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The reconstruction of individual geometries from medical imaging is quite a standard in the framework of patient-specific medicine. A major drawback in such a context is represented by noise inherent to the data acquisition. Low signal-to-noise ratios can negatively impact extraction algorithms, and result in artefacts or poor quality of the reconstructed meshes. Direct application of numerical methods on such meshes can yield misleading results. Indeed, artefacts and badly shaped elements may corrupt numerical simulations or induce relevant errors in the computation of meaningful geometrical quantities, such as the curvature or the geodesic surface distance. In this paper, we propose a denoising procedure to remove artefacts from a triangular mesh of a three-dimensional closed surface which represents a brain cortex. For this purpose, we combine a smoothing technique (i.e., the Taubin or the HC-Laplacian smoothing) with an edge-flipping algorithm. To control the denoising procedure, we introduce a stopping criterion that takes into account both the improvement of the mesh quality and the loss of volume enclosed by the surface. On a brain cortical surface reconstructed from Magnetic Resonance Imaging (MRI) data, we first perform a tuning analysis of the parameters involved in the smoothing algorithm, then we investigate the effectiveness of the denoising procedure. Finally, as an example of relevant geometrical feature, we study the improvement generated by the proposed algorithm on the computation of the cortical curvature.

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\boldsymbol{H}({\rm div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\boldsymbol{H}({\bf curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.

In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importance for the successful subsequent coupling with other physical phenomena such as transport. The virtual element method, as solution scheme, is known to be applicable in problems whose discretizations requires cells of general shape, and the mixed formulation is here preferred to obtain accurate velocity fields. To overcome the issues associated to the complex geometries and, at the same time, retaining the quality of the solutions, we present here the virtual element method to solve the Darcy problem, in mixed form, in presence of curved interfaces in two and three dimensions. The numerical scheme is presented in detail explaining the discrete setting with a focus on the treatment of curved interfaces. Examples, inspired from industrial applications, are presented showing the validity of the proposed approach.

We extend the hybridization procedure proposed in [Arnold, Brezzi, 1985, ESAIM: M2AN] to the Virtual Element Method for linear elasticity problems based on the Hellinger-Reissner principle. To illustrate such a technique, we focus on the 2D case, but other methods and 3D problems can be considered as well. We also show how to design a better approximation of the displacement field using a straightforward post-processing procedure. The numerical experiments confirm the theory for both two and three-dimensional problems., submitted to M3AS

We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities associated with de Rahm sequences of nodal, edge, and face virtual element spaces.

In this paper, we extend the divergence-free VEM of [L. Beirão da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier–Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 56 (2018) 1210–1242] to the Oseen problem, including a suitable stabilization procedure that guarantees robustness in the convection-dominated case without disrupting the divergence-free property. The stabilization is inspired from [N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzman, A. Linke and C. Merdon, A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation, SIAM J. Numer. Anal. 59 (2021) 2746–2774] and includes local SUPG-like terms of the vorticity equation, internal jump terms for the velocity gradients, and an additional VEM stabilization. We derive theoretical convergence results that underline the robustness of the scheme in different regimes, including the convection-dominated case. Furthermore, as in the non-stabilized case, the influence of the pressure on the velocity error is moderate, as it appears only through higher-order terms.

The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al., CMAME 2016] we are able to show an “almost uniform” error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.