Convergence and superconvergence of Hermite bicubic element for eigenvalue problem of the biharmonic equation (Q2725279)

Suppose \(\lambda\) is an eigenvalue of the biharmonic operator with Dirichlet homogeneous boundary conditions in \(H^2_0(\Omega)\) and \(u\) the corresponding eigenfunction, and let \(V_h\subset H^2_0(\Omega)\) be its finite element Hermite bicubic discretization. Assume that \(v_h\) is the corresponding approximation to \(u\) and that \(\lambda_h\) the approximation to \(\lambda\). The author proves that NEWLINE\[NEWLINE0\leq \lambda_h- \lambda\leq Ch^4;NEWLINE\]NEWLINE and if a special uniform partition is used NEWLINE\[NEWLINE0\leq \lambda_h- \lambda\leq C(\varepsilon) h^{8-\varepsilon}.NEWLINE\]