zbMATH — the first resource for mathematics

Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. (English) Zbl 1156.65094
The goal of this paper is to obtain some abstract results of spectral approximation that can be applied to a wide class of nonconforming methods for compact or noncompact operators. The consistency results derived by the authors are extensions of the results developed by J. Descloux , N. Nassif and J. Rappaz [RAIRO, Anal. Numér. 12, 97–112 (1978; Zbl 0393.65024); ibid. 12, 113–119 (1978; Zbl 0393.65025)].
The theory presented here allows the analysis of a large class of discontinuous finite element methods when they are used for the approximation of spectral problems. Two representative eigenvalue elliptical problems are discussed in detail: the Steklov eigenvalue problem (in which the eigenvalue parameter appears in the boundary condition) and an eigenvalue problem for a system of partial differential equations. The analysis is carried out for the lowest order Crouzeix-Raviart finite element space.

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65J10 Numerical solutions to equations with linear operators
47A10 Spectrum, resolvent
35P15 Estimates of eigenvalues in context of PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
[1] Arnold, D.N.; Brezzi, F., Mixed and nonconforming finite element methods implementation, postprocessing and error estimates, RAIRO, model. math. anal. numer., 19, 7-32, (1985) · Zbl 0567.65078
[2] Antonietti, P.; Buffa, A.; Perugia, I., Discontinuous Galerkin approximations of the Laplace eigenproblem, Comput. methods appl. mech. engrg., 195, 3483-3502, (2006) · Zbl 1168.65410
[3] Alonso, A.; Dello Russo, A.; Padra, C.; Rodríguez, R., Accurate pressure post-process of a finite element method for elastoacoustics, Numer. math., 98, 389-425, (2004) · Zbl 1140.76456
[4] Babuˇska, I.; Osborn, J., Eigenvalue problems, () · Zbl 0875.65087
[5] Bramble, J.; Osborn, J., Rate of convergence estimates for non-selfadjoint eigenvalue approximations, Math. comp., 27, 525-549, (1973) · Zbl 0305.65064
[6] Bermúdez, A.; Durán, R.; Muschietti, M.A.; Rodríguez, R.; Solomin, J., Finite element vibration analysis of fluid – solid systems without spurious modes, SIAM J. numer. anal., 32, 1280-1295, (1995) · Zbl 0833.73050
[7] Buffa, A.; Perugia, I., Discontinuous Galerkin approximations of the Maxwell eigenproblem, SIAM J. numer. anal., 44, 2198-2226, (2006) · Zbl 1344.65110
[8] Ciarlet, P.G., Basic error estimates for elliptic problems, () · Zbl 0875.65086
[9] Chen, Z., Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO, model. math. anal. numer., 27, 9-34, (1993) · Zbl 0784.65075
[10] Dauge, M., Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions, () · Zbl 0668.35001
[11] Descloux, J.; Nassif, N.; Rappaz, J., On spectral approximation. part 1: the problem of convergence, RAIRO anal. numer., 12, 97-112, (1978) · Zbl 0393.65024
[12] Descloux, J.; Nassif, N.; Rappaz, J., On spectral approximation. part 2: error estimates for the Galerkin methods, RAIRO anal. numer., 12, 113-119, (1978) · Zbl 0393.65025
[13] Grisvard, P., Elliptic problems for non-smooth domains, (1985), Pitman · Zbl 0695.35060
[14] Kato, T., ()
[15] Mercier, B.; Osborn, J.; Rappaz, J.; Raviart, P.A., Eigenvalue approximation by mixed and hybrid methods, Math. comp., 36, 427-453, (1981) · Zbl 0472.65080
[16] Osborn, J., Spectral approximation for compact operators, Math. comp., 29, 712-725, (1975) · Zbl 0315.35068
[17] Rodríguez, R.; Solomin, J., The order of convergence of eigenfrequencies in finite element approximations of fluid – structure interaction problems, Math. comp., 65, 1463-1475, (1996) · Zbl 0853.65111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.