In this paper, we propose and analyze a Stabilization Free Virtual Element Method (SFVEM), that allows the definition of bilinear forms do not require an arbitrary stabilization term, thanks to exploitation higher-order polynomial projections on divergence free vectors polynomials. The method is introduced in lowest order formulation for Poisson problem. We provide sufficient condition projection space implies well-posedness, proved particular classes polygons, optimal priori error...

We investigate a new numerical approach for the computation of three-dimensional flow in discrete fracture network that does not require conforming discretization partial differential equations on complex systems planar fractures. The within each is performed independently other fractures and their intersections. An independent meshing process very important issue practical large-scale simulations, making mesh generation easier. Some simulations are given to show viability method. resulting...

Following the approach introduced in [SIAM J. Sci. Comput., 35 (2013), pp. B487--B510], we consider formulation of problem fluid flow a system fractures as PDE constrained optimization problem, with discretization performed using suitable extended finite elements; method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to discrete fracture network (DFN) approach. The application DFNs medium complexity is fully analyzed here,...

Abstract In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions degrees affect the convergence rate Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, derive priori error estimate in energy norm between exact solution suitable high-order interpolant computed neural network. Numerical...

In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The method relies modified formulation of discrete diffusion operator that ensures stability preserving all properties differential operator.

In this paper, we present and compare four methods to enforce Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) Variational (VPINNs). Such are usually imposed by adding penalization terms the loss function properly choosing corresponding scaling coefficients; however, practice, requires an expensive tuning phase. We show through several numerical tests that modifying output of neural network exactly match prescribed values leads more efficient accurate solvers. The...

Flows in fractured media have been modeled using many different approaches order to get reliable and efficient simulations for critical applications. The common issues be tackled are the wide range of scales involved phenomenon, complexity domain, huge computational cost. In present paper we propose a parallel implementation PDE-constrained optimization method presented [S. Berrone, S. Pieraccini, Scialò, SIAM J. Sci. Comput., 35 (2013), pp. B487--B510; A908--A935; Comput. Phys., 256 (2014),...

A residual-based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in case of virtual element discretization. The measured considering suitable polynomial projection discrete solution to prove an equivalence between defined and computable residual based estimator that does not involve any term related stabilization. Numerical results display very good behavior ratio estimator, resulting independent meshsize distortion.

In the present paper, a new data-driven model is proposed to close and increase accuracy of Reynolds-averaged Navier–Stokes equations. Among variety turbulent quantities, it has been decided predict divergence Reynolds stress tensor (RST). Recent literature works highlighted potential this choice. The key novelty work obtain through neural network (NN) whose architecture input choice guarantee both Galilean coordinates-frame rotation. former derives from NN while latter expansion RST into...

In the present work a message passing interface (MPI) parallel implementation of an optimization-based approach for simulation underground flows in large discrete fracture networks is proposed. The software capable execution meshing, discretization, resolution, and postprocessing solution. We describe how optimal scalability performances are achieved combining high efficiency computations with optimized use MPI communication protocols. Also, novel graph-topology communications, called...

We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear on a triangulation domain. define an posteriori error estimator, made residual-type term, loss-function and data oscillation terms. prove that estimator is both reliable efficient controlling energy norm between exact VPINN solutions. Numerical results excellent agreement with theoretical predictions.