Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption, Part 2: Lagrange interpolation on tetrahedrons

This is the second lecture note on the error analysis of interpolation on simplicial elements without the shape regularity assumption (the previous one is arXiv:1908.03894). In this manuscript, we explain the error analysis of Lagrange interpolation on (possibly anisotropic) tetrahedrons. The manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.Comment: arXiv admin note: text overlap with arXiv:2102.0476

Hadamard variation of eigenvalues with respect to general domain perturbations

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite dimensional spaces. Then smooth rearrangement of multiple eigenvalues is explicitly given. This result follows from an abstract theory, applicable to general perturbations of symmetric bilinear forms