Author(s)

Isaac Fried, Roberto Riganti, Chen Yu

Department of Mathematics, Boston University, Boston, MA, USA.

The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, predictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.

Finite Elements, Global Stiffness Matrix, Gershgorin and Rayleigh Computed Upper and Lower Bounds on the Extremal Eigenvalues, Similarity Transformations

Fried, I. , Riganti, R. and Yu, C. (2020) Gershgorin and Rayleigh Bounds on the Eigenvalues of the Finite-Element Global Matrices via Optimal Similarity Transformations. Applied Mathematics, 11, 922-941. doi: 10.4236/am.2020.119060.