Xuefeng Liu Monday 28th Nov at 11:00
We are concerned with the guaranteed computation of the eigenfunction (or eigenspace) of differential operators. Upon the setting of eigenvalue problems, three algorithms are proposed to give rigorous and efficient error estimation for the approximate eigenfunctions: Algorithm I: Rayleigh quotient error based algorithm; Algorithm II: Variational residual error based algorithm; Algorithm III: Projection error based algorithm.
The guaranteed eigenfunction computation is applied to solving shape optimization problems. For example, the minimization of the Laplacian eigenvalue over polygonal domains. By explicitly evaluating the Hadamard shape derivative with guaranteed computation of both eigenvalues and eigenfunctions, we provide a computer-assisted proof to declare that the equilateral triangle minimizes the first Laplacian under certain non-homogeneous Neumann boundary condition under the radius constraint condition.
1. R. Endo and X. Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation, https://arxiv.org/abs/2209.13415.
2. X. Liu and T. Vejchodsky, Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions, https://arxiv.org/abs/2211.03218.
3.X. Liu and T. Vejchodsky, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152 (2022), 183–221.