Ricardo Ruiz-Baier

Aparna Bansal, Nicolas A. Barnafi, Dwijendra Narain Pandey et al.

We consider a time-dependent coupled Navier--Stokes/generalized poroelastic flow problem and propose a unified and monolithic finite element discretization based on implicit time stepping. To handle the fluid-structure interface we employ a Nitsche-type formulation. The resulting discrete problem is shown to be well-posed using the theory of differential-algebraic equations (DAEs) and the Banach fixed-point theorem. We prove stability and derive a priori error estimates for the fully discrete scheme. The stability and convergence of the method are ensured by a properly chosen penalty parameter independent of the mesh size. Numerical tests are presented to confirm the theoretical convergence rates and to illustrate the ability of the method to capture the coupled dynamics accurately.

Christian Cherubini, Simonetta Filippi, Alessio Gizzi et al.

We propose a new model to describe diffusion processes within active deformable media. Our general theoretical framework is based on physical and mathematical considerations, and it suggests to use diffusion tensors directly coupled to mechanical stress. A proof-of-concept experiment and the proposed generalised reaction-diffusion-mechanics model reveal that initially isotropic and homogeneous diffusion tensors turn into inhomogeneous and anisotropic quantities due to the intrinsic structure of the nonlinear coupling. We study the physical properties leading to these effects, and investigate mathematical conditions for its occurrence. Together, the experiment, the model, and the numerical results obtained using a mixed-primal finite element method, clearly support relevant consequences of stress-assisted diffusion into anisotropy patterns, drifting, and conduction velocity of the resulting excitation waves. Our findings also indicate the applicability of this novel approach in the description of mechano-electrical feedback in actively deforming bio-materials such as the heart.

Ignacio Muga, Jorge Perera, Sergio Rojas et al.

In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher dimension than the trial space), we approximate the solution by minimizing the variational residual in a discrete dual norm. This minimization is equivalent to a nonlinear saddle-point formulation for the discrete solution in the trial space together with a residual representative in the test space. The latter provides a natural a posteriori error estimator, enabling automatic mesh adaptivity. To solve the resulting nonlinear saddle-point problem, we propose a Newton iteration whose linearized saddle-point system is symmetric, thereby guaranteeing solvability at each step. We take the $p$-Laplacian as a model problem and support the theoretical developments with representative numerical experiments, using standard $H^1$-conforming piecewise linear functions for the trial space, and lowest-order Crouzeix--Raviart functions for the test space.

Santiago Badia, Anne Boschman, Alberto F. Martín et al.

We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass conservation while remaining robust in the presence of arbitrarily small cut cells arising from unfitted meshes. Robustness is achieved by combining an $L^2$-stabilisation of the flux with an additional mixed-term stabilisation that enhances pressure control without destroying the local conservation structure. We consider both cell-wise (bulk) and face-based ghost-penalty realisations of the stabilisation. Mixed boundary conditions are handled by weak imposition of both flux and pressure traces on unfitted boundaries. We prove stability and optimal-order a priori error estimates with constants independent of the cut configuration, and establish pressure-robust flux error bounds in the case of pure pressure boundary conditions. We also introduce an augmented Lagrangian variant that improves control of the conservation constraint and is amenable to efficient preconditioning strategies. Numerical experiments for a range of cut configurations, boundary-condition regimes and parameter choices confirm the theoretical results, demonstrating optimal convergence, cut-independent conditioning and mass conservation up to solver tolerance.

Verónica Anaya, Bryan Gómez-Vargas, David Mora et al.

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity and pressure with non-constant viscosity. The analysis is performed by the classical Babuška-Brezzi theory, and we state that any inf-sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates which are further confirmed through computational examples