This paper introduces a novel Virtual Element Method (VEM) for the coupled Stokes–Darcy system in primal-primal form. In the free-flow Stokes domain, we implement a stream function formulation that inherently satisfies the incompressibility constraint and reduces computational cost. Across the interface, mass conservation, normal stress balance, and the Beavers–Joseph–Saffman slip condition are enforced to couple the biharmonic stream function equation with the Darcy's pressure equation. Leveraging VEM's ability to handle general polygonal meshes, the proposed method naturally accommodates irregular interface geometries without requiring remeshing or adaptive refinement. The accuracy of the method is validated through several numerical simulations that include applications to dead-end filtration, and network flow in bioartificial organs.

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for 𝐇(div, Ω) lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

We analyse a coupled 3D-2D model with a free fluid governed by Stokes flow in the bulk and a poroelastic plate described by the Biot-Kirchhoff equations on the surface. Assuming the form of a double perturbed saddle-point problem, the unique solvability of the continuous formulation is proved using Fredholm's theory for compact operators and the Babu & scaron;ka-Brezzi approach for saddle-point problems with penalty. We propose a stable virtual element method, establishing a discrete inf-sup condition under a small mesh assumption through a Fortin interpolant that requires only H1-regularity for the Stokes problem. We show the well-posedness of the monolithic discrete formulation and introduce an equivalent fixed-point approach employed at the implementation level. The optimal convergence of the method in the energy norm is proved theoretically and is also confirmed numerically via computational experiments. We demonstrate an application of the model and the proposed scheme in the simulation of immune isolation using encapsulation with silicon nanopore membranes.

We study transport phenomena involving chemically reactive species, modeled by advection–diffusion–reaction systems coupled with flow fields governed by Darcy's law. Both the velocity field and the species concentrations are discretized using the Virtual Element Method, while time integration is performed through a discontinuous Galerkin scheme. This work represents a preliminary study, in which we introduce some simplifications of the full model. In particular, we assume a concentration-independent viscosity in the Darcy problem, constant diffusion tensors in the advection–diffusion–reaction systems, and first-order reaction networks with liquid-phase degradation. We derive an abstract error estimate by means of a technique that combines Gauss–Radau interpolation with numerical integration. The theoretical results are supported by numerical experiments that exhibit arbitrary-order accuracy in both space and time.

We propose and analyse residual-based a posteriori error estimates for the virtual element discretisation applied to the thin plate vibration problem in both two and three dimensions. Our approach involves a conforming C^1 discrete formulation suitable for meshes composed of polygons and polyhedra. The reliability and efficiency of the error estimator are established through a dimension-independent proof. Finally, several numerical experiments are reported to demonstrate the optimal performance of the method in 2D and 3D.

We present a proof-of-concept methodology for generating curvilinear polygonal meshes suitable for high-order discretisations by the Virtual Element Method (VEM). A VEM discretisation requires the definition of a set of boundary and internal points used to define basis functions and compute integrals of polynomials. The procedure to locate these points on the boundary borrows ideas from previous work on a posteriori high-order mesh generation in which the geometrical inquiries to a B-rep model of the computational domain are performed via an interface to CAD libraries. Here we describe the steps of the procedure that transforms a straight-sided polygonal mesh, generated using third-party software, into a curvilinear boundary-conforming mesh. We discuss criteria for ensuring and verifying the validity of the mesh. Using an elliptic partial differential equation with Dirichlet boundary conditions as a model problem, we show that VEM discretisations on such meshes achieve the expected rates of convergence as the mesh resolution is increased. This is followed by an illustrative application of the method to the generation of a curvilinear polygonal mesh for an aerofoil geometry. We discuss polygonal curvilinear mesh quality and its enhancement, and use the motion of a cell vertex to appraise three elemental quality metrics, namely convexity, regularity and isotropy, and highlight some of the difficulties associated in their use for mesh quality optimisation. A derivative-free optimisation method is utilised to enhance curvilinear polygonal meshes by maximising a suitable measure of mesh quality. We propose such measure as a combination of the three quality metrics and apply it to optimise a distorted initial mesh for a ring geometry. We show that a suitable version of the convexity metric is effective in untangling invalid meshes. The VEM solution of a model elliptic equation is obtained for a ring geometry where a distorted and an optimised mesh show low errors, indicating that the VEM is robust and relatively insensitive to mesh distortion, and a reduction of the error in the optimised mesh. Finally, we use a more complex geometry, a computational domain for an aerofoil, as a benchmark to further illustrate the ability of the convexity metric to untangle meshes, and also to assess the suitability of two quality measures as optimisation targets to improve the overall quality of curvilinear polygonal meshes.

We introduce a novel residual-based a posteriori error estimator for the conforming C^1 Virtual Element Method (VEM) applied to the buckling eigenvalue problem, incorporating nonlinear plane stress effects in both two and three dimensions. The estimator is fully computable on general polyhedral meshes and implemented within the open-source library. Its reliability is rigorously justified via bounds on the residual equation using polynomial projections, stabilisation contributions, and interpolation estimates, while efficiency is ensured through the use of bubble function arguments. Comprehensive numerical experiments in 2D and 3D illustrate the estimator's optimal accuracy and robustness, highlighting its potential for predictive analysis of complex plate structures.

The Virtual Element Method (VEM) is an extension of the Finite Element Method (FEM) to handle polytopal meshes. After giving a short introduction of the VEM for a two dimensional Laplacian problem, we show the differences between an implementation of a VEM and a FEM code highlighting which are the main issues associated with the VEM framework. Furthermore, this paper will show one of the possible ways to face such issues: Vem++ a C++ library developed to “deal and play” with the VEM discretisation. This C++ library deals with the VEM, since there are several partial differential equations in two/three dimensions coming from both academic and engineering problems. Then, one can “play” with the VEM, since Vem++ has been designed so that one can plug-in new features such as new polytopes quadrature rules, new solvers and new virtual element spaces in a smart way.

We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space- time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full L2(0, T; L2(Q)) norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in (3 + 1)-dimensions that validate our theoretical results.

We propose and analyze two convection quasi-robust and pressure robust finite element methods for a fully nonlinear time-dependent magnetohydrodynamics problem. The schemes make use of suitable upwind and CIP stabilizations to handle the fluid and magnetic convective terms. The developed error estimates are uniform in both diffusion parameters and optimal with respect to the diffusive norm; furthermore, for the second (more complex) method we are able to show a quicker error reduction rate in convection dominated regimes. A set of numerical tests support our theoretical findings.

We carry out a stability and convergence analysis of a fully discrete scheme for the time-dependent Navier–Stokes equations resulting from combining an [Formula: see text]-conforming discontinuous Galerkin spatial discretization, and a discontinuous Galerkin time-stepping scheme. Such a scheme is proven to be pressure robust and Reynolds semi-robust. Standard techniques can be used to analyze only the case of lowest-order approximations in time. Therefore, we use some nonstandard test functions to prove existence of discrete solutions, unconditional stability, and quasi-optimal convergence rates for any degree of approximation in time. In particular, a continuous dependence of the discrete solution on the data of the problem, and quasi-optimal convergence rates for low and high Reynolds numbers are proven in an energy norm including the term [Formula: see text] for the velocity. In addition to the standard discontinuous Galerkin time-stepping scheme, which is fully implicit, we propose and analyze a novel high-order semi-implicit version that avoids the need of solving nonlinear systems of equations after the first time slab, thus improving the efficiency of the method. Some numerical experiments validating our theoretical results are presented for both schemes.

In this work we will present a method of virtual elements to approximate the solution of a polyharmonic problem (−Δ)nu=g. We will consider m+1 auxiliary unknowns when n=2m+1, and m auxiliary unknowns for n=2m. In the first case (n=2m+1), we will solve m fourth order problems and a second order one. In the even case, only m fourth-order problems have to be solved. Virtual element conforming discretizations are written for each fourth-order problem in C1, and a C0-virtual element method is established for the second-order problem. We also carry out the error analysis for both cases. Finally, we report a series of numerical tests to verify the performance of numerical scheme.

The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low- and high-order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary-value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

In the present contribution we propose a novel conforming Finite Element scheme for the time-dependent Navier-Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a "divergence-free" velocity/pressure couple (such as the Scott-Vogelius element), a Discontinuous Galerkin in time approximation, and a suitable SUPG-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included.

We present a proof-of-concept methodology for generating curvilinear polygonal meshes suitable for high-order discretizations by the Virtual Element Method (VEM). A VEM discretization requires the definition of a set of boundary and internal points that are used to interpolate the approximation functions and to evaluate integrals by means of suitable quadratures. The procedure to locate these points on the boundary borrows ideas from previous work on a posteriori high-order mesh generation in which the geometrical inquiries to a B-rep of the computational domain are performed via an interface to CAD libraries. Here we describe the steps of the procedure that transforms a straight-sided polygonal mesh, generated using third-party software, into a curvilinear boundary-conforming mesh. We discuss criteria for ensuring and verifying the validity of the mesh. Finally, using the Laplace equation with Dirichlet boundary conditions as a model problem, we show that VEM discretizations on such meshes achieve the expected rates of convergence as the mesh resolution is increased.

The Virtual Element Method (VEM) is a novel family of numerical methods for approximating partial differential equations on very general polygonal or polyhedral computational grids. This work aims to propose a Balancing Domain Decomposition by Constraints (BDDC) preconditioner that allows using the conjugate gradient method to compute the solution of the saddle-point linear systems arising from the VEM discretization of the three-dimensional Stokes equations. We prove the scalability and quasi-optimality of the algorithm and confirm the theoretical findings with parallel computations. Numerical results with adaptively generated coarse spaces confirm the method's robustness in the presence of large jumps in the viscosity and with high-order VEM discretizations.

We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an H^1-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.

In Beirão da Veiga et al. 2019, the authors proposed a virtual element method for domains with fixed curved boundary that tends to be locally straight when the discretisation gets finer and finer. In this paper, we properly modify this method so that it can also deal with curved mesh edges which are not necessarily located along boundaries and do not tend to be straight when refining the mesh.To achieve this goal, we assume that curved edges are described by polynomials and we increase the dimension of the virtual element space used in Beirão da Veiga et al. 2019.In the numerical experiments, we compare these two methods. Furthermore, we apply the proposed approach to the benchmark “TEAM 25” problem, an optimal shape design problem in magnetostatics characterised by curved edges.

Two numerical schemes are proposed and investigated for the Yang–Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang–Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $ L^2 $-norm of the potential and electrical fields in $ \mathcal O(h^{k+1}) $ (provided that the time step is of that order), where $ k $ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.

In this paper, we are interested in solving optimal shape design problems. A critical challenge within this framework is generating the mesh of the computational domain at each optimisation step according to the information provided by the minimising functional. To enhance efficiency, we propose a strategy based on the Finite Element Method (FEM) and the Virtual Element Method (VEM). Specifically, we exploit the flexibility of the VEM in dealing with generally shaped polygons, including those with hanging nodes, to update the mesh solely in regions where the shape varies. In the remaining parts of the domain, we employ the FEM, known for its robustness and applicability in such scenarios. We numerically validate the proposed approach on the T.E.A.M. 25 benchmark problem and compare the results obtained with this procedure with those proposed in the literature based solely on the FEM. Moreover, since the T.E.A.M. 25 benchmark problem is also characterised by curved shapes, we utilise the VEM to accurately incorporate these “exact” curves into the discrete solution itself.