  By combining concepts from particle-in-cell (PIC) and hybridized
discontinuous Galerkin (HDG) methods, we present a particle-mesh scheme which
allows for diffusion-free advection, satisfies mass and momentum conservation
principles in a local sense, and allows the extension to high-order spatial
accuracy. To achieve this, we propose a novel particle-mesh projection operator
required for the exchange of information between the particles and the mesh.
Key is to cast these projections as a PDE-constrained $\ell^2$-optimization
problem to allow the advective field naturally located on Lagrangian particles
to be expressed as a mesh quantity. By expressing the control variable in terms
of single-valued functions at cell interfaces, this optimization problem
seamlessly fits in a HDG framework. Owing to this framework, the resulting
scheme can be implemented efficiently via static condensation. The performance
of the scheme is demonstrated by means of various numerical examples for the
linear advection-diffusion equation and the incompressible Navier-Stokes
equations. The results show that optimal spatial accuracy can be achieved, and
given the particular time-stepping strategy, second-order time accuracy is
confirmed. The robustness of the scheme is illustrated by considering
benchmarks for advection of discontinuous fields and the Taylor-Green vortex
instability in the high Reynolds number regime.
