Università Cattolica del Sacro Cuore

Numerical analysis: partial differential equations

(Maurizio Paolini - Franco Pasquarelli)

One of the task of the numerical analysis is to find the approximate solutions of some complex problems for which an exact solution does not exist.

Such problems often involve partial differential equations and typically emerge from concrete situations (for example in phisics, engineering and finance) or from a purely theoretical-mathematical context.

The approximate solutions are obtained by means of computers, for this reason exist a natural interdisciplinary connection with computer science which is also present in the Department.

In this framework, the group has recently focus on the numerical simulation of the evolutions of fronts.

The paradigmatic model is the evolution of surfaces in the three dimensional space (or ipersurfaces in high dimensional spaces)  with a velocity proportional to the average local curvature.

In Figure 1 for example is shown a three dimensional section of a torus immersed in the four dimensional space that evolves in time with the law mentioned before.

In Figure 1 the ipersurface is shown at two different times: one of this is the time at which the hole of the torus closes and the surface changes its topology.

In Figure 2 is shown the result of the evolution of a surface (spherical at the beginning) under a similar kind of law but in a anisotropic environment (crystalline anisotropy in this case).

Key words

  • Phase transitions
  • Reactions-diffusion equations
  • Geometric evolution of surfaces
  • Anisotropy

Group leader

Collaborations

  • Università di Roma Tor Vergata (Italia)
  • Hokkaido University (Giappone)
  • University of Warsaw (Polonia)
  • Università di Pisa

Ongoing projects

  • PRIN05 - Metodi variazionali e topologici nello studio di fenomeni non lineari

Main pubblications

  • M. Paolini, F. Pasquarelli,
    Unstable Crystalline Wulff Shapes in 3D, Proceedings Variational methods for discontinuous structures, Progress in Nonlinear Differential Equations and Their Applications 51 (Birkhauser, Basel 2002), 141-153.
  • Y. Giga, M. Paolini, P. Rybka,
    On the motion by singular interfacial energy, Japan J. Indust. Appl. Math., 18, 47-64 (2001).
  • G. Bellettini, M. Novaga, M. Paolini,
    On a crystalline variational problem, part II: BV-regularity and structure of minimizers on facets, Arch. Ration. Mech. Anal. 3, 193-217 (2001).
  • G. Bellettini, M. Novaga, M. Paolini,
    On a crystalline variational problem, part I: first variation and global L-infinityregularity, Arch. Ration. Mech. Anal. 3, 165-191 (2001).
  • G. Bellettini, M. Novaga, M. Paolini,
    Facet-breaking for three dimensional crystals evolving by mean curvature, Interfaces and Free Boundaries 1, 39-55 (1999).