2,753 research outputs found
On simulations of discrete fracture network flows with an optimization-based extended finite element method
No full textFollowing the approach introduced in [Berrone,Pieraccini,Scialò,2013], we consider the formulation of the problem of fluid flow in a system of fractures as a
PDE constrained optimization problem, with discretization performed using suitable extended finite elements; the method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to the DFN approach. The application of the method to discrete fracture networks of medium complexity is fully analyzed here, accounting for several issues related to viable and reliable implementations of the method in complex problems
A posteriori error estimate for a PDE-constrained optimization formulation for the flow in DFNs
No full textFlows in fractured media have been modeled with many different approaches in order to get reliable and efficient simulations for many critical applications. The common issues to be tackled are the wide range of scales involved in the phenomenon, the complexity of the domain, and the huge computational cost. In this paper we introduce residual-based "a posteriori" error estimates for a formulation of the flow in the discrete fracture networks based on a constrained optimization approach (see Berrone, Pieraccini, and Scialo` [SIAM J. Sci. Comput., 35 (2013), pp. B487-B510], [SIAM J. Sci. Comput., 35 (2013), pp. A908-A935], [J. Comput. Phys., 256 (2014), pp. 838-853]), suitable to overcome all the difficulties related to a good quality mesh generation with conformity requirement
A PDE-constrained optimization formulation for discrete fracture network flows
No full textWe investigate a new numerical approach for the computation of the 3D flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex 3D systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. Independent meshing process within each fracture is a very important issue for practical large scale simulations making easier mesh generation. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures
Robustness in a posteriori error estimates for the Oseen equations with general boundary conditions
No full textWe present a residual-based a posteriori error estimator for a stabilized finite element discretization of an incompressible Oseen-like model with general boundary conditions. We focus our attention on the behavior of the effectivity index and we carry out a numerical study of its sensitiveness to the problem and mesh parameters
Adaptive discretization of the stationary and incompressible Navier-Stokes equations by stabilized Finite Element Methods
No full textRobust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
No full textIn this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237; Verf¨urth, Calcolo 40 (2003) 195-212] it is easy to identify a time-discretization error-estimator and a space discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313-348; Petzoldt, Adv. Comput. Math. 16 (2002) 47-75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jump
A local-in-space-timestep approach to a finite element discretization of the heat equation with a posteriori estimates
No full textA new numerical method is presented for the heat equation with discontinuous coefficients based on a Crank-Nicolson scheme and a conforming finite element space discretization. In the proposed method each node of the spatial discretization may have the global timestep split into an arbitrary number of local substeps in order to pursue a local improvement of the time discretization in the regions of the spatial domain where the solution changes rapidly. This method can possibly be used together with adaptive strategies for both the space discretization and the choice of timesteps to suitably produce an efficient space-time discretizaton of the problem. Robust a posteriori upper and lower bounds of the error are proposed to attain this target. Moreover, some indications are given on how to modify the mesh, the timestep, and the number of substeps to improve the discretizatio
A Residual A Posteriori error estimate for the Virtual Element Method
No full textA residual based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in the case of a Virtual Element discretization. The error is measured considering a suitable polynomial projection of the discrete solution to prove an equivalence between the defined error and a computable residual based error estimator that does not involve any term related to the Virtual Element stabilization. Numerical results display a very good behaviour of the ratio between the error and the error estimator, resulting independent of the meshsize and element distortion
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