Trefftz discontinuous Galerkin methods
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Discontinuous Galerkin methods can be seen as finite element methods that allow for discontinuities across mesh elements in the test and trial spaces. Regularity and boundary conditions are enforced in a weak sense, by introducing suitable flux terms on the faces of the mesh. Localizing test and trial functions to single mesh elements leads to a compact discretization stencils, while also offering a lot of flexibility, making the method appealing for multi-physics and multi-domain problems. The price is a larger overall linear system, since the discontinuous Galerkin method uses more shape functions than the standard finite element method.
We can reduce the degrees of freedom used in the discontinuous Galerkin method considerably by using Trefftz spaces for the local test and trial spaces, instead of the standard polynomial spaces. Trefftz functions are chosen to be elementwise in the kernel of the corresponding differential operator, guaranteeing optimal convergence rates with fewer shape functions. We will consider the construction of Trefftz functions and their application to several "standard" PDE problems, as well as, possible extensions of the method to more complex cases.
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