Lower spectral bounds by Wilson’s brick discretization. (English) Zbl 1198.65220

A nonconforming finite element, the Wilson element, is applied for solving the eigenvalue problem of the Laplace operator in polygonal domains in two and three space dimensions. A relationship between the interpolation weak estimate of the Wilson element and the n-linear element is established. The authors prove that the three-dimensional Wilson’s brick eigenvalues approximate the exact eigenvalues from below, which is claimed to be a new progress for the finite element method. A numerical experiment for the eigenvalues of the Laplace operator in three dimensions, considering a unit cube, shows that the results given by the Wilson’s brick yield indeed lower spectral bounds and thus confirm the theoretical results.


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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