An Augmented Two-Scale Finite Element Method for Eigenvalue Problems

In this paper, an augmented two-scale finite element method is proposed for a class of linear and nonlinear eigenvalue problems on tensor-product domains. Through a correction step, the augmented two-scale finite element solution is obtained by solving an eigenvalue problem on a low-dimensional augmented subspace. Theoretical analysis and numerical experiments show that the augmented two-scale finite element solution achieves the same order of accuracy as the standard finite element solution on a fine grid, but the computational cost required by the former solution is much lower than that demanded by the latter. The augmented two-scale finite element method also improves the approximation accuracy of eigenfunctions in the \(L^2(\varOmega )\) $L^2(\Omega )$ norm compared with the two-scale finite element method.