Volume 33, Issue 3

We propose an approach for the L ∞ ${\mathcal{L}}_{\infty }$ model reduction of descriptor systems based on the minimization of the L ∞ ${\mathcal{L}}_{\infty }$ objective by means of smooth optimization techniques. Direct applications of smooth optimization techniques are not feasible even for systems of modest order, since the optimization techniques converge at best at a linear rate requiring too many evaluations of the costly L ∞ ${\mathcal{L}}_{\infty }$ -norm objective to be practical. We replace the original system with a system of smaller order interpolating the original system at points on the imaginary axis, minimize the L ∞ ${\mathcal{L}}_{\infty }$ objective after this replacement, and refine the smaller system based on the minimization. We also describe how asymptotic stability constraints on the reduced system sought can be incorporated into our approach. The numerical experiments illustrate that the approach leads to locally optimal solutions to the L ∞ ${\mathcal{L}}_{\infty }$ model reduction problem, and its capability to deal with systems of order a few ten thousands.

We consider the so-called field–road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions (a field and a road in a population dynamics context) and coupled through exchange terms on the road, which makes its analysis quite involved. We propose a two-point flux approximation (TPFA) finite volume scheme. In both the continuous and the discrete settings, we prove the dissipation of a quadratic entropy, with some entropy dissipation-entropy relation, leading to the exponential decay in time of the solution towards the stationary state selected by the total mass of the initial data. To deal with the problem of different dimensions in the proof of the entropy dissipation-entropy relation, we artificially “thicken” the road and adapt the proof of the Poincaré–Wirtinger inequality. Numerical simulations confirm and complete the analysis, and raise new issues.

In this paper, we derive an improved error estimate of a conforming discontinuous Galerkin (CDG) method for both second and fourth order elliptic problems, assuming only minimal regularity on the exact solutions. The result we established is called a medius error estimate since it relies on both a priori and a posteriori analysis. Compared with the standard interior penalty discontinuous Galerkin (IPDG) method, when choosing the standard DG norm, an additional term ‖∇ u − ∇ w v h ‖ 0 is incorporated in the CDG formulation for second order elliptic equation, while for the case of fourth order equation, this term becomes ‖ Δ u − Δ w v h ‖ 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{v}}_{h}{{\Vert}}_{0}$ . These terms disappear if we directly analyze the nonstandard error formulations ‖∇ u − ∇ w u h ‖ 0 and ‖ Δ u − Δ w u h ‖ 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{u}}_{h}{{\Vert}}_{0}$ . Extensive numerical results are also carried out to validate our theoretical findings.

This work focuses on the development of a novel, strongly-coupled, second-order partitioned method for fluid–poroelastic structure interaction. The flow is assumed to be viscous and incompressible, and the poroelastic material is described using the Biot model. To solve this problem, a numerical method is proposed, based on Robin interface conditions combined with the refactorization of the Cauchy’s one-legged ‘ ϑ -like’ method. This approach allows the use of the mixed formulation for the Biot model. The proposed algorithm consists of solving a sequence of Backward Euler–Forward Euler steps. In the Backward Euler step, the fluid and poroelastic structure problems are solved iteratively until convergence. Then, the Forward Euler problems are solved using equivalent linear extrapolations. We prove that the iterative procedure in the Backward Euler step is convergent, and that the converged method is stable when ϑ ∈ [1/2, 1]. Numerical examples are used to explore convergence rates with varying parameters used in our scheme, and to compare our method to a monolithic method based on Nitsche’s coupling approach.