We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.

This paper provides a provably quasi-optimal preconditioning strategy of the linear Schrödinger eigenvalue problem with periodic potentials for a possibly nonuniform spatial expansion of the domain. The quasi-optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations for different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to bound the number of eigensolver iterations uniformly. Several numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy.

Buoyancy-driven flow of non-Newtonian fluids in annular geometries is relevant to a range of thermal engineering applications. However, the jointly effects of fluid rheology and the Prandtl number on heat transfer and thermodynamic irreversibility in this geometry are still not well characterized. We address this gap through two-dimensional simulations of a power-law fluid undergoing steady laminar buoyancy-driven motion inside a concentric cylindrical annulus held at Ra=104, spanning three rheological states (n=0.6,1.0,1.4) crossed with three thermal-diffusivity regimes (Pr=0.0.026,0.7,10). Circulation patterns, heat transport, and entropy generation are analyzed using a comprehensive set of diagnostics, including stream-function, isotherm, and heat-function contours, along with wall-averaged Nusselt numbers, spatial entropy-generation maps, and the Bejan number. Numerical results show that the effect of Pr depends strongly on the fluid rheology. For the shear-thinning fluid, increasing Pr enhances buoyancy-driven circulation, reduces the thermal boundary-layer thickness, and increasingly confines heat transfer to the heated wall. This shift changes the dominant irreversibility from thermal to fluid-friction entropy generation. In contrast, shear-thickening fluids remain weakly convective, thermally dominated, and only weakly sensitive to the Prandtl number, while Newtonian fluids exhibit an intermediate response. A unified scaling framework is build to connect thermal boundary-layer thickness, the magnitude of heat-transfer enhancement, and irreversibility generation to Ra, Pr, and the power-law index acting together. These outcomes provide a unified physical framework for rationalizing heat transmission and thermodynamic performance for buoyancy-driven flow of shear-dependent fluids.

In this paper, we finally prove the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing us to analyze well-posedness within the abstract LBB framework for saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes’ and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.