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M. Huber:
"Hybrid discontinuous Galerkin methods for th wave equation";
Supervisor, Reviewer: J. Schöberl, S. Rotter, G. Karigl; Institut für Analysis und Scientific Computing, 2013; oral examination: 2013-05-03.

In this thesis, we investigate hybrid discontinuous Galerkin finite element methods (FEM) for the scalar and vector valued wave equation.
In these methods the continuity of basis functions is broken across element facets, i.e., the interfaces between them.
A continuous solution is reinforce via Lagrange multipliers supported only on element facets.
For the wave equation a second set of multipliers is necessary to eliminate the original degrees of freedom cheaply element by element.
This approach allows to reduce the system of equations to a much smaller system just for the Lagrange multipliers.
Apart from this, the work presents an optimized implementation technique of the hybrid FEM for the two dimensional Helmholtz equation, which is based on an eigenfunction basis.
For rectangular meshes the construction of such a basis requires the solution of a one dimensional eigenvalue problem for each pair of edge length and polynomial order.
The eigenvalue problem can be solved for polynomial orders up to thousands. Combining this with the cheap assembly and the computationally inexpensive elimination of the interior degrees of freedom, we are able to use very high order basis functions efficiently.
By allowing for hanging nodes, we can benefit from exponential convergence of [hp]-methods.
A very challenging point is solving the resulting system of equations. Since the hybrid formulation provides appropriate interface conditions, an efficient iterative solution with Krylov space methods combined with domain decomposition preconditioners is possible.
Apart from multiplicative and additive Schwarz block preconditioners with local smoothers or an element wise BDDC preconditioner, a new Robin type domain decomposition preconditioner is constructed.
This preconditioner solves in each iteration step local problems on subdomains by directly inverting the system matrix.
Thus, it is well suited for parallel computations.
Good convergence properties of these iterative solvers are demonstrated by numerical experiments.

http://publik.tuwien.ac.at/files/PubDat_225277.pdf