About

I'm a fourth-year graduate student in applied mathematics at Harvard University interested in numerical analysis and scientific computing. My advisors are professors Alex Townsend and Chris Rycroft. My research focuses on spectral methods, fast methods for PDEs, computational fluid & solid mechanics, and multigrid methods, and is funded by the National Defense Science and Engineering Graduate Fellowship. In the past, I have worked at Walt Disney Animation Studios, Lawrence Berkeley National Laboratory, Wolfram Research, and Apple.

Research

Spectral methods

In collaboration with Alex Townsend, I have developed spectrally accurate Poisson solvers for the square, cylinder, solid sphere, and cube that have optimal complexity (up to polylogarithmic factors). Our solver is based on the ultraspherical polynomials and the alternating direction implicit (ADI) method.

We are currently trying to extend our solver to an optimal complexity spectral \(hp\)-element method.

Computational fluid & solid mechanics

In collaboration with Chris Rycroft and Robert Saye, I am developing a high-order Eulerian method for fluid–solid interaction based on the discontinuous Galerkin and level set methods.

Multigrid methods

In collaboration with Robert Saye and Chris Rycroft, I have developed efficient multigrid schemes for local discontinuous Galerkin methods that are robust with respect to mesh size and polynomial order. The method is based on coarsening the divergence and gradient operators separately.

In collaboration with Rasmus Tamstorf, Steve McCormick, and Toby Jones, I implemented an algebraic multigrid method for cloth simulation based on adaptive smoothed aggregation.

Papers

  1. D. Fortunato, C. Rycroft, and R. Saye, Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods, submitted to SIAM Journal on Scientific Computing, 2018.   PDF
  2. D. Fortunato and A. Townsend, Fast Poisson solvers for spectral methods, submitted to SIAM Journal on Scientific Computing, 2017.   PDF

Talks

  • Efficient geometric multigrid methods for discontinuous Galerkin discretizations
    (ICOSAHOM 2018)   PDF
  • Fast and accurate element methods (2017)   PDF
  • A fast spectrally accurate Poisson solver on rectangular domains (SIAM CSE 2017)   PDF

Software

I contribute to a number of open source software projects:

  • Chebfun — A MATLAB package for numerical computing with functions. I implemented the fast Poisson solver used in chebfun2.poisson.
  • fast-poisson-solvers — A repository of fast spectrally accurate Poisson solvers on a variety of domains.

Projects