Variational multiscale a posteriori error estimation for systems: The Euler and Navier–Stokes equations

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

“…On the LHS of Equations (30), the error pollution is isolated. On the other hand, the RHS of this equations contain the error sources, which are the same as in Equation (22) but with the additional terms related to the internal residual error, Y ′ bub , both in the momentum and continuity equations.…”

“…This decomposition allows the study of the influence between the coarse and fine scales, and therefore, this theory has been widely employed to develop stabilized methods for the Stokes equations, see, for instance, other works . Besides, VMS has been employed to estimate the committed error in numerical methods . For incompressible elasticity, whose equations are similar to those of Stokes flow, relevant works using VMS theory both to stabilization and error estimation can be found in other works …”

“…[19][20][21][22][23][24][25] Besides, VMS has been employed to estimate the committed error in numerical methods. 1,2,5,6,[26][27][28][29][30] For incompressible elasticity, whose equations are similar to those of Stokes flow, relevant works using VMS theory both to stabilization and error estimation can be found in other works. [31][32][33] We present an explicit and implicit elemental error estimation in this work.…”

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

“…The mesh refinement strategy is driven by an error indicator, which identifies regions of the simulation domain where the mesh resolution has to be increased. We use an error indicator based on the scale separation introduced by VMS models, following the ideas of Hauke et al This approach, originally presented for the convection‐diffusion problem, has been extended to the Navier‐Stokes equations in Rossi et al and Hauke et al…”

“…In the current paper, we explore the use of an error indicator motivated by variational multiscale (VMS) stabilization techniques, which provide the basis for the underlying finite element formulation. The forerunner of the technique used here was originally presented for convection‐diffusion problems and more recently has been extended to the Navier‐Stokes equations and used to solve incompressible Newtonian flow problems. The key idea for the proposed error indicator is actually rather simple: VMS solvers are based on the construction of a model for the fine scale variables (representing the part of the solution that is not resolved by the mesh) in terms of the large scale ones (the part that can be described using the finite element mesh).…”