2271:
2270:
2265:
2244:
2242:
2241:
2236:
2212:
2210:
2209:
2204:
2202:
2200:
2186:
2184:
2183:
2168:
2167:
2148:
2146:
2145:
2140:
2127:geometric series
2121:
2119:
2118:
2113:
2111:
2110:
2100:
2095:
2080:
2079:
2064:
2063:
2044:
2042:
2041:
2036:
2034:
2033:
2024:
2023:
2005:
2004:
1985:
1983:
1982:
1977:
1975:
1974:
1956:
1955:
1943:
1942:
1923:
1921:
1920:
1915:
1913:
1912:
1885:
1883:
1882:
1877:
1875:
1874:
1856:
1855:
1837:
1836:
1817:
1815:
1814:
1809:
1794:
1792:
1791:
1786:
1774:
1772:
1771:
1766:
1758:
1757:
1748:
1743:
1742:
1714:
1712:
1711:
1706:
1688:
1686:
1685:
1680:
1678:
1677:
1659:
1658:
1642:
1640:
1639:
1634:
1632:
1631:
1615:
1613:
1612:
1607:
1605:
1604:
1588:
1586:
1585:
1580:
1554:
1553:
1549:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1499:% Post-smoothing
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1064:
1061:
1058:
1054:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1004:% Post-smoothing
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
569:
566:
563:
559:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
509:% Post-Smoothing
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
275:
272:
269:
265:
251:
246:condition number
89:Fourier analysis
53:multigrid method
33:
31:Multigrid method
21:
3901:
3900:
3896:
3895:
3894:
3892:
3891:
3890:
3866:
3865:
3864:
3859:
3828:Galerkin method
3771:Method of lines
3748:
3716:Neumann–Neumann
3670:
3627:
3569:
3536:High-resolution
3513:
3484:
3446:
3413:
3408:
3356:
3346:
3329:
3283:(April 1977), "
3259:
3254:
3248:
3244:
3222:
3221:
3217:
3207:
3205:
3203:
3186:
3185:
3181:
3174:
3170:
3163:
3148:
3147:
3143:
3136:
3119:
3118:
3114:
3107:
3090:
3089:
3085:
3078:
3063:
3062:
3058:
3051:
3036:
3035:
3031:
3024:
3003:
3002:
2998:
2991:
2974:
2973:
2969:
2962:
2941:
2936:
2921:
2919:
2915:
2908:
2891:
2890:
2886:
2881:
2877:
2872:
2868:
2863:
2859:
2813:
2812:
2808:
2799:
2795:
2788:
2773:
2772:
2768:
2761:
2740:
2739:
2735:
2717:
2716:
2712:
2705:
2686:
2685:
2681:
2674:
2659:
2658:
2654:
2647:
2632:
2631:
2627:
2623:
2614:Poisson's ratio
2607:is a symmetric
2589:
2588:
2561:
2560:
2541:
2540:
2512:
2511:
2504:
2476:
2464:sparse matrices
2456:
2411:
2402:
2335:
2334:
2327:
2276:
2275:
2247:
2246:
2218:
2217:
2190:
2175:
2159:
2154:
2153:
2131:
2130:
2102:
2071:
2055:
2050:
2049:
2025:
2009:
1996:
1991:
1990:
1966:
1947:
1934:
1929:
1928:
1904:
1899:
1898:
1866:
1841:
1828:
1823:
1822:
1800:
1799:
1777:
1776:
1749:
1728:
1717:
1716:
1691:
1690:
1669:
1650:
1645:
1644:
1623:
1618:
1617:
1596:
1591:
1590:
1571:
1570:
1555:
1551:
1547:
1545:
1544:
1535:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1073:% Pre-smoothing
1072:
1069:
1066:
1062:
1059:
1056:
1052:
1040:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
578:% Pre-smoothing
577:
574:
571:
567:
564:
561:
557:
545:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
284:% Pre-Smoothing
283:
280:
277:
273:
270:
267:
263:
234:Krylov subspace
156:discretizations
144:
93:preconditioners
81:multiple scales
73:discretizations
28:
23:
22:
15:
12:
11:
5:
3899:
3897:
3889:
3888:
3883:
3878:
3868:
3867:
3861:
3860:
3858:
3857:
3852:
3847:
3842:
3837:
3836:
3835:
3825:
3820:
3815:
3810:
3805:
3800:
3799:
3798:
3788:
3783:
3778:
3773:
3768:
3765:Pseudospectral
3762:
3756:
3754:
3750:
3749:
3747:
3746:
3741:
3735:
3729:
3723:
3718:
3713:
3708:
3707:
3706:
3701:
3691:
3686:
3680:
3678:
3672:
3671:
3669:
3668:
3662:
3656:
3650:
3644:
3637:
3635:
3629:
3628:
3626:
3625:
3619:
3614:
3608:
3603:
3597:
3591:
3585:
3579:
3577:
3575:Finite element
3571:
3570:
3568:
3567:
3561:
3555:
3553:Riemann solver
3550:
3544:
3538:
3533:
3527:
3525:
3519:
3518:
3515:
3514:
3512:
3511:
3505:
3499:
3492:
3490:
3486:
3485:
3483:
3482:
3477:
3472:
3467:
3462:
3460:Lax–Friedrichs
3456:
3454:
3448:
3447:
3445:
3444:
3442:Crank–Nicolson
3439:
3432:
3430:
3421:
3415:
3414:
3409:
3407:
3406:
3399:
3392:
3384:
3378:
3377:
3372:
3367:
3362:
3355:
3354:External links
3352:
3351:
3350:
3344:
3327:
3324:
3317:
3296:
3278:
3267:
3258:
3255:
3253:
3252:
3242:
3231:(9): 585–595.
3215:
3201:
3179:
3168:
3161:
3141:
3134:
3112:
3105:
3083:
3076:
3056:
3049:
3029:
3022:
2996:
2989:
2967:
2960:
2934:
2913:
2906:
2884:
2875:
2866:
2857:
2806:
2793:
2786:
2766:
2759:
2733:
2710:
2703:
2679:
2672:
2652:
2645:
2624:
2622:
2619:
2596:
2568:
2548:
2528:
2525:
2522:
2519:
2503:
2500:
2475:
2472:
2455:
2452:
2410:
2407:
2401:
2398:
2383:preconditioned
2351:
2348:
2345:
2342:
2331:preconditioner
2326:
2323:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2263:
2260:
2257:
2254:
2234:
2231:
2228:
2225:
2214:
2213:
2199:
2196:
2193:
2189:
2182:
2178:
2174:
2171:
2166:
2162:
2138:
2123:
2122:
2109:
2105:
2099:
2094:
2091:
2088:
2084:
2078:
2074:
2070:
2067:
2062:
2058:
2032:
2028:
2022:
2019:
2016:
2012:
2008:
2003:
1999:
1987:
1986:
1973:
1969:
1965:
1962:
1959:
1954:
1950:
1946:
1941:
1937:
1911:
1907:
1895:
1894:
1873:
1869:
1865:
1862:
1859:
1854:
1851:
1848:
1844:
1840:
1835:
1831:
1807:
1784:
1764:
1761:
1756:
1752:
1747:
1741:
1738:
1735:
1731:
1727:
1724:
1704:
1701:
1698:
1676:
1672:
1668:
1665:
1662:
1657:
1653:
1630:
1626:
1603:
1599:
1578:
1543:
1540:
1537:
1536:
1298:% Re-smoothing
1051:
1041:
803:% Re-smoothing
556:
546:
262:
238:preconditioned
230:PĂ©clet numbers
213:
212:
206:
194:
184:
181:residual error
174:
143:
140:
128:Lamé equations
104:coarse problem
43:
42:
37:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3898:
3887:
3884:
3882:
3879:
3877:
3874:
3873:
3871:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3834:
3831:
3830:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3797:
3794:
3793:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3766:
3763:
3761:
3758:
3757:
3755:
3751:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3705:
3702:
3700:
3697:
3696:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3679:
3677:
3673:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3638:
3636:
3634:
3630:
3623:
3620:
3618:
3615:
3612:
3609:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3584:
3581:
3580:
3578:
3576:
3572:
3565:
3562:
3559:
3556:
3554:
3551:
3548:
3545:
3542:
3539:
3537:
3534:
3532:
3529:
3528:
3526:
3524:
3523:Finite volume
3520:
3509:
3506:
3503:
3500:
3497:
3494:
3493:
3491:
3487:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3457:
3455:
3453:
3449:
3443:
3440:
3437:
3434:
3433:
3431:
3429:
3425:
3422:
3420:
3416:
3412:
3405:
3400:
3398:
3393:
3391:
3386:
3385:
3382:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3357:
3353:
3347:
3341:
3337:
3333:
3328:
3325:
3322:
3318:
3315:
3314:0-89871-462-1
3311:
3307:
3303:
3302:
3297:
3294:
3290:
3286:
3282:
3279:
3276:
3272:
3268:
3265:
3261:
3260:
3256:
3246:
3243:
3238:
3234:
3230:
3226:
3219:
3216:
3204:
3202:9780444875976
3198:
3194:
3190:
3183:
3180:
3177:
3172:
3169:
3164:
3158:
3154:
3153:
3145:
3142:
3137:
3131:
3127:
3123:
3116:
3113:
3108:
3102:
3098:
3094:
3087:
3084:
3079:
3073:
3069:
3068:
3060:
3057:
3052:
3046:
3042:
3041:
3033:
3030:
3025:
3019:
3015:
3011:
3007:
3000:
2997:
2992:
2986:
2982:
2978:
2971:
2968:
2963:
2957:
2953:
2949:
2945:
2937:
2931:
2927:
2926:
2920:For example,
2917:
2914:
2909:
2903:
2899:
2895:
2888:
2885:
2879:
2876:
2870:
2867:
2861:
2858:
2853:
2849:
2844:
2839:
2834:
2829:
2825:
2821:
2817:
2810:
2807:
2803:
2797:
2794:
2789:
2783:
2779:
2778:
2770:
2767:
2762:
2756:
2752:
2748:
2744:
2737:
2734:
2729:
2725:
2721:
2714:
2711:
2706:
2700:
2696:
2692:
2691:
2683:
2680:
2675:
2669:
2665:
2664:
2656:
2653:
2648:
2642:
2638:
2637:
2629:
2626:
2620:
2618:
2615:
2610:
2594:
2586:
2582:
2579:is symmetric
2566:
2546:
2526:
2523:
2520:
2517:
2509:
2501:
2499:
2497:
2493:
2489:
2485:
2481:
2473:
2471:
2467:
2465:
2461:
2453:
2451:
2449:
2445:
2441:
2439:
2434:
2432:
2428:
2424:
2420:
2416:
2408:
2406:
2399:
2397:
2395:
2391:
2387:
2384:
2380:
2376:
2371:
2369:
2365:
2346:
2340:
2332:
2324:
2322:
2321:complexity.
2302:
2296:
2293:
2290:
2287:
2281:
2258:
2252:
2229:
2223:
2197:
2194:
2191:
2187:
2180:
2176:
2172:
2169:
2164:
2160:
2152:
2151:
2150:
2136:
2128:
2107:
2103:
2097:
2092:
2089:
2086:
2082:
2076:
2072:
2068:
2065:
2060:
2056:
2048:
2047:
2046:
2030:
2026:
2020:
2017:
2014:
2010:
2006:
2001:
1997:
1971:
1967:
1963:
1960:
1957:
1952:
1948:
1944:
1939:
1935:
1927:
1926:
1925:
1909:
1905:
1889:
1871:
1867:
1863:
1860:
1857:
1852:
1849:
1846:
1842:
1838:
1833:
1829:
1821:
1820:
1819:
1805:
1796:
1782:
1762:
1759:
1754:
1750:
1745:
1739:
1736:
1733:
1729:
1725:
1722:
1702:
1699:
1696:
1674:
1670:
1666:
1663:
1660:
1655:
1651:
1628:
1624:
1601:
1597:
1576:
1567:
1559:
1550:
1541:
1364:% Restriction
1139:% Restriction
1049:
1047:
1042:
869:% Restriction
644:% Restriction
554:
552:
547:
350:% Restriction
260:
258:
253:
249:
247:
241:
239:
235:
231:
227:
223:
219:
210:
207:
204:
200:
199:
198:Interpolation
195:
192:
188:
185:
182:
178:
175:
172:
168:
164:
161:
160:
159:
157:
148:
141:
139:
137:
133:
129:
125:
121:
116:
111:
109:
105:
101:
96:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
41:
38:
34:
19:
3775:
3647:Peridynamics
3465:Lax–Wendroff
3335:
3299:
3292:
3288:
3245:
3228:
3224:
3218:
3206:. Retrieved
3192:
3182:
3171:
3151:
3144:
3125:
3115:
3096:
3086:
3066:
3059:
3039:
3032:
3013:
3009:
2999:
2980:
2970:
2951:
2948:Eitan Tadmor
2924:
2916:
2897:
2887:
2878:
2869:
2860:
2823:
2819:
2809:
2796:
2776:
2769:
2750:
2746:
2736:
2719:
2713:
2694:
2689:
2682:
2662:
2655:
2635:
2628:
2581:semidefinite
2505:
2477:
2468:
2459:
2457:
2443:
2442:
2435:
2412:
2403:
2372:
2328:
2215:
2124:
1988:
1896:
1797:
1568:
1564:
1487:prolongation
1286:prolongation
1045:
1043:
992:prolongation
791:prolongation
550:
548:
497:prolongation
256:
254:
242:
214:
208:
203:prolongation
202:
196:
186:
179:– computing
176:
162:
153:
112:
99:
97:
63:for solving
56:
52:
46:
3781:Collocation
3281:Achi Brandt
3195:: 189–197.
2826:: 276–285.
2492:concurrency
2484:Runge–Kutta
1373:restriction
1148:restriction
878:restriction
653:restriction
359:restriction
187:Restriction
3870:Categories
3470:MacCormack
3452:Hyperbolic
3257:References
2585:null space
2370:problems.
2368:eigenvalue
2125:Using the
209:Correction
167:iterations
132:elasticity
3786:Level-set
3776:Multigrid
3726:Balancing
3428:Parabolic
3295:: 333–90.
3271:Bakhvalov
3152:Multigrid
3097:Multigrid
3040:Multigrid
2833:1212.6680
2780:. Wiley.
2663:Multigrid
2547:ε
2524:ε
2446:exhibits
2198:ρ
2195:−
2104:ρ
2083:∑
2018:−
2011:ρ
1961:ρ
1861:ρ
1723:ρ
1664:ρ
1508:smoothing
1307:smoothing
1082:smoothing
1013:smoothing
812:smoothing
587:smoothing
518:smoothing
293:smoothing
222:overheads
163:Smoothing
142:Algorithm
69:hierarchy
61:algorithm
57:MG method
3886:Wavelets
3760:Spectral
3699:additive
3622:Smoothed
3588:Extended
3273:(1966),
3208:1 August
2950:(eds.).
2852:51978658
2587:, while
2496:Parareal
2438:wavelets
2045:) gives
1715:. Here,
1340:residual
1115:residual
1053:function
845:residual
620:residual
558:function
326:residual
264:function
191:residual
67:using a
59:) is an
3744:FETI-DP
3624:(S-FEM)
3543:(MUSCL)
3531:Godunov
2724:Bibcode
2559:. Here
1439:W_cycle
1238:W_cycle
1065:phi,f,h
1060:W_cycle
944:V_Cycle
743:F_Cycle
570:phi,f,h
565:F_Cycle
449:V_Cycle
276:phi,f,h
271:V_Cycle
169:of the
134:or the
3753:Others
3740:(FETI)
3734:(BDDC)
3606:Mortar
3590:(XFEM)
3583:hp-FEM
3566:(WENO)
3549:(AUSM)
3510:(FDTD)
3504:(FDFD)
3489:Others
3475:Upwind
3438:(FTCS)
3342:
3312:
3269:N. S.
3250:(2007)
3199:
3159:
3132:
3103:
3074:
3047:
3020:
2987:
2958:
2932:
2904:
2850:
2784:
2757:
2701:
2670:
2643:
2394:LOBPCG
1546:": -->
100:global
3767:(DVR)
3728:(BDD)
3667:(PIC)
3661:(MPM)
3655:(MPS)
3643:(SPH)
3613:(GDM)
3602:(SEM)
3560:(ENO)
3498:(ADI)
2848:S2CID
2828:arXiv
2621:Notes
2377:(CG)
2364:Hypre
1924:that
1166:zeros
671:zeros
377:zeros
36:Class
3649:(PD)
3596:(DG)
3340:ISBN
3310:ISBN
3210:2015
3197:ISBN
3157:ISBN
3130:ISBN
3101:ISBN
3072:ISBN
3045:ISBN
3018:ISBN
2985:ISBN
2956:ISBN
2940:and
2930:ISBN
2902:ISBN
2782:ISBN
2755:ISBN
2699:ISBN
2668:ISBN
2641:ISBN
2388:and
2170:<
1760:<
1548:edit
1430:else
1229:else
1172:size
935:else
734:else
677:size
440:else
383:size
51:, a
3287:",
3233:doi
2838:doi
2486:or
1532:end
1514:phi
1502:phi
1493:eps
1481:phi
1475:phi
1469:end
1451:rhs
1445:eps
1433:eps
1412:rhs
1406:eps
1394:eps
1367:rhs
1346:phi
1313:phi
1301:phi
1292:eps
1280:phi
1274:phi
1268:end
1250:rhs
1244:eps
1232:eps
1211:rhs
1205:eps
1193:eps
1181:));
1178:rhs
1160:eps
1142:rhs
1121:phi
1088:phi
1076:phi
1055:phi
1037:end
1019:phi
1007:phi
998:eps
986:phi
980:phi
974:end
956:rhs
950:eps
938:eps
917:rhs
911:eps
899:eps
872:rhs
851:phi
818:phi
806:phi
797:eps
785:phi
779:phi
773:end
755:rhs
749:eps
737:eps
716:rhs
710:eps
698:eps
686:));
683:rhs
665:eps
647:rhs
626:phi
593:phi
581:phi
560:phi
542:end
524:phi
512:phi
503:eps
491:phi
485:phi
479:end
461:rhs
455:eps
443:eps
422:rhs
416:eps
404:eps
392:));
389:rhs
371:eps
353:rhs
332:phi
299:phi
287:phi
266:phi
201:or
130:of
71:of
47:In
3872::
3334:.
3308:,
3293:31
3291:,
3227:.
3191:.
3124:.
3095:.
3016:.
3014:ff
2846:.
2836:.
2824:51
2822:.
2818:.
2753:.
2751:ff
2745:.
2697:.
2695:ff
2433:.
2149:)
1818::
1529:);
1496:);
1466:);
1427:);
1388:if
1382:);
1361:);
1328:);
1295:);
1265:);
1226:);
1187:if
1157:);
1136:);
1103:);
1034:);
1001:);
971:);
932:);
893:if
887:);
866:);
833:);
800:);
770:);
731:);
692:if
662:);
641:);
608:);
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506:);
476:);
437:);
398:if
368:);
347:);
314:);
259::
240:.
138:.
95:.
3403:e
3396:t
3389:v
3348:.
3316:.
3239:.
3235::
3229:8
3212:.
3165:.
3138:.
3109:.
3080:.
3053:.
3026:.
2993:.
2964:.
2938:.
2910:.
2854:.
2840::
2830::
2790:.
2763:.
2730:.
2726::
2707:.
2676:.
2649:.
2595:M
2567:A
2527:M
2521:+
2518:A
2350:)
2347:N
2344:(
2341:O
2309:)
2306:)
2303:N
2300:(
2297:g
2294:o
2291:l
2288:N
2285:(
2282:O
2262:)
2259:N
2256:(
2253:O
2233:)
2230:N
2227:(
2224:O
2192:1
2188:1
2181:1
2177:N
2173:K
2165:1
2161:W
2137:n
2108:p
2098:n
2093:0
2090:=
2087:p
2077:1
2073:N
2069:K
2066:=
2061:1
2057:W
2031:1
2027:N
2021:1
2015:k
2007:=
2002:k
1998:N
1972:1
1968:N
1964:K
1958:+
1953:2
1949:W
1945:=
1940:1
1936:W
1910:1
1906:N
1872:k
1868:N
1864:K
1858:+
1853:1
1850:+
1847:k
1843:W
1839:=
1834:k
1830:W
1806:k
1783:K
1763:1
1755:i
1751:N
1746:/
1740:1
1737:+
1734:i
1730:N
1726:=
1703:1
1700:+
1697:i
1675:i
1671:N
1667:K
1661:=
1656:i
1652:W
1629:i
1625:N
1602:i
1598:N
1577:i
1552:]
1526:h
1523:,
1520:f
1517:,
1511:(
1505:=
1490:(
1484:+
1478:=
1463:h
1460:*
1457:2
1454:,
1448:,
1442:(
1436:=
1424:h
1421:*
1418:2
1415:,
1409:,
1403:(
1397:=
1379:r
1376:(
1370:=
1358:h
1355:,
1352:f
1349:,
1343:(
1337:=
1334:r
1325:h
1322:,
1319:f
1316:,
1310:(
1304:=
1289:(
1283:+
1277:=
1262:h
1259:*
1256:2
1253:,
1247:,
1241:(
1235:=
1223:h
1220:*
1217:2
1214:,
1208:,
1202:(
1196:=
1175:(
1169:(
1163:=
1154:r
1151:(
1145:=
1133:h
1130:,
1127:f
1124:,
1118:(
1112:=
1109:r
1100:h
1097:,
1094:f
1091:,
1085:(
1079:=
1067:)
1063:(
1057:=
1031:h
1028:,
1025:f
1022:,
1016:(
1010:=
995:(
989:+
983:=
968:h
965:*
962:2
959:,
953:,
947:(
941:=
929:h
926:*
923:2
920:,
914:,
908:(
902:=
884:r
881:(
875:=
863:h
860:,
857:f
854:,
848:(
842:=
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830:h
827:,
824:f
821:,
815:(
809:=
794:(
788:+
782:=
767:h
764:*
761:2
758:,
752:,
746:(
740:=
728:h
725:*
722:2
719:,
713:,
707:(
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680:(
674:(
668:=
659:r
656:(
650:=
638:h
635:,
632:f
629:,
623:(
617:=
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605:h
602:,
599:f
596:,
590:(
584:=
572:)
568:(
562:=
536:h
533:,
530:f
527:,
521:(
515:=
500:(
494:+
488:=
473:h
470:*
467:2
464:,
458:,
452:(
446:=
434:h
431:*
428:2
425:,
419:,
413:(
407:=
386:(
380:(
374:=
365:r
362:(
356:=
344:h
341:,
338:f
335:,
329:(
323:=
320:r
311:h
308:,
305:f
302:,
296:(
290:=
278:)
274:(
268:=
173:.
55:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
