Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in $H^m$. Moreover, we show the existence of a unique $H^m$-solution if the coefficients and the data are accordingly regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence conditions in the Maxwell equations in order to derive the energy-type $H^m$-estimates. The combination of these estimates with several regularization techniques then yields the existence of solutions in $H^m$.Comment: 44 pages; typo correcte

Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the Brezzi-Pitk\"aranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the $L^2$-convergence of the components of the velocity, and, in the steady case, the weak $L^2$-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov's theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.Comment: submitted September 0

Classical solutions of drift-diffusion equations for semiconductor devices: the 2d case

We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing.--This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation