Computational Mathematics and Scientific Computing Seminar

Time and Location:

Speaker:

Link:

In the first part of the talk, we present a Newton-based free-boundary Grad–Shafranov (GS) solver using adaptive finite elements and preconditioning strategies. The free-boundary coupling introduces a domain-dependent nonlinear form, with Jacobian contributions derived via shape calculus. Key components include the treatment of global constraints, a nonlocal reformulation, and adaptive mesh refinement. The resulting Newton solver is robust, reducing the nonlinear residual below 1e-6 in a few iterations, even for challenging configurations where Picard-based methods often fail.

In the second part, we turn to kinetic modeling and particle dynamics on finite element fields. We propose a structure-preserving reduction of the Vlasov–Poisson system based on a novel macro-particle representation that maintains the Hamiltonian structure by construction. Building on ideas of Scovel and Weinstein (1994), we formulate reduced models through an associated Poisson bracket, enabling a systematic transition from infinite- to finite-dimensional dynamics. Theoretical results on model reduction are presented alongside numerical experiments that highlight the advantages of this approach over traditional particle-in-cell methods.