Haslinger, J.; Neittaanmäki, P. On different finite element methods for approximating the gradient of the solution to the Helmholtz equation. (English) Zbl 0574.65123 Comput. Methods Appl. Mech. Eng. 42, 131-148 (1984). We consider the numerical solution of the Helmholtz equation by different finite element methods. In particular, we are interested in finding an efficient method for approximating the gradient of the solution. We first approximate the gradient by the standard Ritz-Galerkin method. As a second method a two-stage method due to A. Aziz and A. Werschulz [SIAM J. Numer. Anal. 17, 681-686 (1980; Zbl 0466.65061)] is presented. It is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case. Finally, a direct method with asymptotic error estimates is given. It turns out that the presented direct method is of lowest computational complexity. Test examples are presented to illustrate the accuracy of the methods. Cited in 5 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:Helmholtz equation; finite element methods; gradient; Ritz-Galerkin method; asymptotic error estimates; computational complexity; Test examples Citations:Zbl 0466.65061 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aziz, A. K.; Werschulz, A., On numerical solution of Helmholtz’s equation by the finite element method, SIAM J. Numer. Anal., 17, 681-686 (1980) · Zbl 0466.65061 [2] Ciarlet, P. 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