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Plane Wave Discontinuous Galerkin Methods
(2013)Research ReportWe 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 ...Report -
Fast Convolution Quadrature Based Impedance Boundary Conditions
(2013)We consider an eddy current problem in time-domain relying on impedance boundary conditions on the surface of the conductor(s). We pursue its full discretization comprising (i) a finite element Galerkin discretization by means of lowest order edge elements in space, and (ii) temporal discretization based on Runge-Kutta convolution quadrature (CQ) for the resulting Volterra integral equation in time. The final algorithm also involves the ...Report -
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Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation
(2012)Research ReportThe nonlinear spatially homogeneous integro-differential Boltzmann equation is a uniquely challenging task for numerical solvers due to the difficulty of efficiently computing the collision operator. A popular method is to expand the solution in Fourier modes and to truncate the collision operator. We present an approach based on the hyperbolic cross, whereby the performance can be greatly enhanced in some situations, as well as an offset ...Report -
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Dispersion Analysis of Plane Wave Discontinuous Galerkin Methods
(2012)Research ReportThe plane wave discontinuous Galerkin (PWDG) method for the Helmholtz equation was introduced and analyzed in [Gittelson, C., Hiptmair, R., and Perugia, I. Plane wave discontinuous Galerkin methods: Analysis of the h-version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so-called ultra-weak variational formulation, see [O. Cessenat and B. Despr´es, Application of an ultra weak variational formulation of elliptic ...Report -
Multi-trace boundary integral equations
(2012)We consider the scattering of acoustic or electromagnetic waves at a penetrable object composed of different homogeneous materials. This problem can be recast as a firstkind boundary integral equation posed on the interface trace spaces through what we call a single trace boundary integral equation formulation (STF). Its Ritz-Galerkin discretization by means of low-order piecewise polynomial boundary elements on fine interface triangulations ...Report