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Convergence of lowest order semi-Lagrangian schemes
(2011)SAM Research ReportWe consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in $R^n$. We provide comprehensive a priori convergence estimates for their spatio-temporal discretization by means of a semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic $L^2$-estimate $O(\tau + h^r + h^{r+1} ...Report -
Shape derivatives in differential forms I: An intrinsic perspective
(2011)SAM Research ReportWe treat Zolesio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher order shape derivatives of domain and boundary integrals and skirts the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective ...Report -
Boundary integral formulation of the first kind for acoustic scattering by composite structures
(2011)SAM Research ReportWe study the scattering of an acoustic wave by an object composed of several adjacent sub-domains with different material properties. For this problem we derive an integral formulation of the first kind. This formulation involves two Dirichlet data and two Neumann data at each point of each interface of the diffracting object. This formulation is immune to spurious resonances, and it satisfies a stability property that ensures quasi optimal ...Report -
Semi-lagrangian methods for advection of differential forms
(2011)SAM Research ReportWe study the discretization of linear transient transport problems for differential forms on bounded domains. The focus is on semi-Lagrangian methods that employ finite element approximation on fixed meshes combined with tracking of the flow map. They enjoy unconditional stability. We derive these methods as finite element Galerkin approach to discrete material derivatives and discuss further approximations. An a priori convergence ...Report