To give accurate eigenvalue evaluation for elliptic operator, the classical methods, such as Nokao's method, intermediate method, homotopy method and so on, are not so easy to be implemented in cases of general domain. In this paper, starting from an early work of G. Birkhoff (1966), I propose a new framework to give guaranteed estimation for leading n-th eigenvalue on arbitrary polygonal domain, where the finite element method (FEM) is used to give approximate eigenvalues with computable error bounds. At the end of this paper, we show several computation results, including the case of L-shaped domain, to demonstrate the efficiency of the proposed method.
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