About

I'm a Flatiron Research Fellow in the Center for Computational Mathematics at the Flatiron Institute interested in numerical analysis and scientific computing. I received my Ph.D. in applied mathematics from Harvard University, where I was advised by professors Alex Townsend and Chris Rycroft. My research focuses on spectral methods, fast methods for PDEs, computational fluid & solid mechanics, and multigrid methods. 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 and Nick Hale, I have developed the ultraspherical spectral element method, a sparse spectral element method suitable for \(hp\)-adaptivity with very high \(p\). The method is based on the hierarchical Poincare–Steklov scheme and has a computational complexity of \(\mathcal{O}(p^4/h^3)\).

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). The solver is based on the ultraspherical polynomials and the alternating direction implicit (ADI) 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, N. Hale, and A. Townsend, The ultraspherical spectral element method, submitted (2020).   arXiv   PDF
  2. D. Fortunato, C. Rycroft, and R. Saye, Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods, SIAM J. Sci. Comput., 41 (2019), pp. A3913–A3937.   arXiv   DOI   PDF
  3. D. Fortunato and A. Townsend, Fast Poisson solvers for spectral methods, IMA J. Numer. Anal., 40 (2019), pp. 1994–2018.   arXiv   DOI   PDF
  4. A. Mijailovic, B. Qing, D. Fortunato, and K. Van Vliet, Characterizing viscoelastic mechanical properties of highly compliant polymers and biological tissues using impact indentation, Acta Biomaterialia, 71 (2018), pp. 388-397.   DOI   PDF

Talks

  • The ultraspherical spectral element method (2019)   PDF
  • Fast Poisson solvers for spectral methods (2019) [Awarded a second place Leslie Fox Prize]   PDF
  • Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods
    (2018)   PDF
  • Fast and accurate element methods (2017)   PDF
  • A fast spectrally accurate Poisson solver on rectangular domains (2017)   PDF

Software

I contribute to a number of open source software projects:

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

Projects