In this work we present a technique to numerically solve partial differential equations
(PDE’s) defined in general domains . It basically consists in approximating the domain
by polyhedral subdomains Dh and suitably defining extensions of the solution from
Dh to . More precisely, we solve the PDE in Dh by using a numerical method for
polyhedral domains. In order to do that, the boundary condition is transferred from
¡ := ∂ to ¡h, the boundary of Dh, by integrating the gradient of the scalar variable
along a path. That is why, in principle, any numerical method that provides an accurate
approximation of the gradient can be used. In this work we consider a hybridizable
discontinuous Galerkin (HDG) method.
This technique has two main advantages over other methods in the literature. First
of all, it only requires the distance between ¡ and ¡h to be of the order of the meshsize.
This allows us to easily mesh the computational domain. Moreover, high degree
polynomial approximations can be used and still obtain optimal orders of convergence
even though ¡h is “far” from ¡.
We numerically explore this approach by considering three types of steady-state
equations. As starting point, we deal with Dirichlet boundary problems for second
order elliptic equations. For this problem we fully explain how to properly transfer the
boundary condition and how to define the paths, as well. We then apply this technique
to exterior diffusion problems. Herein, the HDG method is used for solving the so-called
interior problem on a bounded region whereas a boundary element method (BEM) is
used for solving the problem exterior to that region. Both methods are coupled at
the smooth interface that divides the two regions. Finally, we consider convectiondiffusion
problems where we explore how the magnitude of the convective field affects
the performance of our method.
University of Minnesota Ph.D. dissertation. January 2012. Major: Scientific Computation. Advisor: Bernardo Cockburn. 1 computer file (PDF); vii, 75 pages.
Solano Palma, Manuel Esteban.
Hybridizable discontinuous Galerkin method for curved domains.
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