Silvia Bertoluzza

Daniele Prada, Silvia Bertoluzza, Micol Pennacchio et al.

We analyze the performance of a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method, for the efficient solution of very large linear systems arising from elliptic problems discretized by the Virtual Element Method. We provide numerical experiments on a model linear elliptic problem with highly heterogeneous diffusion coefficients on arbitrary Voronoi meshes, which we modify by adding nodes and edges deriving from the intersection with an unrelated coarse decomposition. The experiments confirm also in this case that the FETI-DP method is numerically scalable with respect to both the problem size and number of subdomains, and its performance is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements.

Silvia Bertoluzza

We present BioNetFlux, an open-source Python framework for the numerical simulation of coupled systems of partial differential equations (PDEs) on one-dimensional multi-arc networks by the Hybridized Discontinuous Galerkin method. Its design targets biological transport phenomena on graph-like geometries that arise naturally in microfluidic organ-on-chip (OoC) devices, vascular networks, and in-vitro cell-migration assays.

Silvia Bertoluzza, Astrid Decoene, Loïc Lacouture et al.

The solutions of elliptic problems with a Dirac measure in right-hand side are not H1 and therefore the convergence of the finite element solutions is suboptimal. Graded meshes are standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in L2-norm. Optimal (or quasi-optimal for the lowest order case) convergence has been shown in L2-seminorm, where the L2-seminorm is defined as the L2-norm on a subdomain which excludes the singularity. Here we show a quasi-optimal convergence for the Hs-seminorm, s \textgreater{} 0, and an optimal convergence in H1-seminorm for the lowest order case, on a family of quasi- uniform meshes in dimension 2. This question is motivated by the use of the Dirac measure as a reduced model in physical problems, and a high accuracy at the singularity of the finite element method is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.