Event Type:

Department Colloquium

Date/Time:

Friday, May 27, 2016 - 12:00 to 13:00

Location:

GILK 113

Guest Speaker:

Institution:

Los Alamos National Laboratory

Abstract:

This is a joint Colloquium/Applied Math and Computation seminar talk.

Finite Element (FE) methods constitute a popular numerical technique for finding approximate solutions to partial differential equations (PDE). It starts by partitioning the domain into smaller parts, called elements, approximating the PDE on each element and assembling these approximations into a single finite dimensional problem. The elements typically are simple polygons and polyhedra, most often - triangles and tetrahedra. Extension of the FE construction to more general polygonal and polyhedral elements is highly non-trivial. Virtual Element (VE) methods represent a new approach to construction of discretizations on general polygonal and polyhedral meshes. Historically VE methods are an evolution of ideas coming from Mimetic Finite Difference (MFD) methods recast into the FE framework. The resulting methods can be viewed as a generalization of FE methods.

In this talk, after a general introduction to VE methods, we will focus on our recent work on Virtual Element (VE) discretization for incompressible steady Stokes equations in the velocity-pressure formulation. The present approach makes it possible to construct nonconforming VE spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. The non-conforming formulation allows for a simpler construction of the discretization in 3D compared to a conforming one. We will show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Finally we will present numerical examples that confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

This is a joint work with Andrea Cangiani from the Department of Mathematics, University of Leicester and Gianmarco Manzini who has a joint affiliation with Los Alamos National Laboratory and the Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI).

Host: