Chang, C. L. A least-squares finite element method for the Helmholtz equation. (English) Zbl 0726.65121 Comput. Methods Appl. Mech. Eng. 83, No. 1, 1-7 (1990). The author proposes a new finite element method in order to solve the Helmholtz equation for both the solution and its gradient. He modifies an idea proposed by G. J. Fix, M. D. Gunzburger and R. A. Nicolaides [Comput. Math. Appl. 5, 87-98 (1979; Zbl 0422.65064)] consisting in the transformation of the original problem into a first order system; the system is then dealt with an appropriate Sobolev space, where a convenient bilinear operator is shown to be coercive, thus allowing the application of well-known results leading to a priori error estimates. A numerical example is included to demonstrate the convergence properties of the proposed method. Reviewer: J.P.Milaszewicz (Buenos Aires) Cited in 27 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 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:finite element method; Helmholtz equation; first order system; Sobolev space; error estimates; numerical example; convergence Citations:Zbl 0422.65064 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] G.J. Fix, M.D. Gunzburger and R.A. Nicolaides, On finite element methods of the least squares type, Comput. Math. Appl.5, 87-98.; G.J. Fix, M.D. Gunzburger and R.A. Nicolaides, On finite element methods of the least squares type, Comput. Math. Appl.5, 87-98. · Zbl 0422.65064 [2] Aziz, A. K.; Werschulz, A., On the numerical solution of Helmholtz’s equation by the finite element method, SIAM J. Numer. Anal., 17, 5, 681-686 (October 1980) · Zbl 0466.65061 [3] Haslinger, J.; Neittaanmäki, P., On different finite element methods for approximating the gradient of the solution to the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 42, 131-148 (1984) · Zbl 0574.65123 [4] Ciarlet, P. G., The Finite Element Method for Elliptic Problem (1977), North-Holland: North-Holland Amsterdam [5] Oden, J. T.; Carey, G. F., Finite Elements (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0558.73064 [6] Wendland, W. L., Elliptic Systems in the Plane (1979), Pitman: Pitman London · Zbl 0396.35001 [7] Chang, C. L., A finite element method for first order elliptic systems in three dimensions, Applied Math. Comput., 23, 171-184 (1987) · Zbl 0631.65108 [8] C.L. Chang and M.D. Gunzburger, A subdomain collocation/least-squares method for first order elliptic systems in the plane, SIAM J. Numer. Anal., to appear.; C.L. Chang and M.D. Gunzburger, A subdomain collocation/least-squares method for first order elliptic systems in the plane, SIAM J. Numer. Anal., to appear. · Zbl 0717.65087 [9] Agmon, A.; Douglis, A.; Nirenbert, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17, 35-92 (1964) · Zbl 0123.28706 [10] Dikanskij, A. S., Conjugate problems of elliptic differential and pseudo-differential boundary value problems in a bounded domain, Math. USSR-Sb., 20, 67-83 (1973) · Zbl 0282.35075 [11] Douglis, A.; Nirenberg, L., Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8, 503-538 (1955) · Zbl 0066.08002 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.