zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
$H^1$-Galerkin mixed finite element method for the regularized long wave equation. (English) Zbl 1098.65096
The regularized long wave equation in one space dimension, a scalar valued third-order hyperbolic partial differential equation, is discretized with a mixed finite element method. For time discretization a first-order backward Euler scheme is used. The equation describes nonlinear dispersive waves and has solitary wave solutions. It is for example encountered in modelling shallow water waves or ion acoustic plasma waves. The paper describes an $H^1$-Galerkin mixed finite element method for the above equation. The numerical analysis yields existence and uniqueness of the semi-discrete and the fully discrete solution. Optimal error estimates are also established. It is shown that no Ladyshenskaya-Babuška-Brezzi consistency condition, as necessary for classical mixed finite element methods, is required to approximate simultaneously the solution and the vector valued flux. Numerical results illustrate for a test problem for which an analytical solution is given the properties of the method. It is shown that the $H^1$ error is of order one while $L^2$ and $L^\infty$ errors are of second-order.

65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35L75Nonlinear hyperbolic PDE of higher $(>2)$ order
65M15Error bounds (IVP of PDE)
35Q51Soliton-like equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76X05Ionized gas flow in electromagnetic fields; plasmic flow
76M10Finite element methods (fluid mechanics)
Full Text: DOI
[1] · Zbl 0915.65107 · doi:10.1137/S0036142995280808
[2] · Zbl 0325.65054 · doi:10.1016/0021-9991(75)90115-1
[3] · Zbl 0631.65107 · doi:10.1007/BF01396752
[4] · Zbl 0757.65103 · doi:10.1002/num.1690080407
[5] · Zbl 0599.65072 · doi:10.1007/BF01389710
[6] · doi:10.1090/S0025-5718-1985-0771029-9
[8] · Zbl 0724.65087 · doi:10.1016/0045-7825(90)90165-I
[10] Pani, A. K., Das, P. C.: An H1-Galerkin method for quasilinear parabolic differential equations. In: Methods of functional analysis in approximation theory (Micchelli, C.A., Pai, D.V., Limaye, B.V., eds), Basel: Birkhäuser, pp. 357--370 (1986).
[12] · Zbl 1008.65101 · doi:10.1093/imanum/22.2.231
[13] · Zbl 0361.65100 · doi:10.1016/0021-9991(77)90088-2
[14] · Zbl 0283.65052 · doi:10.1007/BF01438256
[15] · Zbl 0578.65120 · doi:10.1016/0021-9991(85)90001-4
[16] · Zbl 0545.76131 · doi:10.1016/0045-7825(84)90048-3
[18] · Zbl 0407.76014 · doi:10.1016/0021-9991(79)90124-4
[21] · doi:10.1017/S0022112066001678
[22] · Zbl 0451.65086 · doi:10.1016/0021-9991(81)90138-8
[23] · Zbl 0777.76049 · doi:10.1002/cnm.1640090706
[24] · Zbl 0717.65072 · doi:10.1016/0021-9991(90)90047-5
[25] · doi:10.1090/S0025-5718-1985-0777266-1
[26] · doi:10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-O
[28] · Zbl 0927.65123 · doi:10.1137/S0036142996312999
[29] · Zbl 0819.65125 · doi:10.1002/cnm.1640110109
[30] · Zbl 0258.35041 · doi:10.1137/0710062