New expansions of numerical eigenvalues for -Δu = λρu by nonconforming elements

The paper explores new expansions of the eigenvalues for −∆u = λρu in S with Dirichlet boundary conditions by the bilinear element (denoted Q1) and three nonconforming elements, the rotated bilinear element (denoted Qrot 1 ), the extension of Q rot 1 (denoted EQ rot 1 ) and Wilson’s elements. The expansions indicate that Q1 and Qrot 1 provide upper bounds of the eigenvalues, and that EQrot 1 andWilson’s elements provide lower bounds of the eigenvalues. By extrapolation, the O(h4) convergence rate can be obtained, where h is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.