Date/time: Thursday Feb 22, 1-2pm

Room: FAB 1146 (Nelson Library)

Speaker: Xuefeng Liu (Tokyo Woman's Christian University)

 Title: Guaranteed computation for Hadamard derivative and its application in finite element error analysis

 Abstract: In error analysis of finite element methods, various error constants are present, whose optimal values are determined by solving shape optimization problems of eigenvalues.

The derivative of eigenvalues with respect to domain variation, known as the shape derivative, plays a crucial role in addressing challenges in these shape optimization problems.

Hadamard’s foundational work in the early 20th century laid the theoretical groundwork for these shape derivatives.

However, the direct estimation of shape derivatives remains a significant challenge due to the requirements for rigorous estimation of eigenvalues and eigenfunctions,especially in cases of repeated or closely spaced eigenvalues. 

This presentation introduces a novel computational method for rigorously evaluating the Hadamard derivative of Laplacian eigenvalues.

The method employs recently developed rigorous computation algorithms for both eigenvalues (Liu'2013,2015) and eigenfunctions (Liu-Vejchodský'2022) using the finite element method.

As an application of this method, we provide a computer-assisted proof for a type of shape optimization problem for the Laplacian eigenvalues.

This proof determines the optimal interpolation error constant for the Crouzeix-Raviart finite element (Endo-Liu'2023).

References:

1) Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, Vol. 267, 2015, pp. 341-355, https://doi.org/10.1016/j.amc.2015.03.048.

2) Xuefeng Liu and Tomas Vejchodský, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152, 183–221 (2022). https://doi.org/10.1007/s00211-022-01304-0

3) Ryoki Endo, Xuefeng Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis, Journal of Differential Equations, Volume 376, 2023, pp.750-772, https://doi.org/10.1016/j.jde.2023.09.016.