A posteriori space error indicator for the finite element approximation of the incompressible flow equation based on the variational multiscale concept
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One key factor for improving the accuracy and efficiency of numerical simulations is the use of space error indicators. A posterior space error indicator specifically evaluates errors arising from space discretization. This work introduces a space a posteriori error estimator for stabilized finite element methods in flow problems. For reliable error estimation, we use the orthogonal subgrid scale (OSGS) approach. This approach is based on the variational multiscale (VMS) theory [1]. The VMS method splits the continuous part of the problem into a coarse scale and a fine scale. The fine scale, also known as the sub-grid scale, is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space. In our approach, the error estimator is designed as a suitable norm of the modeled sub-grid scale. This methodology has been used for the stationary convection-diffusion-reaction equation [2]. In this work, we extend it to incompressible Navier-Stokes flow equations. Based on the space error indicator, we use hierarchical octree-based h-refinement. This technique allows us to coarsen and refine the mesh as needed. Numerical experiments are conducted on manufactured solutions with various flow regimes. Benchmark problems include incompressible flow over a backward-facing step and flow over a cylinder. These tests demonstrate the effectiveness of the proposed space-error indicator and mesh-size refinement strategy.