8 research outputs found
Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods
We present a numerical study to investigate the conditioning of the plane
wave discontinuous Galerkin discretization of the Helmholtz problem. We provide
empirical evidence that the spectral condition number of the plane wave basis
on a single element depends algebraically on the mesh size and the wave number,
and exponentially on the number of plane wave directions; we also test its
dependence on the element shape. We show that the conditioning of the global
system can be improved by orthogonalization of the local basis functions with
the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES
iterations for solving the discrete problem iteratively.Comment: Submitted as a conference proceeding; minor revisio
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A high frequency boundary element method for scattering by convex polygons
In this paper we propose and analyze a hybrid boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods
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Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version
We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns