For more information about this meeting, contact Mary Anne Raymond.
|Title:||Approximations on general geometries|
|Seminar:||Marker Lecture Series|
|Speaker:||Professor Franco Brezzi, Istituto Universitario di Studi Superiori (IUSS)|
|The typical approach of numerical methods for partial differential equations is to choose a finite dimensional space and look, in this space, for an element that solves a problem that is similar to the original “physical” one (or, rather, to its mathematical model). All this is more easily done when the computational domain has a simple geometry and/or is decomposed into sub domains of simple geometry (squares, triangles, and the like). On more complicated geometries many additional difficulties arise, and several of them are related to the choice of convenient finite dimensional functional spaces (mostly polynomials or trigonometric polynomials suitably adapted to the shape of each subdomain).
In recent times a new approach appeared on the scene. It consists, initially, in considering that one is using cochains (as vertex values, works along edges, fluxes through faces, etc) as already done in finite difference and in finite volumes. However it then combines this with weak formulations that would require the knowledge of the functions (or vector valued functions) all over the domain. Apparently this brings back the original problem of finding convenient finite dimensional spaces in each subdomain. However one could get away without doing that, taking just care to produce in each subdomain a scheme that is exact only on small classes of polynomials, as constants vectors or linear functions or similar simple objects, depending on the problem.
All this could be seen as a possible way of putting the celebrated patch test (a clever engineering trick that gives reliable indications on the convergence of a given numerical approach) on solid mathematical foundations|
Room Reservation Information
|Date:||04 / 19 / 2010|
|Time:||08:00pm - 10:00pm|