报告摘要：
We present new finite elements for solving the Riesz maps of the de Rham complex on triangular and tetrahedral meshes at high order. The finite elements discretize the same spaces as usual, but with different basis functions, so that the resulting matrices have desirable properties. These properties mean that we can solve the Riesz maps to a given accuracy in a p-robust number of iterations with O(p⁶) flops in three dimensions, rather than the naive O(p⁹) flops.
The degrees of freedom build upon an idea of Demkowicz et al., and consist of integral moments on an equilateral reference simplex with respect to a numerically computed polynomial basis that is orthogonal in two different inner products. As a result, the interior-interface and interior-interior couplings are provably weak, and we devise a preconditioning strategy by neglecting them.
The combination of this approach with a space decomposition method on vertex and edge star patches allows us to efficiently solve the canonical Riesz maps at high order. We apply this to solving the Hodge Laplacians of the de Rham complex with novel augmented Lagrangian preconditioners.
报告人介绍：
Patrick Farrell is a Professor in the Numerical Analysis group at the Mathematical Institute, University of Oxford, and a Tutorial Fellow at Oriel College, Oxford. For the 2025–2026 academic year, he also holds the Donatio Universitatis Carolinæ Chair at the Faculty of Mathematics and Physics at Charles University in Prague. His research concerns the numerical solution of partial differential equations, with a particular focus on finite element methods, bifurcation analysis of nonlinear PDEs, adjoint techniques (including their application and automation), preconditioners, and fast solvers. He has applied these numerical techniques to diverse fields, including mixtures, renewable energy, cardiac electrophysiology, glaciology, magnetohydrodynamics, quantum mechanics, and liquid crystals. Professor Farrell is an Invited Lecturer for the 2026 International Congress of Mathematicians (ICM). His awards include the Germund Dahlquist Prize (2025), the Whitehead Prize (2021), the Broyden Prize in Optimization (2021), and the Wilkinson Prize for Numerical Software (2015). He serves as an Associate Editor for Mathematics of Computation and the SIAM Journal on Scientific Computing, and is the Editor of the Numerical Mathematics and Scientific Computing book series (OUP).
