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Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class \(C^0\) in Hilbert spaces. (English) Zbl 0967.65067
This paper presents a stabilized Galerkin technique for approximating linear contraction semi-groups of class \(C^0\) in Hilbert space. The technique proposed is based on a hierarchical 2-level decomposition of the approximation space. The stability in the graph norm is obtained by introducing an artificial diffusion on the subgrid scales. As a result, optimal convergence in the graph norm has been proved.
The author also considers the singular perturbation problem. In terms of partial differential equations, this situation corresponds in practice to hyperbolic equations perturbed by a small elliptic term or a degenerate elliptic operator. Some numerical tests show the application of the theoretical results.

MSC:
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
34G10 Linear differential equations in abstract spaces
65L05 Numerical methods for initial value problems
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
47D06 One-parameter semigroups and linear evolution equations
35L15 Initial value problems for second-order hyperbolic equations
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References:
[1] Brenner, RAIRO Anal Numer 16 pp 5– (1982) · Zbl 0477.65040 · doi:10.1051/m2an/1982160100051
[2] and ?On a finite element method for solving the neutron transport equation,? Mathematical aspects of finite elements in partial differential equations, (Editor), Academic, New York, 1974, pp. 89-123. · doi:10.1016/B978-0-12-208350-1.50008-X
[3] Johnson, Math Comp 46 pp 1– (1986) · doi:10.1090/S0025-5718-1986-0815828-4
[4] Johnson, Comp Meth Appl Mech Eng 45 pp 285– (1984) · Zbl 0526.76087 · doi:10.1016/0045-7825(84)90158-0
[5] and Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968.
[6] Guermond, Mod Math Anal Num 33 pp 1293– (1999) · Zbl 0946.65112 · doi:10.1051/m2an:1999145
[7] Guermond, ZAMM 79 pp 29– (1999) · doi:10.1002/zamm.19990791308
[8] Smagorinsky, J Atmos Sci 32 pp 680– (1963)
[9] Tadmor, SIAM J Numer Anal 26 pp 30– (1989) · Zbl 0667.65079 · doi:10.1137/0726003
[10] Foias, Math Mod Num Anal 22 pp 93– (1988) · Zbl 0663.76054 · doi:10.1051/m2an/1988220100931
[11] Marion, Numer Math 57 pp 1– (1990) · Zbl 0702.65081 · doi:10.1007/BF01386407
[12] Arnold, Calcolo 21 pp 337– (1984) · Zbl 0593.76039 · doi:10.1007/BF02576171
[13] Brezzi, Comp Meth Appl Mech Eng 96 pp 117– (1992) · Zbl 0756.76044 · doi:10.1016/0045-7825(92)90102-P
[14] Baiocchi, Comp Meth Appl Mech Eng 105 pp 125– (1993) · Zbl 0772.76033 · doi:10.1016/0045-7825(93)90119-I
[15] Crouzeix, RAIRO 3 pp 33– (1973) · Zbl 0302.65087 · doi:10.1051/m2an/197307R300331
[16] Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983.
[17] Functional analysis, SCSM 123, Sixth edition, Springer-Verlag, New York, 1980. · Zbl 0830.46001 · doi:10.1007/978-3-642-61859-8
[18] and Finite element methods for Navier-Stokes equations, Springer Series in Comp Math 5, Springer-Verlag, New York, 1986. · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[19] Maday, SIAM J Numer Anal 30 pp 321– (1993) · Zbl 0774.65072 · doi:10.1137/0730016
[20] Zhou, Math Comp 66 pp 31– (1997) · Zbl 0854.65094 · doi:10.1090/S0025-5718-97-00788-6
[21] Shu, J Comp Phys 83 pp 32– (1989) · Zbl 0674.65061 · doi:10.1016/0021-9991(89)90222-2
[22] Goodman, Nonlinearity 12 pp 247– (1999) · Zbl 0946.35039 · doi:10.1088/0951-7715/12/2/006
[23] Guermond, J Comp Phys
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