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An a posteriori error estimate and a comparison theorem for the nonconforming \(P_{1}\) element. (English) Zbl 1192.65142

Author’s abstract: A posteriori error estimates for the nonconforming \(P_{1}\) element are easily determined by the hypercircle method via L. Donatella Marini’s observation on the relation to the mixed method of Raviart-Thomas [SIAM J. Numer. Anal. 22, 493–496 (1985; Zbl 0573.65082)]. Another tool is M. Ainsworth’s application of the hypercircle method to mixed methods [SIAM J. Sci. Comput. 30, No. 1, 189–204 (2007; Zbl 1159.65353)]. The relation on the finite element solutions is also extended to an a priori relation of the errors, and the errors of four different finite element methods can be compared.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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