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Computing the lower and upper bounds of Laplace eigenvalue problem by combining conforming and nonconforming finite element methods. (English) Zbl 1261.65112

The authors consider the two-dimensional Laplace eigenvalue problem in a domain guaranteeing a generalized eigenfunction in \(H^{1+\gamma},\,0<\gamma\leq 1\). They propose to solve it using the extended Crouzeix-Raviart element (on triangles) or its rotated version (extended Rannacher-Turek element, on rectangles) and prove (relying on an identity for \(\lambda-\lambda_h\) obtained by Z. Zhang et al. [Math. Numer. Sin. 29, No. 3, 319–321 (2007; Zbl 1142.65435)] that both lead to lower approximations \(\underline\lambda_h\) of the eigenvalues. For small enough \(h\), the convergence order is \(2\gamma\). Upper bounds can be obtained taking conforming elements, but the authors propose a cheaper approach, to take the results \(\lambda_hu_h\) of the nonconforming elements as right-hand sides in a source problem solved in the (linear) conforming space. This method is generalized to higher order conforming elements where a Ritz step is added on the space spanned by the first \(m\) eigenfunctions. For the first resp. second, higher order approach (then assuming \(u\in H^{1+2\gamma}\)) they prove the convergence order of \(\overline\lambda_h-\underline\lambda_h\) to be \(2\gamma\) resp. \(4\gamma\) assuming small enough \(h\) and \(\frac12<\gamma\). Numerical results illustrate the theory and show also that the methods cope with multiple eigenvalues.

MSC:

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
65N15 Error bounds for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Citations:

Zbl 1142.65435
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References:

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