Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: Application to convection–diffusion problems

Abstract

In this work, we present a new a posteriori error estimator based on the Variational Multiscale method for anisotropic adaptive fluid mechanics problems. The general idea is to combine the large scale error based on the solved part of the solution with the sub-mesh scale error based on the unresolved part of the solution. We compute the latter with two different methods: one using the stabilizing parameters and the other using bubble functions. We propose two different metric tensors H iso and H new aniso . They are both defined by the recovered Hessian matrix of the solution and rely on the sub-grid scale error estimator. Thus, we develop a new anisotropic local error indicator and we test it for mesh adaptation on convection-dominated benchmarks in 2D and 3D. The results show that the proposed error indicator leads to enhanced and accurate solutions while using a drastically reduced number of elements.

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

“…is the Euclidean norm of the gradient of the solution on the element T k . Notice that the error indicator (20) uses the gradient at the next time level t n+1 instead of the current time t n . This implicit formulation is possible due to the use of modified method of characteristics which computes the solution backwards in time as in (5).…”

“…As a consequence, error accumulations occur due to the coarse mesh used in the approximation and the computational cost becomes prohibitive due to multiple interpolations between adaptive meshes. In the current work, we make use of the modified method of characteristics to develop a highly efficient algorithm for estimating the gradient in the error indicator (20) at the next time level using known solutions at the current time level. This implicit approximation is accurate and it does not require an extra background mesh or intermediate solutions.…”

“…One other objective is also to implement a multilevel adaptive algorithm for enrichments using the gradient of the solution as an error indicator. Automatic adaptation is an useful tool to improve the accuracy and efficiency of the numerical solution and various adaptation techniques have been used for convection-diffusion problems which can be generally categorized as methods based on error estimates or methods based on physical features such as gradients, see for example [20,21,3,22]. This second approach is the most popular strategy of evaluating error behavior and it is adopted in the present work.…”

“…with X n h is the departure point associated to x h and x * i are the nodal points of the element T * h where the departure point X n h belongs. Note that in practice, the gradient in the adaptive criterion (20) can be evaluated at any point within the element T k . In our simulations, the gradient is calculated at the barycenter xk of the element T k .…”

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

“…for modeling small scales in LES turbulence models [14,24,1,71], and for estimating the numerical error. Application of the latter can be found in many applications, like fluid mechanics [39,40,42,41,43,38,37,54,53,55,6,5,68,73,33,12,13,64], elliptic problems [58,59,52], and elasticity [63,45,8]. A recent application of VMS error estimation to the propagation of error in uncertainty quantification has been published in [25].…”