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- ItemError estimates of B-spline based finite-element method for the wind-driven ocean circulation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Rotundo, Nella; Kim, Tae-Yeon; Jiang, Wen; Heltai, Luca; Fried, EliotWe present the error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale winddriven ocean circulation. In particular, we derive optimal error estimates for h-refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and StommelMunk models. Numerical results on rectangular and embedded geometries confirm the error analysis.
- ItemOn thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Koprucki, Thomas; Rotundo, Nella; Farrell, Patricio; Doan, Duy Hai; Fuhrmann, JürgenDriven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.
- ItemDoping optimization for optoelectronic devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Peschka, Dirk; Rotundo, Nella; Thomas, MaritaWe present a mathematical and numerical framework for the optimal design of doping profiles for optoelectronic devices using methods from mathematical optimization. With the goal to maximize light emission and reduce the thresholds of an edge-emitting laser, we consider a driftdiffusion model for charge transport and include modal gain and total current into a cost functional, which we optimize in cross sections of the emitter. We present 1D and 2D results for exemplary setups that point out possible routes for device improvement.
- ItemMathematical modeling of semiconductors: From quantum mechanics to devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Kantner, Markus; Mielke, Alexander; Mittnenzweig, Markus; Rotundo, NellaWe discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.
- ItemNumerical methods for drift-diffusion models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farrell, Patricio; Rotundo, Nella; Doan, Duy Hai; Kantner, Markus; Fuhrmann, Jürgen; Koprucki, ThomasThe van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions.
- ItemTowards doping optimization of semiconductor lasers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Peschka, Dirk; Rotundo, Nella; Thomas, MaritaWe discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.
- ItemGradient structure for optoelectronic models of semiconductors(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Peschka, Dirk; Rotundo, Nella; Thomas, MaritaWe derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon- densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.
- ItemError estimates in weighted Sobolev norms for finite element immersed interface methods(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heltai, Luca; Rotundo, NellaWhen solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.
- ItemExistence and uniqueness of solution for multidimensional parabolic PDAEs arising in semiconductor modeling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alì, Giuseppe; Rotundo, NellaThis paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with multi-dimensional parabolic-elliptic boundary value problems for the devices. We prove an existence and uniqueness result, and the asymptotic behavior of this mixed initial boundary value problem of partial differential-algebraic equations.