## Research Abstracts Online

January 2009 - March 2010

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### University of Minnesota Twin Cities

Institute of Technology

School of Mathematics

# PI: Bernardo Cockburn, Fellow

### Hybridized Discontinuous Galerkin Methods for Curved Domains

In order to develop new techniques that contribute to the improvement of numerical analysis methods to solve partial differential equations, these researchers have developed a new way to handle finite element approximations for problems where the domain eventually could have a curved boundary, i.e., when it has no piecewise flat boundary. The main idea is to approximate the actual domain by a polyhedral subdomain and to compute the solution inside of the polyhedral subdomain using the hybridizable discontinuous Galerkin method (HDG). The solution is then extended, in a suitable way, to the whole domain.

The researchers use a two-dimensional second-order elliptic equation as a model problem. The method works properly in this case, which means that they can obtain an accurate solution with the expected rates of convergence. The next step is to extend the technique to other kinds of problems like convection-diffusion, coupling between discontinuous Galerkin and boundary-element methods for exterior problems, or the immersed boundary method.

### Group Members

Ivan Merev, Graduate Student

Ke Shi, Graduate Student

Manual Solano, Graduate Student

Wujun Zhang, Graduate Student