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Prikazchikov, V.

Efficient two-sided estimates for the spectrum of some elliptic operators. (English. Ukrainian original) Zbl 1505.65285

Cybern. Syst. Anal. 58, No. 3, 417-428 (2022); translation from Kibern. Sist. Anal. 58, No. 3, 111-122 (2022).
Summary: Using the principle of maximum, we establish the upper and lower bounds for the spectrum of some elliptic operators and their grid analogs. More accurate estimates of the spectrum of differential operators are obtained from the exact formulas for the error of the eigenvalues by the finite-difference method. Two-sided estimates of the eigenvalues of difference analogs of spectral problems give a majorant and a minorant for the error of the phase velocities of grid waves in vibration problems for various objects.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
49K35 Optimality conditions for minimax problems
35P15 Estimates of eigenvalues in context of PDEs
35B50 Maximum principles in context of PDEs
35J15 Second-order elliptic equations

Keywords:

elliptic operators; maximum principle; finite-difference method; exact formulas for errors of eigenvalues; two-sided estimates of spectrum; oscillation equations; error of phase velocities of grid waves
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References:

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