×

zbMATH — the first resource for mathematics

Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: Computation of the constant and new measures of error estimator quality. (English) Zbl 0887.76040
This paper continues the study of an explicit residual-based a posteriori error estimator developed for finite element discretizations of the Helmholtz equation, in the context of time-harmonic exterior acoustics problems. A methodology is established for computing error estimates; these error estimates are then analyzed in detail. The error estimates are made possible by computing the (scaling) constant, which exists in error estimators of this type. In our case, this constant is difficult to obtain. An algorithm for computing the constant is described. (It is noted that the error indicators used for adaptive computations do not require knowledge of this constant.) Several measures are used to analyze the quality of the error estimates, providing a complete description of the error estimator in the context of these problems. We compute global effectivity indices, and utilize the local (element-wise) effectivity index in computing three additional measures. The analysis of the error estimator is carried out on relatively coarse meshes, typical of those used in practical engineering computations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76Q05 Hydro- and aero-acoustics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.R. Stewart and T.J.R. Hughes, An a posteriori error estimator and hp-adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains, Finite Elem. Anal. Des., to appear. · Zbl 0897.65065
[2] J.R. Stewart and T.J.R. Hughes, h-adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions, J. Acoust. Soc. Am., in press. · Zbl 0901.76038
[3] Stewart, J.R.; Hughes, T.J.R., Adaptive finite element methods for the Helmholtz equation in exterior domains, ()
[4] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. methods appl. mech. engrg., 101, 143-181, (1992) · Zbl 0778.73071
[5] Johnson, C., Finite element methods for flow problems, (), I-1-I-47
[6] Eriksson, K.; Johnson, C., An adaptive finite element method for linear elliptic problems, Math. comput., 50, 361-383, (1988) · Zbl 0644.65080
[7] Johnson, C., Adaptive finite element methods for diffusion and convection problems, Comput. methods appl. mech. engrg., 82, 301-322, (1990) · Zbl 0717.76078
[8] Eriksson, K.; Johnson, C., Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. numer. anal., 24, 12-23, (1987) · Zbl 0618.65104
[9] Wriggers, P.; Scherf, O.; Carstensen, C., Adaptive techniques for the contact of elastic bodies, (), 78-86, A book dedicated to Robert L. Taylor · Zbl 1122.74528
[10] Harari, I.; Hughes, T.J.R., Design and analysis of finite element methods for the Helmholtz equation in exterior domains, Appl. mech. rev., ASME, 43, 2, 366-373, (1990)
[11] Harari, I., Computational methods for problems of acoustics with particular reference to exterior domains, ()
[12] Harari, I.; Hughes, T.J.R., Galerkin least squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains, Comput. methods appl. mech. engrg., 98, 411-454, (1992) · Zbl 0762.76053
[13] Harari, I.; Hughes, T.J.R., Finite element methods for the Helmholtz equation in an exterior domain: model problems, Comput. methods appl. mech. engrg., 87, 59-96, (1991) · Zbl 0760.76047
[14] Thompson, L.L.; Pinsky, P.M., A multi-dimensional Galerkin least squares finite element method for time-harmonic wave propagation, (), 444-451, Chapter 47 · Zbl 0814.65103
[15] Thompson, L.L.; Pinsky, P.M., A Galerkin least squares finite element method for the two-dimensional Helmholtz equation, Int. J. numer. methods engrg., 38, 371-397, (1995) · Zbl 0844.76060
[16] Thompson, L.L., Design and analysis of space-time and Galerkin least squares finite element methods for fluid-structure interaction in exterior domains, ()
[17] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation of computational fluid dynamics: VIII. the Galerkin least squares method for advective-diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[18] Shakib, F.; Hughes, T.J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin least squares algorithms, Comput. methods appl. mech. engrg., 87, 35-58, (1991) · Zbl 0760.76051
[19] Babuška, I.; Sauter, S., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers, () · Zbl 0894.65050
[20] Babuška, I.; Strouboulis, T.; Mathur, A.; Upadhyay, C.S., Pollution error in the h-version of the finite element method and the local quality of a posteriori error estimators, () · Zbl 0924.65098
[21] Babuška, I.; Strouboulis, T.; Upadhyay, C.S.; Gangaraj, S.K., A posteriori estimation and adaptive control of the pollution-error in the h-version of the finite element method, () · Zbl 0844.65078
[22] Oden, J.T.; Feng, Y., Local and pollution error estimation for finite element approximations of elliptic boundary value problems, () · Zbl 0871.65094
[23] Babuška, I.; Strouboulis, T.; Upadhyay, C.S.; Gangaraj, S.K.; Copps, K., Validation of a posteriori error estimators by numerical approach, Int. J. numer. methods engrg., 37, 1073-1123, (1994) · Zbl 0811.65088
[24] Babuška, I.; Strouboulis, T.; Upadhyay, C.S., A model study of the quality of a posteriori error estimators for linear elliptic problems. error estimation in the interior of patchwise uniform grids of triangles, Comput. methods appl. mech. engrg., 114, 307-378, (1994)
[25] Keller, J.B.; Givoli, D., An exact nonreflecting boundary condition, J. comput. phys., 82, 172-192, (1989) · Zbl 0671.65094
[26] Harari, I.; Hughes, T.J.R., Studies of domain-based formulations for computing exterior problems of acoustics, Int. J. numer. methods engrg., 37, 2935-2950, (1994) · Zbl 0818.76040
[27] Ciarlet, P.G., The finite element methods for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[28] Oñate, E.; Bugeda, G., A study of mesh optimality criteria in adaptive finite element analysis, Engrg. comput., 10, 307-321, (1993)
[29] Peraire, J.; Vahdati, M.; Mohan, K.; Zienkiewicz, O.C., Adaptive remeshing for compressible flow computations, J. comput. phys., 72, 449-466, (1987) · Zbl 0631.76085
[30] Ihlenburg, F.; Babuška, I., Dispersion analysis and error estimation of Galerkin finite element methods for the numerical computation of waves, ()
[31] P. Hansbo, Private communication.
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.